Combinatorial Addition Formulas and Applications

Pierre Auger* Gilbert Labelle* Pierre Leroux�

May 23, 2001

Abstract

We derive addition formulas at the combinatorial level, that is equations of the form F (X1+X2 + � � � +Xk) = �F (X1; X2; : : : ; Xk), where F = F (X) is a given combinatorial species and�F is a species on k sorts of singletons X1; X2; : : : ; Xk; depending on F . General results aregiven in the case of a molecular species M = Xn=H. Speci�c formulas are also presented in thecases of the species Ln of n-lists, Chan of n-chains, En of n-sets, E�n of oriented n-sets, Cn of(oriented) n-cycles and Pn of n-gons (unoriented cyles). These formulas are useful for the com-putation of molecular expansions of species de�ned by functional equations. Applications to thecomputation of cycle index series and asymmetry index series, to the extension of substitutionto virtual species (and K-species), and to the analysis of generalized binomial coe�cients

�M

N

�k

for molecular species are also given.

R�esum�e

Nous obtenons des formules d'addition combinatoires, c'est-�a-dire des �equations de la formeF (X1 + X2 + � � � + Xk) = �F (X1; X2; : : : ; Xk), o�u F = F (X) est une esp�ece de structuresdonn�ee et �F est une esp�ece, d�ependant de F , sur k sortes de singletons X1; X2; : : : ; Xk. Nousdonnons des formules g�en�erales pour les esp�eces mol�eculaires M = Xn=H et des r�esultats plussp�eci�ques dans le cas des esp�eces Ln, des n-listes (listes de longueur n), Chan, des n-cha�nes,En, des n-ensembles, E�n , des n-ensembles orient�es, Cn, des n-cycles (orient�es) et Pn, des n-gones (cycles non orient�es). Ces formules sont utiles pour calculer le d�eveloppement mol�eculaired'esp�eces d�e�nies par des �equations fonctionnelles. Nous pr�esentons �egalement des applicationsau calcul des s�eries indicatrices des cycles ou d'asym�etrie, �a l'extension de la substitution auxesp�eces virtuelles (et au K-esp�eces) et �a l'analyse des coe�cients binomiaux g�en�eralis�es

�M

N

�k

pour les esp�eces mol�eculaires.

1 Introduction

Addition formulas are well known in classical analysis. They involve trigonometric, elliptic andother special functions and algebraic identities. Examples include

ex+y = ex � ey ; sin(x+ y) = sin(x) cos(y) + cos(x) sin(y) ;

(x+ y + z)n =P

i+j+k=n

n!i!j!k!x

iyjzk :

�LaCIM et D�epartement de math�ematiques; Universit�e du Qu�ebec �a Montr�eal; Case postale 8888, succursaleCentre-Ville; Montr�eal, Qu�ebec; Canada; H3C 3P8; www.lacim.uqam.ca/�leroux; with the partial support of FCAR(Qu�ebec) and NSERC (Canada).

1

The purpose of this paper is to study addition formulas at the combinatorial level. In thiscontext, the functions are replaced by combinatorial species and the variables are replaced byvarious sorts of singletons representing various sorts of elements (labels) in the underlying sets ofthe structures.

Informally, a combinatorial species is a class, G, of �nite labelled structures which is closed underrelabeling along bijections. When the labels are of di�erent sorts, X1; : : : ; Xk, the relabellings mustbe made along bijections that preserve each sort Xi of elements. We write G = G(X1; : : : ; Xk) toexpress the fact that G is a species of structures on the sorts X1; : : : ; Xk. The theory of specieswas founded by Joyal (see [Joyal (1981)]) and a comprehensive English presentation of the theorycan be found in [Bergeron et al. (1998)].

De�nition 1.1 Let F = F (X) be a combinatorial species of one sort. A combinatorial additionformula for the species F is a combinatorial equation of the form

F (X1 +X2 + � � �+Xk) = �F (X1; X2; : : : ; Xk) (1.1)

where �F is a species, depending on F , on the k sorts of singletons X1; X2; : : : ; Xk; k � 2. 4

Note that the species �F in (1.1) is necessarily symmetric in X1; X2; : : : ; Xk; that is: for everypermutation � 2 Sk we have a corresponding combinatorial equation

�F (X1; X2; : : : ; Xk) = �F (X�(1); X�(2); : : : ; X�(k)): (1.2)

One of the simplest instances of a combinatorial addition formula is the exponential formulasatis�ed by the species, E, of sets

E(X1 +X2 + � � �+Xk) = E(X1) �E(X2) � � �E(Xk): (1.3)

Indeed, any set U made up of elements of various sorts X1; : : : ; Xk can be canonically identi�edwith a k-uple, (U1; U2; : : : ; Uk), of sets where

Ui = fu 2 U j u is of sort Xig; i = 1; : : : ; k: (1.4)

Another example, for k = 2, is the addition formula satis�ed by the species, C, of orientedcycles [Bergeron et al. (1998)], namely

C(X + Y ) = C(X) + C(L(X)Y ); (1.5)

where L is the species of (possibly empty) lists. This formula expresses the fact that if a C(X+Y )-structure has at least one element of sort Y , then it can be identi�ed, in a natural manner, to anoriented cycle formed of disjoint lists of the form x1x2 � � �x�y, where the xi are distinct elements ofsort X and y is an element of sort Y (see Figure 1b).

There exists, however, a more \symmetrical" version of (1.5), that is

C(X + Y ) = C(X) + C(Y ) + C(L+(X) �L+(Y )); (1.6)

where L+ is the species of non empty lists (see Figure 1c).Since any species can be written as a sum of irreducible ones (molecular species) and since the

transformationF (X) 7�! F (X1 +X2 + � � �+Xk) (1.7)

2

a) b) c)

Figure 1: C(X + Y )

is obviously linear in F , we focus our attention, in the present paper, on addition formulas of theform

M(X1 +X2 + � � �+Xk) =XN

cM;NN(X1; X2; : : : ; Xk) (1.8)

whereM =M(X) is a molecular species on the sortX;N = N(X1; : : : ; Xk) runs through molecularspecies on the sorts X1; X2; : : : ; Xk and the coe�cients cM;N are nonnegative integers. Finitemolecular decompositions such as (1.8) can be considered as generalized multinomial formulassince they give

(X1 + � � �+Xk)n =

Xn1+���+nk=n

n!

n1! � � �nk !Xn1

1 � � �Xnkk ; (1.9)

in the case M(X) = Xn, the species of n-lists (lists of length n).The knowledge of an addition formula is obviously an interesting asset for a species F . It can be

used in various computations. As an illustration, consider the species A of rooted trees, recursivelyde�ned by the functional equation

A = XE(A) : (1.10)

Using this equation, a simple iterative procedure can be given in order to compute A explicitely:

A0 := 0An+1 := (XE(A0+A1 + � � �+An))n+1 ;

(1.11)

where the notation Fn is used for the restriction of the species F to degree n, that is to sets ofcardinality n. One �nds

A0 = 0 ;

A1 = X ;

A2 = X2 ;

A3 = X3 +XE2(X) ;

A4 = 2X4 +X2E2(X) +XE3(X) ;

A5 = 3X5 + 3X3E2(X) +X2E3(X) +XE4(X) +XE2(X2) ;

A6 = 6X6 + 6X4E2(X) + 3X3E3(X) +X2E2(X)2 +X2E4(X) + 2X2E2(X2) +XE5(X) ;

etc.

3

and this gives A = A0 + A1 + A2 + � � �, expressed in its molecular form. For example, the fourterms of degree 4 in the molecular decomposition of A are illustrated in Figure 2. It is then

4X

4X

X2E (X) E3(X)X2

Figure 2: The molecules of A4

possible to compute also the molecular decomposition of the species a of unrooted trees, using theDissymmetry theorem for trees (see [Bergeron et al. (1998), x 4.1]),

a = A+ E2(A)�A2 ; (1.12)

and the addition formula for E2. One �nds

a = X + E2(X) +XE2(X) +XE3(X) +E2(X2) +X3E2(X) +XE4(X) +XE2(X

2)

+X4E2(X) +X3E3(X) +X2E2(X2) +XE5(X) + E2(XE2(X)) +E2(X

3) + � � �

See Tables 7 and 8 of Appendix 2 of [Bergeron et al. (1998)] for the terms up to degree 10 fora and A. Note that an iterative procedure of Newton-Raphson, with quadratic convergence, canbe developped for the computation of A; see [Bergeron et al. (1998), algorithm 3:3:2]. Moreover,the method can be adapted to the computation of the species a R and AR of R-enriched trees androoted trees, provided that addition formulas are known for the species R and its derivative R0.

In Section 2, we make use of suitable weighted versions of Burnside's lemma, in the context ofspecies, to develop some general addition formulas of type (1.8) when M is written in the canonicalform M(X) = Xn=H , H being a subgroup of the symmetric group Sn(H � Sn).

In Section 3, we analyze speci�c instances of addition formulas for special molecular speciesM(X): the species Xn, of n-lists; Chan(X), of n-chains; En(X); of n-sets; E�

n (X), of orientedn-sets; Cn(X), of oriented n-cycles, and Pn(X), of n-gons (unoriented n-cycles).

In Section 4, we give speci�c applications of combinatorial addition formulas to various combi-natorial setting, including:

1. the computation of cycle index series and asymmetry index series [Labelle (1992)] of speciesand their associated canonical q-series [D�ecoste (1993)];

2. the extension of substitution (partional composition) to virtual species, C -species and moregenerally K-species, where K is a binomial half-ring, via Yeh's formulas [Yeh (1986)] (see also[Joyal (1985b)])

F (m1X1 + � � �+mkXk) = F (X1 + � � �+Xk)�E(X1)m1 � � �E(Xk)

mk (1.13)

where \�" denotes the cartesian product of species and

F (X � Y ) = F (X + kY )��k:=�1

: (1.14)

4

3. the analysis of generalized binomial coe�cients�MN

�kfor molecular species, introduced by the

authors in [Auger (2000)] and [Auger et al. (2000)], de�ned by the equations

M(X + k) =M(X + kY )��Y :=1

=XN

M

N

!k

N(X): (1.15)

2 General addition formulas

In order to develop general addition formulas of type (1.8), we need to recall �rst some basic notionsfrom the theory of group actions in the context of sets weighted in a commutative ring A . A �niteweighted set is an ordered pair (W; w) where W is a �nite set and w : W �! A is an arbitraryfunction, called a weight function. The total weight of W is denoted by,

jWjw =Xs2W

w(s): (2.1)

Let G be a �nite group. A group action

G�W �! W; (g; s) 7�! g � s; (2.2)

is said to be weight preserving if w(g � s) = w(s), for every g 2 G and s 2 W. The G-orbit of anelement s 2 W is de�ned by Gs = fg � s j g 2 Gg. The set of all G-orbits is denoted W=G. It is aweighted set with the weight function w� : W=G �! A; w�(Gs) = w(s).

Lemma 2.1 (Burnside's (Cauchy-Frobenius) Lemma) For any weight-preserving �nite groupaction (2.2), the total weight of the G-orbits is given by

jW=Gjw� =1

jGj

Xg2G

jFix(g)jw; (2.3)

whereFix(g) = fs 2 W j g � s = sg: (2.4)

2

Now, our �rst general addition theorem for molecular species involves the Young subgroupsSn1;n2;:::;nk ' Sn1 � Sn2 � : : :� Snk of the symmetric group Sn which are de�ned by

Sn1;n2;:::;nk = f� 2 Sn j �(�ni) = �ni; i = 1; : : : ; kg; (2.5)

with n = n1 + n2 + � � �+ nk and

�n1 = [n1]; �n2 = [n1 + n2]\[n1]; �n3 = [n1 + n2 + n3]\[n1 + n2]; : : : (2.6)

where [n] denotes the set f1; 2; : : : ; ng. More generally, for any subgroup H of Sn (H � Sn), wewrite

Hn1;n2 ;:::;nk = H \ Sn1;n2;:::;nk : (2.7)

The general addition formulas of this section are obtained by analysis of the k-sort moleculardecomposition of the k-species M(X1 +X2 + � � �+Xk), where M is a given molecular species (of

5

one sort). Note that k-sort molecular species can be characterized as follows. For any subgroupK of Sn1;:::;nk , let X

n11 � � �Xnk

k =K denote the k-sort species whose structures on (U1; U2; : : : ; Uk)are K-classes of bijections, of the form �K, � = (�1; �2; : : : ; �k), �i : [ni]

��! Ui, the transport of

strutures along a bijection � = (�1; �2; : : : ; �k), �i : Ui��! Vi being de�ned by �(�H) = (� � �)H .

It is easily checked thatXn11 � � �Xnk

k =K is a molecular k-species (there is only one isomorphism classof structures) and that Xn1

1 � � �Xnkk =K1 = Xn1

1 � � �Xnkk =K2, where equality denotes isomorphism of

k-species, if and only if K1 and K2 are conjugate subgroups of Sn1;:::;nk . See also [Bergeron et al.(1998)], Exercises 2:6:15 and 2:6:16.

Theorem 2.1 (Generalized multinomial expansion) Let M = M(X) be a molecular speciesof degree n, then

M(X1 + � � �+Xk) =X

n1+���+nk=n

Xs2M [n]

jAut(s)n1;:::;nk j

n1! � � �nk !

Xn11 � � �Xnk

k

Aut(s)n1;:::;nk(2.8)

where

M [n] = the set of all M�structures on [n]; (2.9)

Aut(s) = f� 2 Sn j � is automorphism of sg � Sn: (2.10)

Proof

AnyM(X1+� � �+Xk)-structure can be expressed in the form (s; �n1; : : : ; �nk), up to isomorphism,where s 2M [n]; n1+ � � �+nk = n, the elements of �ni being interpreted as elements of sort Xi. Themultisort molecular species associated to such a structure is then

w(s; �n1; : : : ; �nk) = Xn11 � � �Xnk

k

�Aut(s)n1;:::;nk : (2.11)

Any � 2 Sn1;:::;nk acts on such structures by transport (relabelling) of structures and the sub-groups Aut(� � s)n1 ;:::;nk and Aut(s)n1;:::;nk are conjugate in Sn1;:::;nk , so that the correspondingmolecular species are isomorphic and considered equal. In order to �nd the multisort moleculardecomposition of the species M(X1; : : : ; Xk) we have to enumerate isomorphism types of struc-tures (s; n1; : : : ; nk) under the action of Sn1;:::;nk , regrouped according to their associated molecu-lar species. The result then follows by applying, for each (n1; : : : ; nk), Burnside's Lemma withG = Sn1;:::;nk , W = f(s; �n1; : : : ; �nk) j s 2 M [n]g ' M [n], A = ZkX1; : : : ; Xkk, the ring of virtualspecies [Joyal (1985a), Joyal (1985b), Joyal (1986), Yeh (1986), Bergeron et al. (1998)], and theweight function given by (2.11) and the action described above. 2

Here is a variant of the generalized multinomial expansion formula which uses the standardform Xn=H for the molecular species M . It also uses the concept of k-composition (U1; U2; : : : ; Uk)of a set U , de�ned as a list of disjoint (possibly empty) subsets Ui of U , i = 1; : : : ; k, such that[ki=1Ui = U . We write U1+U2 + � � �+Uk = U to denote this. There is a closely related concept ofk-coloring of U de�ned as a function (coloring) c : U ! [k]. The relationship with k-compositionsis given simply by

x 2 Ui , c(x) = i : (2.12)

In words, the elements of U which belong to Ui are those of color i.

6

For example, we have the standard k-composition (�n1; �n2; : : : ; �nk) of U = [n] determined by theintegers (n1; n2; : : : ; nk) such that n1 + n2 + � � �+ nk = n, which is given by (2.6).The associated(standard) k-coloring is the unique (weakly) increasing function c : [n]! [k] such that jc�1(i)j = ni,for all i.

In analogy with (2.5) and (2.7), we introduce, for any k-composition (U1; : : : ; Uk) of [n], theassociated Young subgroup SU1;:::;Uk of Sn de�ned by

SU1;:::;Uk = f� 2 Sn j �(Ui) = Ui; i = 1; : : : ; kg ;

and, for any H � Sn, the subgroup HU1;:::;Uk = H \ SU1;:::;Uk . While Theorem 2.1 classi�es theM(X1+ � � �+Xk)-structures by keeping a �xed k-composition (the standard k-composition) for anyn1+ � � �+nk = n and letting theM -structures vary, the following variant keeps a �xedM -structures0 2M [n] and lets the k-compositions (U1; : : : ; Uk) of [n] vary.

Theorem 2.2 (k-colored molecular expansion) Let M =M(X) = Xn=H be a molecular spe-cies with H � Sn. Then

M(X1 + � � �+Xk) =X

U1+���+Uk=[n]

jHU1;:::;Uk j

jH j

XU11 � � �XUk

k

HU1;:::;Uk

(2.13)

where XU11 XU2

2 � � �XUkk =HU1;U2;:::;Uk is the molecular species whose set of structures, on [V1; V2;

: : : ; Vk], is de�ned to be the set

f�HU1;U2;:::;Uk j � = (�1; �2; : : : ; �k); �i : Ui��! Vig: (2.14)

Proof

Let s0 be aM -structure on [n], having H as its stabilizer. Then anyM(X1+ � � �+Xk)-structurecan be expressed, up to isomorphism, in the form (s0; U1; : : : ; Uk) where U1 + � � �+ Uk = [n]. Theaction of an element h 2 H on this structure is given by

h � (s0; U1; : : : ; Uk) = (s0; h(U1); : : : ; h(Uk)): (2.15)

and the stabilizer of (s0; U1; : : : ; Uk) is clearly HU1;:::;Uk . Since we wish to classify the isomorphismtypes of M(X1 + � � �+Xk)-structures according to their stabilizers, the result follows by applyingBurnside's Lemma with

G = H; W = f(s0; U1; : : : ; Uk) j U1 + � � �+ Uk = [n]g ;

A =ZjjX1; : : : ; Xkjj; w(s0; U1; : : : ; Uk) = XU11 � � �XUk

k =HU1;:::;Uk :

2

Our last general addition formula involves words on the ordered alphabet

X = Xk = f1; 2; : : : ; kg ; (2.16)

with the letter i associated with the sort Xi. Words of length n, � = �1�2 � � ��n, in this alphabetencode k-compositions (U1; : : : ; Uk) of [n] or, equivalently, k-colorings c : [n]! [k] of [n] accordingto the following rule, for j 2 [n]:

j 2 Ui () �j = i() c(j) = i : (2.17)

7

Hence the word � 2 Xn can be considered as a color word encoding the k-composition. For anyword �, let �� denote some bijection of [n] such that ��(Ui) = �ni where ni = jUij is the numberof occurences of the letter i in �. Note that any subgroup H � Sn acts on the set Xn of words� = �1�2 : : :�n of length n on X by

h � (�1�2 : : :�n) = �h�1(1)�h�1(2) : : :�h�1(n): (2.18)

This action can be equivalently de�ned on the k-compositions of [n] by

h � (U1; U2; : : : ; Uk) = (h(U1); h(U2); : : : ; h(Uk)) (2.19)

or on the k-colorings of [n] byh � c = c � h�1 : (2.20)

An orbit of this action on Xn is called a H-class (or H-orbit) of words of length n on X. For anyword �, we set

AutH

� = fh 2 H j h � � = �g: (2.21)

Theorem 2.3 (Word class expansion) Let M = M(X) = Xn=H be a molecular species withH � Sn. Then

M(X1 + � � �+Xk) =X

n1+���+nk=n

X�2n1;:::;nk

Xn11 � � �Xnk

k

��(AutH�)��1�

(2.22)

where n1;n2;:::;nk is a set of representatives (for example, lexicographically smallest words) of theH-classes of words on X = f1; 2; : : : ; kg having exactly ni occurences of letter i for i = 1; 2; : : : ; k.

Proof

Note that under the correspondence (2.17) we have AutH � = HU1;:::;Uk . Also note that in thering of k-sort species, we have

XU11 � � �XUk

k =HU1;:::;Uk = Xn11 : : :Xnk

k =��(AutH�)��1� ; (2.23)

The result then follows from Theorem 2.2 by regrouping the k-compositions (U1; : : : ; Uk) of [n]which belong to the same H-orbit, since the size of such an orbit is jH j=jHU1;:::;Uk j. 2

3 Speci�c addition formulas

3.1 Lists, chains, sets and oriented sets

The speci�c addition formulas of this subsection, for the species Xn, Chan, En and E�n , are conse-

quences of the general addition Theorems 2.1{2.3.

3.1.1 Lists. Let M = Xn be the molecular species of n-lists. Here H = f1g, the trivial oneelement subgroup of Sn, and any one of the general addition formulas (2.8), (2.13) or (2.22) leadsto the classical multinomial formula (1.9) mentioned in the Introduction.

8

123 37 26122524

11 12 23 24 25 26 37

12 11 26 25 24 23 37

1

Figure 3: Transformation Cha(X1 + � � �+Xk)! Xn11 � � �Xnk

k =Z2 (palindromic case).

3.1.2 Chains. Since chains can be seen as lists up to \reversal", the molecular species M =M(X) = Chan(X) of n-chains is characterized by the combinatorial equation

Chan(X) = Xn=Z2; (3.1)

where Z2 is interpreted as the subgroup H = h�i � Sn generated by the \global" transposition

� = �n = (1; n)(2; n� 1)(3; n� 2) : : : :

In this case, addition formula (2.22), takes the following form:

Theorem 3.1 We have

Chan(X1 + � � �+Xk) =X

n1+���+nk=n

paln1;:::;nkXn1

1 � � �Xnkk

Z2(3.2)

+1

2

�n!

n1! � � �nk !� paln1;:::;nk

�Xn1

1 � � �Xnkk

where paln1;:::;nk is the number of palindromic words on X = f1; 2; : : : ; kg having ni occurences of let-ter i for i = 1; : : : ; k and the action of Z2 on X

n11 � � �Xnk

k -structures corresponds to the simulaneousreversal of the induced ni-lists on �ni.

ProofWe illustrate the use of the word class expansion theorem (Theorem 2.3). The action of the

generator � ofH 'Z2 on a word � = �1�2 � � ��n is given by � �� = ����1 = ��� = �n � � ��2�1=:�.Suppose �rst that � is a palindrome, i.e. that � � � = �. In this case, the H-class of � is f�g,and we have AutH(�) = H and ��H���1 is generated by �����1� = �n1�n2 � � ��nk 2 Sn1;n2;:::;nk ,giving the �rst term in the right hand side of (3.2). The correspondence between a Cha7(X1 +X2+X3)-structure on U1 = f11 ; 12 g, U2 = f23 ; 24 ; 25 ; 26 g and U3 = f37 g (this is a variant of thenotation �n1; �n2; �n3, which puts emphasis on the colors), and the associated X2

1X42X3=Z2-structure

is illustrated in Figure 3 in the case of the palindromic color word � = 2123212.If the word � is not palindromic, its H-class is f�; �g, where � = � �� and we have AutH(�) = 1.

We choose the lexicographically minimum among � and � to obtain a canonical orientation andthen regroup the elements in increasing order of color. The number of such terms is one half of thenumber of non palindromic words and we obtain the second part of the right side of (3.2). Thiscorrespondence is illustrated in Figure 4 with � = 1223212. 2

9

1 23 7324 25 26

26 12 25 73 24 11

11 12

1

2

12

23

24 25 26 733=

Figure 4: Transformation Cha(X1 + � � �+Xk)! Xn11 � � �Xnk

k (non palindromic case).

The following expressions are easily checked:

paln1;:::;nk =

8>>>>>><>>>>>>:

(n2 )!(n12 )!���(

nk2 )!

; if every ni is even;

(n�12 )!

(n12 )!����nj�1

2

�!���(

nk2 )!

; if only one nj is odd;

0; otherwise:

(3.3)

3.1.3 Sets. Since sets are unordered lists, the molecular species M = M(X) = En(X) of n-setscan be written in the form

En(X) = Xn=Sn: (3.4)

There is only one En-structure s on [n] and Aut(s)n1;:::;nk = Sn1;:::;nk . Addition formula (2.8) takesthe form

En(X1 + � � �+Xk) =X

n1+���+nk=n

En1(X1) � � �Enk(Xk); (3.5)

since Xn11 � � �Xnk

k =Sn1;:::;nk = (Xn11 =Sn1) � � �(X

nkk =Snk).

3.1.4 Oriented sets. By de�nition [Bergeron et al. (1998)], an oriented set is a list up to aneven permutation of its elements. This kind of structures has been used by P�olya in [P�olya andRead (1976)]. The molecular species M =M(X) = E�

n (X) of oriented n-sets is characterized by

E�n (X) = Xn=An; (3.6)

where An is the alternating subgroup of Sn. For n � 2, there are exactly two oriented set-structureson [n] each with stabilizer An (since An is normal in Sn) and exactly one for n = 0 or 1. Observethat E�

0 (X) = 1, E�1 (X) = X , E�

2 (X) = X2 and E�3 (X) = C3(X). This time, addition formula

(2.8) can be written in the following form.

Theorem 3.2 Let n � 2. Then we have

E�n (X1 + � � �+Xk) =

Xn1+���+nk=n

"n1;:::;nkE�n1(X1) � � �E�

nk(Xk)

�; (3.7)

where

"n1;:::;nk =

(2 if ni � 1, for all i,1 if ni � 2, for some i,

(3.8)

10

and two E�n1(X1) � � �E�

nk(Xk)-structures are congruent (�) if one can be obtained from the other by

an even number of local non-trivial orientation reversals.

Proof

Since n � 2, there are exactly two oriented sets on [n], s1 and s2, each having the stabilizerAn � Sn. Hence, Theorem 2.1, with M = E�

n , gives

E�n (X1 + � � �+Xk) =

Xn1+���+nk=n

2jAn1;:::;nk j

n1! � � �nk !

Xn11 � � �Xnk

k

An1;:::;nk; (3.9)

where An1;:::;nk = An \Sn1;:::;nk is the group of all even permutations � of [n] such that �( �ni) = �ni,i = 1; : : : ; k. Now, given any (n1; : : : ; nk) with n1 + � � �+ nk , let

J = fj j nj � 2g : (3.10)

We have two cases to consider: J = ; or J 6= ;.

Case 1 If J = ;, then An1;:::;nk = fidg and n1! � � �nk ! = 1. This gives the contribution 2Xn11 � � �Xnk

k

in (3.9).

Case 2 If J 6= ;, then

An1;:::;nk =[

(h1;:::;hk)2An1 ;:::;nk

(h1An1)� � � � � (hkAnk) ; (3.11)

which can be rewritten as a disjoint union of 2jJj�1 sets each havingQj2J(nj !=2) elements, since

(h1; : : : ; hk) 2 An1;:::;nk if and only if an even number of hi are odd permutations. Hence,

J 6= ; ) jAn1;:::;nk j =1

2

Yj2J

nj ! :

Summing up Cases 1 and 2 shows that (3.9) can be rewritten in the form

E�n (X1 + � � �+Xk) =

Xn1+���+nk=n

J=;

2Xn11 � � �Xnk

k +X

n1+���+nk=nJ 6=;

Xn11 � � �Xnk

k

An1;:::;nk: (3.12)

There remains to show that

J 6= ; )Xn1

1 � � �Xnkk

An1;:::;nk=

�Xn1

1

An1

�� � �

Xnkk

Ank

!!,� : (3.13)

This follows from the fact that, in view of (3.11), any coset (�1; : : : ; �k)An1;:::;nk of An1;:::;nk inSn1;:::;nk can be written as a disjoint union of products of cosets of the form (�1h1An1) � � � � �(�khkAnk) where (h1; : : : ; hk) corresponds to an even number of local non trivial orientation rever-sals in the E�

n1 � � �E�nk-structures (�1An1 ; : : : ; �kAnk ). 2

11

3.2 Cycles and polygons

3.2.1 Cycles. The molecular species M =M(X) = Cn(X) of (oriented) n-cycles is characterizedby

Cn(X) = Xn=H; H = h�i � Sn; (3.14)

where � = �n is the n-cycle (1; 2; � � � ; n). Note that C0 = 0, C1 = X and C2 = E2. As in theword class addition theorem, we can classify the Cn(X1 + � � �+ Xk)-structures according to theirassociated H-classes of words on the alphabet X = f1; 2; : : : ; kg. Here H = Cn = < � > andan H-class is called a circular class (or conjugacy class) of words. The circular class of a word� = �1 � � ��n 2 Xn is denoted by [�]. Hence

[�] = f�; � � �; �2 � �; : : : ; �n�1 � �g : (3.15)

The words � and �0 2 Xn are conjugate (denoted by � � �0) if �0 = �j �� for some j = 0; 1; : : : ; n�1.A word is primitive if it is not a positive power (for concatenation) of another word. A Lyndonword is a primitive word which is lexicographically minimum in its conjugacy class. Here we usethe total order of the alphabet X.

The distribution of a word � 2 X� is the vector = ( 1; 2; : : : ; k) where j is the number ofoccurences of the letter j. We set

� = number of Lyndon words with distribution :

Proposition 3.1 The number � of Lyndon words on X = f1; 2; : : : ; kg with distribution =( 1; : : : ; k) is given by

� =1

1 + � � �+ k

X�jgcd( 1;���; k)

�(�)

1+���+ k

� 1� ; � � � ;

k�

!: (3.16)

ProofThis is a simple application of M�obius inversion. 2

Theorem 3.3 ([Labelle (1992)]) For n � 2, we have

Cn(X1 + � � �+Xk) =Xdjn

X 1+���+ k=d

� Cn=d

�X 1

1 � � �X kk

�: (3.17)

ProofLet U = U1+U2+� � �+Uk and consider any Cn(X1+� � �+Xk)-structure s on this k-colored set. It

determines a circular class of color words [�] obtained by reading the colors along the oriented cycle.In fact there is a unique representative � of this class which is a power of a Lyndon word � : � = �n=d

where d = j�j, the length of �. The power m = n=d corresponds to the circular symmetry of the

color pattern. From this decomposition, we naturally deduce a Cn=d(X 11 + � � �+ X k

k )-structure�(s) on U , where is the distribution of �, by regrouping the elements of the same color in eachof the n=d blocks of type �. It is clear that s and �(s) have the same stabilizer and that s canbe recovered from �(s) and �. The theorem follows. This correspondence is illustrated with the

12

4

310

11

25

1

2

262

11

27

13

3

8

312

292

Figure 5: A C12(X1 +X2 +X3)-structure.

C12(X1 + X2 + X3)-structure s (see Figure 5) where U1 = f11 ; 12 ; 13 g, U2 = f24 ; 25 ; � � � ; 29 g,U3 = f310 ; 311 ; 312 g, � = 123212321232, and � = 1232. In fact, we have

s = [11 24 310 25 12 26 311 27 13 28 312 29 ]

7! [11 24 310 25 , 12 26 311 27 , 13 28 312 29 ]

7! [11 24 25 310 , 12 26 27 311 , 13 28 29 312 ] = �(s) ;

with s 2 C12(X1 +X2 +X3)[U ] and �(s) 2 C3(X1X22X3)[U ]. 2

3.2.2 Polygons. By de�nition, a polygon is an unoriented cycle. The molecular species M =M(X) = Pn(X) of n-gons (polygons on n vertices) is characterized by

Pn(X) = Xn=Dn; (3.18)

where Dn is the dihedral 2n-element subgroup of Sn with generators � = �n = (1; 2; � � � ; n) and� = �n = (1; n)(2; n� 1)(3; n� 2) � � �.

Note that P3 = E3. Similarly to Cn, our classi�cation of the Pn(X1+� � �+Xk)-structures is basedon the analysis of the associated Dn-classes of color words on the alphabet X = f1; 2; : : : ; kg. To dothis, we introduce the concept of a dexterpalindromic word. Recall that a word � = �1�2 � � ��n 2 X

n

is a palindrome if � = �, where � = � � � = �n � � ��2�1.

De�nition 3.1 A word � 2 Xn is called a dexterpalindrome if � = x� where x 2 X and � is apalindrome. We say that the letter x is the pivot of �. 4

Note that in the odd length case, a circular class can contain both a palindrome and a dexter-palindrome; for example the palindrome 12321 is conjugate to the dexterpalindrome 32112. In theeven case, a circular class can contain two di�erent palindromes or dexterpalindromes; for example

123321 � 321123 and 112321 � 321112 :

The enumeration of dexterpalindromes is an interesting question in itself. Observe that the class� of dexterpalindromes (of all possible lengths) is iterative in the sense that for any word �, and

13

any integer m > 0, � 2 � if and only if �m 2 �. It follows that the union of all circular classesof dexterpalindromes forms a cyclic language in the sense of [Berstel and Reutenauer (1990)]. Thesame is true for palindromes. The cyclic languages have the property that their zeta functions arerational. See also [Reutenauer (1997)].

A Dn-orbit of words is called a dihedral class. The dihedral class of a word � is denoted by [[�]].We have [[�]] = [�] [ [�]. A dihedral class c = [[�]] is called skew if [�] 6= [�]. The situation where[�] = [�], corresponding to a re exive symmetry of the color pattern, occurs exactly in one of two(not mutually exclusive) cases:

1. c is palindromic, i.e. c = [�], where � is a plalindrome;

2. c is dexterpalindromic, i.e. c = [�], where � is a dexterpalindrome.

Finaly, we say that c = [�] is primitive if � is primitive and the distribution of c is de�ned to bethe distribution of �.

The numbers � ; � ; � ; and � (i) are then de�ned as follows:

� � is the number of skew primitive dihedral classes having distribution ,

� � is the number of palindromic primitive dihedral classes having distribution ,

� � is the number of dexterpalindromic primitive dihedral classes having distribution ,

� � (i) is the number of dexterpalindromic primitive dihedral classes having distribution , forwhich the letter i is a pivot.

These numbers, which represent the various multiplicities in the molecular decomposition ofPn(X1 + � � �+ Xk) can be explicitely computed, using M�obius inversion. For a given distribution = ( 1; : : : ; k) we write dj to denote that dj 1, . . . , dj k. Also let ei 2 Nk denote the ith

unit vector. Recall that the numbers pal = pal 1;:::; k of palindromes with distribution is givenby (3.3) and � , of Lyndon words with distribution , by (3.16).

Proposition 3.2 The numbers of dihedral classes � , � , � and � (i) de�ned above are given by

� =1

1 + �(j j even)

Xdj

�(d) pal =d ; (3.19)

� (i) =1

1 + �(j ij even)

Xdj

�(d) pal =d�ei ; (3.20)

� =1

1 + �2( )

Xi2Supp

� (i) ; (3.21)

where �2( ) = �( has 2 odd components), and

� =1

2(� � � � �(j j even) � ) : (3.22)

2

14

We also need two families of quotient species, denoted by

Cm(X 11 : : :X k

k )�i0Z2 and Cm(X

11 : : :X k

k )��i0Z2; (3.23)

where 0 � i0 � k, i0 > 0, according to actions "�i0" and "�i0" of Z2 = f1; �g on the species

Cm(X 11 � � �X k

k ), de�ned as follows: Given any s = [�1; �2; : : : ; �m] 2 Cm(X 11 � � �X k

k )[U ], where

each �j = �1;j�2;j � � ��k;j is a X 11 � � �X k

k -structure, i.e. �ij is a list of elements of Ui, we de�ne� �i0 s by

� �i0 s = [�0m; �0m�1; : : : ; �

01] ; (3.24)

with �0j = �1;j � � ��i0;j+1 (mod m) � � ��k;j . Moreover, if �i0;j = aj�i0 ;j , with aj 2 Ui0 and �i0;j a

X i0�1i0

-structure, we de�ne � �i0 s by

� �i0 s = [�00m; �00m�1; : : : ; �

001] ; (3.25)

with �00j = �1;j � � �aj+1 (mod m)�i0;j � � ��k;j .

Given a distribution 2 Nk, we introduce the following notations:

j j = 1 + 2 + � � �+ k; Supp = fi j i 6= 0g;O( ) = fi j i is oddg;�m( ) = �(jO( )j= m) = �( has m odd components);

(3.26)

m = 0; 1; 2; : : : ; where � denotes the truth function.

Theorem 3.4 For n � 3, the species Pn(X) = Xn=Dn of n-gons satis�es the addition formula

Pn(X1 + � � �+Xk) =Xdjn

X 2Nk

j j=d

5Xj=1

� j (X1; : : : ; Xk) (3.27)

where the � j are species of k sorts, de�ned as follows:

� 1 (X1; : : : ; Xk) = � Cn=d(X 11 � � �X k

k ); (3.28)

� 2 (X1; : : : ; Xk) = �1( )� Cn=d(X 11 � � �X k

k ).i0Z2; (3.29)

where i0 is the unique element of O( );

� 3 (X1; : : : ; Xk) = �2( )� (i0)Cn=d(X 11 � � �X k

k ).i0Z2; (3.30)

where O( ) = fi0 < i1g;

� 4 (X1; : : : ; Xk) = �0( )� Cn=d(X 11 � � �X k

k ).j0Z2; (3.31)

where j0 = min Supp ;

� 5 (X1; : : : ; Xk) = �0( )X

i2Supp

� (i) Cn=d(X 11 � � �X k

k )..iZ2: (3.32)

15

1

24

313

25

26

12228

314

13

29

2

7

2113

10

212

11

24

313

25

26

1227

28

15

14

13

29

210

2

3

315 212

111

b)a)

Figure 6: Correspondence s 7! �(s) (skew case).

Proof

Let s be any Pn(X1 + � � �+ Xk) on U , with k-composition U = U1 + � � �+ Uk . It determinesa dihedral class c = [[�]] of color words, obtained by reading the colors along the polygon in onedirection (to get [�]) or the other (to get [�]). There exists a unique integer m � 1 and a uniqueprimitive word � such that � = �m. We distinguish �ve cases which depend on the symmetries of[[�]] and on the distribution = ( 1; : : : ; k) of the primitive root � of �. In each case j = 1; 2; : : : ; 5,

there is a canonical corresponding � j (X1; : : : ; Xk)-structure �(s) on U .

Case 1. The diheral class c = [[�]] is skew. Then � � �m1 and � � �m2 where �1 and �2 are distinctLyndon words. We can assume without loss of generality that �1 is lexicographically smaller that�2. This gives a canonical orientation to the polygon. Moreover, the elements of U can be regroupedin increasing order of the colors within each block of type � to obtain a Cn=d(X

11 � � �X 2

2 )-structure�(s) on U where d = n=m is the length of �1. The word �1 is needed as information to recover s from�(s). Observe that the structures s and �(s) have the same stabilizer K � SU1;:::;Uk . This gives theterm (3.28). An illustration of this correspondence is given in Figure 6, with a P15(X1+X2+X3)-structure on U , where U1 = f11 ; 12 ; 13 g, U2 = f24 ; 25 ; � � � ; 212 g, U3 = f313 ; 314 ; 315 g, �1 = 12232,�2 = 12322 and m = n=d = 3. The correspondence can also be described in linear notation asfollows:

s = [[11 24 25 313 26 12 27 28 314 29 13 210 211 315 212 ]]

7! [11 24 25 313 26 , 12 27 28 314 29 , 13 210 211 315 212 ]

7! [11 24 25 26 313 , 12 27 28 29 314 , 13 210 211 212 315 ] = �(s) ;

with s 2 C15[U ] and �(s) 2 C3(X1X32X3)[U ].

Case 2. The class c is palindromic and � � �n=d, where � is a primitive palindrome of odd length.This implies that the distribution of � has only one odd component i0 where i0 is the central ele-

ment of �. There is a canonical algorithm which transforms s into �(s) 2 Cn=d(X 11 � � �X 2

2 )�i0Z2

such that s and �(s) have the same stabilizer and s can be recovered from �(s) and �. Thisyields (3.29). The algorithm is decribed here on an example where n = 27, � = 132121231, = (4; 3; 2) and m = n=d = 3. Note that the central letter i0 is 2.

16

1

12

215

323

1

16

2

5

112

322

2

17

14324216

1

13

218

325

18

19

3262

7

110 220 111221

327

11

317

1

19

12

2921013

211

212

319

318320

321

14

5

213

1323 324

16

215

316

27

214

b)a)

82

22

214

3

13

Figure 7: a) primitive root = odd palindrome (Case 2); b) root = even dexterpalindrome withdistinct pivots (Case 3).

Start with the P27(X1 +X2 +X3)-structure (see Figure 7 a))

s = [[11 322 213 12 214 13 215 323 14

15 324 216 16 217 17 218 325 18

19 326 219 110 220 111 221 327 112 ]] :

First, regroup the elements of the central color i0 (i.e. 2) on each side of the central element ofeach line, i.e. 214 , 217 and 220 , to get

s1 = [[11 322 12 213 214 215 13 323 14

15 324 16 216 217 218 17 325 18

19 326 110 219 220 221 111 327 112 ]] :

Then, take the conjugate of s1, which puts the elements of color 2 �rst to get the circular class

s2 = [213 214 215 13 323 14 15 324 16

216 217 218 17 325 18 19 326 110

219 220 221 111 327 112 11 322 12 ] :

Finally, regroup the elements in increasing order of colors along each line to obtain

s3 = [13 14 15 16 213 214 215 323 324 ,

17 18 19 110 216 217 218 325 326 ,

111 112 11 12 219 220 221 327 322 ] ;

which is a C3(X41X

32X

23)-structure on U . This construction depends on the orientaion chosen to

describe the polygon structure s on U as a dihedral class s = [[�]]. The reader will easily check thatif we start with the opposite orientation �, the result will be the C3(X4

1X32X

23)-structure

s03 = [12 11 112 111 215 214 213 322 327 ,

110 19 18 17 221 220 219 326 325 ,

16 15 14 13 218 217 216 324 323 ] ;

17

and that we have s03 = � �2 s3 under the action � given by (3.24), with i0 = 2. The �nal result is

then the (unordered) pair �(s) = fs3; s03g 2

�C3(X

41X

32X

23)�2Z2

�[U ].

Case 3. The class c = [[�]] is dexterpalindromic, � � �m, where � is of even length, and thetwo possible pivots of the primitive root � are distinct letters (colors) i0 < i1. In this case, thedistribution of � has exactly two odd components i0 and i1 . We choose to write � = i1�i0� sothat the smallest pivot i0 is \central". The algorithm to construct �(s) from s, to give (3.30), is thensimilar to Case 2 and is illustrated in the following example, where i0 = 2, i1 = 3, � = 32132312, = (2; 3; 3),m = 3 and n = 24. Starting with s 2 P24(X1+X2+X3) (see Figure 7 b)), we obtain

s3 2 C3(X21X

32X

33)[U ] and s

03 = � �2 s3 so that �(s) = fs3; s

03g 2

�C3(X

21X

32X

33)�2Z2

�[U ]. Note

that s can be recovered from �(s) and �.

s = [[316 27 11 317 28 318 12 29319 210 13 320 211 321 14 212322 213 15 323 214 324 16 215 ]] ;

s1 = [[316 11 317 27 28 29 318 12319 13 320 210 211 212 321 14322 15 323 213 214 215 324 16 ]] ;

s2 = [27 28 29 318 12 319 13 320210 211 212 321 14 322 15 323213 214 215 324 16 316 11 317 ] ;

s3 = [12 13 27 28 29 318 319 320 ,14 15 210 211 212 321 322 323 ,16 11 213 214 215 324 316 317 ] ;

while

s03 = [11 16 29 28 27 317 316 324 ,

15 14 215 214 213 323 322 321 ,

13 12 212 211 210 320 319 318 ] :

Case 4. The class c = [[�]] is palindromic, with � � �m, where � is a primitive palindrome of evenlength. Then � is of the form � = �� and we can assume that � < � (lexicographically) to obtaina canonical representative in the dihedral class. In this case, all components of the distribution of � are even. The algorithm to obtain �(s) from s is similar to the two preceeding cases, exceptthat we have the choice of the \centralizing" color. Hence we choose the minimum color j0 whichoccurs in �: j0 = min Supp . This is illustrated in the following example, where � = 21322312,with = (2; 4; 2), j0 = 1, m = 3 and n = 24. We have s 2 P24(X1 +X2 +X3) (see Figure 8 a)),

s3 2 C3(X21X

42X

23)[U ] and s

03 = � �1 s3 so that �(s) = fs3; s

03g 2

�C3(X

21X

42X

23)�1Z2

�[U ].

s = [[27 11 319 28 29 320 12 210211 13 321 212 213 322 14 214215 15 323 216 217 324 16 218 ]] ;

s1 = [[27 319 28 11 12 29 320 210211 321 212 13 14 213 322 214215 323 216 15 16 217 324 218 ]] ; ;

s2 = [11 12 29 320 210 211 321 21213 14 213 322 214 215 323 21615 16 217 324 218 27 319 28 ] ;

s3 = [11 12 29 210 211 212 320 321 ,13 14 213 214 215 216 322 323 ,15 16 217 218 27 28 324 319 ] ;

while

s03 = [12 11 28 27 218 217 319 324 ,

16 15 216 215 214 213 323 322 ,

14 13 212 211 210 29 321 320 ] :

18

11

13

15323

27

319

320

29

12210211

321

212

213

4

2

14

214

215 16

216 217

324

218

323

27

11

18

319

320

210

1221113212

321

213

1

214

15 16

216217 324

b)a)

22

3

29

2

2

83

8

152

22

Figure 8: a) primitive root = even palindrome (Case 4); b) root = even dexterpalindrome withequal pivots (Case 5).

Case 5. The class c = [[�]] is dexterpalindromic, with � � �m, where � is a primitive palindromeof even length, with the two pivots equal, say equal to i 2 Supp . In this case also, has no oddcomponent. We can write � = i�i� and assume, without loss of generality, that � < � to obtain acanonical representative in the dihedral class. The algorithm to obtain �(s) from s is described inthe following example, where � = 21232321, with = (2; 4; 2), i = 2, m = 3 and n = 24. Startingwith the P24(X1 +X2 +X3)-structure (see Figure 8 b))

s = [[27 11 28 319 29 320 210 12

211 13 212 321 213 322 214 14

215 15 216 323 217 324 218 16 ]] ;

we immediately regroup the elements of each line in increasing order of colors, to obtain

s1 = [11 12 27 28 29 210 319 320 ,

13 14 211 212 213 214 321 322 ,

15 16 215 216 217 218 323 324 ] ;

which is a C3(X21X

42X

23)-structure on U . If the structure s is \read" in the opposite direction, one

obtains

s01 = [16 15 27 218 217 216 324 323 ,

14 13 215 214 213 212 322 321 ,

12 11 211 210 29 28 320 319 ] ;

and it is readily checked that s01 = � �2 s1 with the action de�ned by (3.25), with i0 = 2. Hence we

obtain �(s) = fs1; s01g 2�C3(X2

1X42X

23)��2Z2

�[U ], with the same stabilizer as s. Since s can be

recovered uniquely from �(s) and �, we have (3.32) and the proof is complete.

2

19

4 Some speci�c applications

We conclude our analysis of combinatorial addition formulas by applications to some fundamentalcomputation in the theory of species.

4.1 Applications to cycle and asymmetry index series and associated q-series

Let F = F (X) be a species and t = (t1; t2; t3; : : :) be a countable family of (commuting) formalvariables. By giving weights, taken arbitrarily among the ti's, to each singleton of the underlyingset of F -structures we obtain the folowing weighted species

Ft(X) = F (Xt1 +Xt2 + � � �) ; (4.1)

where, for i = 1; 2; 3; : : :, Xti denotes the species of singletons with weight ti. The weight of aFt-structure is, by de�nition, the product of the weights of all the singletons in its underlying set.This weight is a monomial in the ti's. Figure 9 shows, for F = a , the species of trees, a typical a t-structure on the set [8] = f1; 2; 3; 4; 5; 6; 7; 8g. By unlabelling the Ft-structures, while keeping their

13t

2

31t

1t1t

t3

t

t4t8

3

1

7

4

8

5

6

Figure 9: A typical a t-structure on the set [8] (of weight t31t23t4t8t13)

weights, two symmetric functions e�F (t1; t2; t3; : : :) and �F (t1; t2; t3; : : :) in the ti's can be associatedto the species F as follows:

e�F (t1; t2; t3; : : :) = total t-weight of all unlabelled Ft-structures : (4.2)

�F (t1; t2; t3; : : :) = total t-weight of all unlabelled asymmetric Ft-structures : (4.3)

Figures 10 and 11 describe respectively a typical unlabelled a t-structure and a typical unlabelledasymmetric a t-structure having, incidentally, the same weight t31t

23t4t8t13.

It is well-known, from the theory of symmetric functions [Macdonald (1995)] that the powersums of the ti's, de�ned by

p� = p�(t1; t2; t3; : : :) = t�1 + t�2 + t�3 + � � � ; � = 1; 2; 3; : : : ; (4.4)

form an algebraic basis of the ring of symmetric functions in t1; t2; t3; : : :. This means that anysymmetric function in the t0is can be written in a unique way as a power series in p1; p2; p3; : : :.

In particular ([Bergeron et al. (1998), Labelle (1992)]), the cycle index series, ZF (x1; x2; x3; : : :),and the asymmetry index series, �F (x1; x2; x3; : : :), of the species F can be de�ned as being theunique series satisfying

ZF (p1; p2; p3; : : :) = e�F (t1; t2; t3; : : :) ; (4.5)

20

1

13tt

1t1t

t3

t

t4t8

3

Figure 10: Unlabelled a t-structure

t3

t3 t8

t1

13t1t

4t1t

Figure 11: Unlabelled asymmetric a t-structure

and�F (p1; p2; p3; : : :) = �F (t1; t2; t3; : : :) : (4.6)

Hence, in view of (4.1){(4.3), any addition formula F (X1+X2+ � � �) = �F (X1; X2; : : :) can be usedto analyse and/or compute ZF and �F via substitutions of the form

Ft(X)���X:=1

= F (Xt1 +Xt2 + � � �)���X := 1

= F (X1 +X2 + � � �)���Xi :=Xti ; i=1;2;:::

���X := 1

= �F (X1; X2; : : :)���Xi :=Xti ; i=1;2;:::

���X := 1

(4.7)

since the substitution X :=1 in a weighted species corresponds to unlabelling the structures whilekeeping track of their weights. In the case of �F , one must restrict (4.7) to asymFt(X), thesubspecies of Ft(X) consisting of asymmetric Ft-structures.

As an illustration, let us compute, using this method, the series ZCn and �Cn for the speciesCn of n-cycles. These series are known (see [Bergeron et al. (1998)]) to be given by

ZCn(p1; p2; p3; : : :) =1

n

Xdjn

�(d)pn=dd ;

�Cn(p1; p2; p3; : : :) =1

n

Xdjn

�(d)pn=dd ;

(4.8)

21

where � is the Euler function and � is the M�obius function. We will compute �rst �Cn , via theformula �Cn = �Cn , and then establish the special relation

e�Cn(t1; t2; t3; : : :) =Xdjn

�Cd(tn=d1 ; t

n=d2 ; t

n=d3 ; : : :) ; (4.9)

from which the formula for ZCn will follow. By the addition formula (3.17) for Cn, we have

(Cn)t(X) = Cn(Xt1 +Xt2 + � � �)

=Xdjn

X 1+ 2+���=d

� Cn=d

�(Xt1)

1(Xt1) 1 � � �

�: (4.10)

Observe that, for 1 + 2 + � � � = d, (Xt1) 1(Xt2)

2 � � � =�X 1+ 2+���

�t1 1 t2 2 ���

=�Xd�t1 1 t2 2 ���

and that a Cn=d(Xd)-structure is asymmetric if and only if d = n. This implies that

�Cn(p1; p2; : : :) = �Cn(t1; t2; : : :)

=X

1+ 2+���=n

� t 11 t 22 � � �

=1

n

Xdjn

�(d)(td1 + td2 + � � �)n=d

=1

n

Xdjn

�(d)pn=dd : (4.11)

making use of the explicit formula (3.16) for � . On the other hand, using (4.10), we �nd that

ZCn(p1; p2; : : :) = e�Cn(t1; t2; : : :)=

Xdjn

X 1+ 2+���=d

� tnd 1

1 tnd 2

2 � � �

=Xdjn

�Cd

�tnd1 ; t

nd2 ; : : :

�

=1

n

Xdjn

�(d)pn=dd ; (4.12)

since the weight of an unlabelled Cn=d�(Xt1)

1(Xt2) 2 � � �

�-structure is t

nd 1

1 tnd 2

2 � � �. This compu-

tation also establishes (4.9). As a by-product, we have the identity

ZCn(p1; p2; p3; : : :) =Xdjn

�Cn(pn=d; p2n=d; p3n=d; : : :) ; (4.13)

which implies thatZC(p1; p2; p3; : : :) =

Xk�1

�C(pk; p2k; p3k; : : :) ; (4.14)

where C = C1+C2+C3+ � � � is the species of all oriented cycles. Using the M�obius function �(k),formula (4.14) can be rewritten as

�C(p1; p2; p3; : : :) =Xk�1

�(k)ZC(pk; p2k; p3k; : : :) ; (4.15)

22

which can be used to prove the compact formula (�rst found in [Labelle (1992)])

�S(p1; p2; p3; : : :) =1� p21� p1

; (4.16)

where S = E(C) is the species of all permutations.For an arbritrary molecular species M , the computation of the series ZM and �M can be done

in a similar way by making use of the general expansion Theorems 2.1, 2.2 and 2.3:

Theorem 4.1 Let M = M(X) = Xn=H be a molecular species of degree n, with H � Sn. Thenthe cycle index series ZM and the asymmetry index series �M of M can be written as follows, interms of monomials in the ti's,

ZM (p1; p2; : : :) = e�M (t1; t2; : : :)

=X

n1+n2+���=n

Xs2M [n]

jAut(s)n1;n2;:::jtn11 t

n22 � � �

n1!n2! � � �(4.17)

=1

jH j

XU1+U2+���=[n]

jHU1;U2;:::j tjU1j1 t

jU2 j2 � � � (4.18)

=X

n1+n2+���=n

X�2n1;n2 ;:::

tn11 tn22 � � � ; (4.19)

�M (p1; p2; : : :) = �M(t1; t2; : : :)

=X

n1+n2+���=n

Xs2M [n]

jAut(s)n1;n2 ;::: j=1

tn11 tn22 � � �

n1!n2! � � �(4.20)

=1

jH j

XU1+U2+���=[n]jHU1;U2 ;:::

j=1

tjU1j1 t

jU2j2 � � � (4.21)

=X

n1+n2+���=n

X�2n1;n2 ;:::jAutH �j=1

tn11 tn22 � � � : (4.22)

2

Combinatorial addition theorems can also be applied to compute the canonical q-series, F (x; q)and F hx; qi, associated with any species F , which were introduced by H�el�ene D�ecoste in [D�ecoste(1993)]. These series are the q-exponential power series in x which arise from the substitutionsti :=(1� q)qi�1x, i = 1; 2; : : :, in the symmetric functions e�F (t1; t2; : : :) and �F (t1; t2; : : :):De�nition 4.1 (Canonical q-series) [D�ecoste (1993)] F (x; q) and F hx; qi, the canonical q-seriesassociated to a species F , are de�ned by

F (x; q) =Xn�0

fn(q)xn

n!q

:= e�F ((1� q)x; (1� q)qx; (1� q)q2x; : : :) (4.23)

= ZF

(1� q)x

1� q;(1� q)2x2

1� q2;(1� q)3x3

1� q3; : : :

!;

23

F hx; qi =Xn�0

fnhqixn

n!q

:= �F ((1� q)x; (1� q)qx; (1� q)q2x; : : :) (4.24)

= �F

(1� q)x

1� q;(1� q)2x2

1� q2;(1� q)3x3

1� q3; : : :

!;

where

n!q =nYk=1

1� qk

1� q(4.25)

is the q-analogue of n!. 4

D�ecoste has shown that fn(q) and fnhqi are polynomials in q with integral coe�cients, of degreeless or equal to n(n� 1)=2 and that the coe�cients of fn(q) are non negative. Moreover

limq!1 fn(q) = the number of labelled F -structures on [n] ;limq!0 fn(q) = the number of unlabelled F -structures on n \nodes" ;limq!1 fnhqi = the number of labelled F -structures on [n] ;limq!0 fnhqi = the number of unlabelled asymmetric F -structures on n \nodes" :

(4.26)

In the case of a molecular species M , both q-series, M(x; q) and Mhx; qi are monomials in x,and the general expansion Theorems 2.1, 2.2 and 2.3 imply the following:

Theorem 4.2 Let M = M(X) = Xn=H be a molecular species of degree n, with H � Sn. Thenthe canonical q-series M(x; q) and Mhx; qi are monomials in x,

M(x; q) = mn(q)xn

n!q; Mhx; qi = mnhqi

xn

n!q;

whose coe�cients, mn(q) and mnhqi, are given by

mn(q) = (1� q)(1� q2) � � �(1� qn)X

n1+n2+���=n

Xs2M [n]

jAut(s)n1;n2;:::j

n1!n2! � � �qP

i�1(i�1)ni (4.27)

=(1� q)(1� q2) � � �(1� qn)

jH j

XU1+U2+���=[n]

jHU1;U2;:::j qP

i�1(i�1)jUij (4.28)

= (1� q)(1� q2) � � �(1� qn)X

n1+n2+���=n

X�2n1;n2 ;:::

qP

i�1(i�1)ni ; (4.29)

mnhqi = (1� q)(1� q2) � � �(1� qn)X

n1+n2+���=n

Xs2M [n]

jAut(s)n1;n2 ;::: j=1

1

n1!n2! � � �qP

i�1(i�1)ni (4.30)

=(1� q)(1� q2) � � �(1� qn)

jH j

XU1+U2+���=[n]jHU1 ;U2 ;:::

j=1

qP

i�1(i�1)jUij (4.31)

= (1� q)(1� q2) � � �(1� qn)X

n1+n2+���=n

X�2n1;n2 ;:::jAutH �j=1

qP

i�1(i�1)ni : (4.32)

2

24

4.2 Applications to substitution of K-species

Combinatorial addition formulas can also be applied to the analysis and computation of the substi-tution (partitional composition), F �G, of a species G into a species F , with G(0) = 0. For example,if G = A4 = 2X4 + X2E2(X ) + XE3(X ) is the species of 4-element rooted trees and F = C3 isthe species of oriented 3-cycles, then the molecular decomposition of the species F � G = C3 �A4

of \circular hags" consisting of three four-element rooted trees, can be computed as follows. Note�rst that

C3(A4(X )) = C3(2X4+X2E2(X ) +XE3(X ))

= C3(X4 +X4 +X2E2(X ) +XE3(X ))

= C3(X1 +X2 +X3 +X4)���X1 :=X4; X2 :=X4

X3 :=X2E2(X );X4 :=XE3(X )

; (4.33)

then, the addition formula (3.17) for n-cycles with n = 3, k = 4, gives

C3(X1 +X2 +X3 +X4) = C3(X1) + C3(X2) + C3(X3) + C3(X4)

+2X21X2 + 2X1X

22 + 2X2

1X3 + 2X1X23

+2X21X4 + 2X1X

24 + 2X2

2X3 + 2X2X23

+2X22X4 + 2X2X

24 + 2X2

3X4 + 2X3X24

+2X1X2X3 + 2X1X2X4 + 2X1X3X4 + 2X2X3X4 : (4.34)

Finally, making the substitution for the Xi's described in (4.33), and collecting similar terms, weobtain the decomposition

C3(A4(X )) = 2C3(X4) + C3(X

2E2(X )) + C3(XE3(X ))

+ 4X12+ 6X10E2(X ) + 4X8E2(X )2

+ 6X9E3(X ) + 4X6E3(X )2

+ 2X5E2(X )2E3(X ) + 2X4E2(X )E3(X )2

+ 4X7E2(X )E3(X ) : (4.35)

Note that in the above computation, both the variables X1 and X2 have been replaced by the samespecies, X4. Hence, the computations would have been simpler if, instead of (4.34), we had usedthe shorter addition formula

C3(2X1 +X2 +X3) = 2C3(X1) + C3(X2) + C3(X3)

+ 4X31 + 6X2

1X2 + 4X1X22 + 6X2

1X3

+ 4X1X23 + 2X2

2X3 + 2X2X23 + 4X1X2X3 ; (4.36)

together with the substitutions X1 :=X4, X2 :=X

2E2(X ), X3 :=XE3(X ).From this example, it should be clear to the reader that general addition formulas of the form

F (m1X1 +m2X2 +m3X3 + � � �) =XN

cF;N (m1; m2m3; : : :)N(X1; X2; X3; : : :) (4.37)

25

involving \multiplicities" m1; m2; m3; : : :, (mi 2 N) and a �nite or in�nite number of variablesX1; X2; X3; : : :, can be used in a similar way to compute F � G in general, given the moleculardecomposition of G. A fundamental instance of (4.37) arises when F = E, the species of sets. Inthis case, the computation can be done as follows

E(m1X1 +m2X2 +m3X3 + � � �)

= E(m1X1) �E(m2X2) �E(m3X3) � � �

= E(X1)m1 �E(X2)

m2 �E(X3)m3 � � �

= (1 +E+(X1))m1 � (1 + E+(X2))

m2 � (1 + E+(X3))m3 � � �

=X

�1;�2;�3;:::

m1

�1

! m2

�2

! m3

�3

!� � �E+(X1)

�1E+(X2)�2E+(X3)

�3 � � � ; (4.38)

where E+(X) = E(X) � 1 is the species of non empty sets. Now, for any i � 1, the classicalmultinomial formula gives

E+(Xi)�i = (E1(Xi) +E2(Xi) + E3(Xi) + � � �)�i

=X

�i1+�i2+�i3+���=�i

�i!

�i1!�i2!�i3! � � �E1(Xi)

�i1E2(Xi)�i2E3(Xi)

�i3 � � � : (4.39)

Substituting (4.39) in (4.38), we �nally have the following molecular decomposition:

Theorem 4.3 . For the species E = E(X ) of sets and for non negative integers m1; m2; m3; : : :,we have the addition formula

E(m1X1 +m2X2 +m3X3 + � � �) =Xn�0

X�P

i;j�1j�ij=n

p�(m1; m2; m3; : : :)Yi;j�1

E�ijj (Xi) ; (4.40)

where � = (�ij)i;j�1 runs through matrices with non negative integer entries �ij which are 0 foralmost all i; j and the coe�cients p�(m1; m2; m3; : : :) are polynomials in each mi given explicitelyby

p�(m1; m2; m3; : : :) =m

(�11+�12+�13+���)1

�11!�12!�13! � � ��m

(�21+�22+�23+���)2

�21!�22!�23! � � ��m

(�31+�32+�33+���)3

�31!�32!�33! � � �� � � ; (4.41)

with m(k) = m(m� 1)(m� 2) � � �(m� k + 1). 2

As observed by Yeh (see [Yeh (1986)]), this example, combined with the general formula

F (m1X1 +m2X2 + � � �) = F (X1 +X2 + � � �)� E(m1X1 +m2X2 + � � �)

= F (X1 +X2 + � � �)� E(m1X1) �E(m2X2)) � � � ; (4.42)

where \�" denotes the cartesian (superposition) product of species, imply the following fundamen-tal result.

26

Theorem 4.4 [Yeh (1986)] Given any species F = F (X), the coe�cients cF;N (m1; m2; : : :) in themolecular decomposition

F (m1X1 +m2X2 + � � �) =XN

cF;N(m1; m2; : : :)N(X1; X2; : : :) (4.43)

are polynomials in m1; m2; : : :. More precisely, these polynomials are �nite N-linear combinationsof �nite products of binomial coe�cients

�m1�1

��m2�2

�� � �.

An important consequence of this polynomiality result is that the multiplicities mi, i = 1; 2; : : :,which were, up to now, restricted to be non negative integers, can now be replaced by arbitraryelements of any half-rings K for which the notion of binomial coe�cient

�m�

�, m 2 K, � 2 N, have

a meaning. Such half-rings have been called binomial half-rings by Yeh [Yeh (1986)]. Technically,these are half-ring K such that

(a) there exists a Q-algebra L containing K, and

(b) for every m 2 K and � 2 N,�m�

�= m(m� 1) � � �(m� � + 1)=�! 2 K.

For example, N, Z, Q, R, C and Q[i] are binomial half-rings, but Z[i] is not.When K = Z, K-species have been called virtual species by Joyal in [Joyal (1985a), Joyal

(1985b), Joyal (1986)]. A virtual species can also be seen as a formal di�erence � = P � Q ofordinary species P and Q. The molecular decomposition of the virtual species � is de�ned as thecoe�cientwise di�erence of the molecular decompositions of P and Q. Independantly of Yeh, Joyal(in [Joyal (1985a), Joyal (1985b), Joyal (1986)]) found the correct de�nition of the substitution�� of a virtual species � into a virtual species . This de�nition involves the following two steps(i) and (ii):

(i) Given a species F = F (X) and two sorts, X1 and X2, of singletons, then F (X1 � X2) isde�ned by

F (X1 �X2) = F (X1 +X2)��E(X1) �E(X2)

�1�; (4.44)

where E(X2)�1 is de�ned via the geometric series

E(X2)�1 = (1 + E+(X2))

�1 =X��0

(�1)�E+(X2)� : (4.45)

(ii) Given two virtual species � = P � Q and = F � G, where P = P (X), Q = Q(X),F = F (X), G = G(X) are species, then, the virtual species �� = ( ��)(X) is de�ned by

�� = F (P �Q)�G(P � Q) ; (4.46)

where, for any species R = R(X),

R(P � Q) = R(X1 �X2)���X1 :=P; X2 :=Q

: (4.47)

27

Notice that, using Yeh's setting, we can write

F (X1 �X2) = F (m1X1 +m2X2)���m1 := 1; m2 :=�1

= F (X1 +mX2)���m :=�1

; (4.48)

which is legitimate since the coe�cients of the molecular decomposition of F (X1 + mX2) arepolynomials in m.

Another interesting special case of Yeh's method is the \multiplication" formula

F (kX) =XN

cF;N (k)N(X) ; (4.49)

where the coe�cients cF;N(k) are polynomials in k which can be obtained via F (kX) = F (X) �

E(X)k or via F (X1 + � � �+Xk) = �F (X1; : : : ; Xk)���Xi :=X; i=1;:::;k

.

As an illustration of these methods, let us consider the case where F = Cn is the species oforiented n-cycles.

Theorem 4.5 ([Pineau (1995)]) Given a binomial half-ring K and a family m1; m2; : : : of ele-ments of K, we have, for n � 2, the expansion

Cn(m1X1 +m2X2 + � � �) =Xdjn

X 1+ 2+���=d

� (m1; m2; : : :)Cn=d(X 11 X

22 � � �) ; (4.50)

where � (m1; m2; : : :) is polynomial in m1; m2; : : : given by

� (m1; m2; : : :) =1

1 + 2 + � � �

X�j

�(�)

1+ 2+����

1� ;

2� ; : : :

!m 1=�1 m

2=�2 � � � : (4.51)

Proof

The proof follows by the combination of the multiplication formula (4.49) with the additionformula (3.17) for the species Cn. Making the substitution Xi :=X , i = 1; : : : ; k in (3.17) gives(see [Pineau (1995)])

Cn(kX) =Xdjn

�d(k)Cn=d(Xd) ; (4.52)

where �d(k) = 1=dP�jd �(�)k

d=� is the number of Lyndon words of length d in the alphabet Xk =f1; 2; : : : ; kg. Hence,

Cn(m1X1 +m2X2 + � � �) = Cn(X1 +X2 + � � �)���Xi :=miXi;i=1;2;:::

=Xdjn

X 1+ 2+���=d

� Cn=d�(m1X1)

1(m2X2) 2 � � �

�=

Xdjn

X 1+ 2+���=d

� Cn=d

�m 1

1 m 22 X 1

1 X 22 � � �

�; (4.53)

and the result follows by applying (4.52) with k :=m 11 m

22 � � � and X :=X 1

1 X 22 � � �. 2

28

Note that � (m1; m2; : : :) can be interpreted combinatorially as follows: Consider the alphabetXm1;m2;::: which is formed with a �rst group of m1 letters, a second group of m2 letters, etc. It isassumed that each group of letters is totally ordered and that letters of the jth group are smallerthan those of the (j + 1)th. Then � (m1; m2; : : :) is the number of Lyndon words in the alphabetXm1;m2;::: which have 1 occurences of the letters of the �rst group, 2 occurences of the letters ofthe second group, etc.

Summation over n in (4.50) gives the following expansion for the species C = C1 + C2 + � � � ofall oriented cycles:

C(m1X1 +m2X2 + � � �) =X

0< 1+ 2+���<1

� (m1; m2; : : :)C(X 11 X 2

2 � � �) : (4.54)

Formula (4.54) can be applied to compute successive approximations (compare with the exampleof rooted trees, A = XE(A), given in the Introduction) of the species

B = X � (1 + C(B)) (4.55)

of cyclically structured rooted trees, also called mobiles in [Bergeron et al. (1998)].The special case n = 2 in (4.50) corresponds to the following expansion formula for the species

E2 = C2 of 2-element sets,

E2(m1X1 +m2X2 + � � �) =Xi�1

miE2(Xi) +Xi�1

mi

2

!X2i +

Xi>j�1

mimjXiXj ; (4.56)

which is basic for the theory of symmetric square roots initiated by Bouchard et al. (see [Bouchardet al. (1995)]).

Another special case of (4.50) is the following (making use of m1 = 1, m2 = �1).

Theorem 4.6 Let C = C1 + C2 + � � �Cn � � � be the species of oriented cycles. Then

Cn(X � Y ) =Xdjn

Xi+j=d

�i;jCn=d(XiY j) ; (4.57)

C(X � Y ) =Xi+j>0

�i;jC(XiY j) ; (4.58)

where

�i;j = �i;j(1;�1) =1

i+ j

X�j(i;j)

�(�)

d=�

i=�

!(�1)j=� : (4.59)

Note that the numbers �i;j are integers (in Z). In particular, for n = 1; : : : ; 6, we have (withC1(X) = X and C2 = E2) the expansions

C1(X � Y ) = �C1(Y ) + C1(X) ;C2(X � Y ) = �C2(Y ) + C2(X) + C1(Y 2)� C1(XY ) ;C3(X � Y ) = �C3(Y ) + C3(X) + C1(XY

2)� C1(X2Y ) ;

C4(X � Y ) = �C4(Y ) + C4(X) + C2(Y 2)� C2(XY )� C1(XY 3)+ 2C1(X

2Y 2)� C1(X3Y ) ;

C5(X � Y ) = �C5(Y ) + C5(X) + C1(XY 4)� 2C1(X2Y 3) + 2C1(X3Y 2)� C1(X

4Y ) ;C6(X � Y ) = �C6(Y ) + C6(X) + C3(Y 2)� C3(XY ) + C2(XY 2)� C2(X2Y )

� C1(XY 5) + 2C1(X2Y 4)� 3C1(X3Y 3) + 3C1(X4Y 2)� C1(X5Y ) :

(4.60)

29

Apart from enabling us to compute Cn(�) and C(�) for virtual species � = P � Q, formulassuch as (4.57) and (4.58) can also be applied to obtain various algebraic identities. For example

Corollary 4.1 Let x and y be two (commuting) formal variables. Then

1� x+ y =Y

i+j>0

(1� xiyj)��i;j : (4.61)

ProofApplying (4.58) to the species S = E(C) of permutations, we obtain the following successive

combinatorial equalities:

S(X � Y ) = E(C(X � Y ))

= E� Xi+j>0

�i;jC(XiY j)

�=

Yi+j>0

E�C(X iY j)

��i;j=

Yi+j>0

S(X iY j)�i;j : (4.62)

Now, since the exponential generating series of the species S(X) is 1=(1�x), we immediately obtainfrom (4.62)

1

1� (x� y)=Y�

1

1� xiyi

��i;j; (4.63)

which is equivalent to (4.61) 2

4.3 Substitution of species with non-zero constant term

The original de�nition by Joyal [Joyal (1981)] of the substitution F �G of a species G into a speciesF requires G to have no constant term, that is: G(0) = 0. Later, Joyal [Joyal (1986)] dropped thislast condition by de�ning F �G by

(F �G)(X) = F (TG(X))���T :=1

(4.64)

in the case where F is a polynomial species (that is, 9n such that there is no F -structure on anyset U with jU j > n). Another approach to this extended substitution has been taken by [Auger(2000)] and [Auger et al. (2000)] who replaced (4.64) by the equivalent de�nition

(F �G)(X) = F (k +X) �G+(X); (4.65)

where k = G(0) 2 N and G+(X) = G(X)� k has no constant term. The computation of F (k+X)can be done in the following way

F (k +X) = �F (X0; X1; : : : ; Xk)���X0:=X

Xi:=1;i=1;:::;k

(4.66)

= �F (X; 1; : : : ; 1) : (4.67)

30

The substitution X := 0 in (4.66) and (4.67) gives, in particular,

F (k) = �F (0; 1; : : : ; 1) = ZF (k; k; k; : : :) : (4.68)

In the case when F is a molecular species, we have the following expansion formula.

Theorem 4.7 Let M =M(X) be a molecular species of degree n. Then, for any k 2 N, we have

M(k +X) =nX�=0

1

�!

Xs2M [n]

n1+���+nk=n��

j�� Aut(s)�;n1;:::;nk j

n1! � � �nk!

X�

�� Aut(s)�;n1;:::;nk; (4.69)

where, for any subgroup H�;n1;:::;nk � S�;n1;:::;nk ,

��H�;n1 ;:::;nk = fhj[�] j h 2 H�;n1;:::;nkg : (4.70)

ProofTake F =M in formulas (4.66) and (4.67) and use the generalized multinomial expansion (2.8)

with k+1 variables X0; X1; : : : ; Xk. The result then follows from the fact that, if �+n1+� � �+nk = n,then

X�0X

n11 � � �Xnk

k

H�;n1;:::;nk

����� X0:=X

Xi:=1;i=1;:::;k

=X�

��H�;n1;:::;nk

: (4.71)

2

Of course, other expansions forM(k+X), similar to (4.69), can be obtained using the k-coloredor word class expansions (Theorems 2.2 and 2.3). By collecting similar terms in addition formulas

such as (4.69), we can analyse the generalized binomial coe�cients�MN

�kfor molecular species,

introduced by the authors in [Auger (2000)] and [Auger et al. (2000)], de�ned by the equations

M(k +X) =XN

�M

N

�kN(X) ; (4.72)

where N runs through (a system of representatives of) the molecular species of degree less or equal

to degM . Each coe�cient�MN

�kis a polynomial in k.

For instance, in the case M = E�n , the species of oriented n-sets, we obtain the following result.

Theorem 4.8 For n � 2 and k 2 N, we have

E�n (k +X) = a0(k) + a1(k)X +

nX�=2

b�(k)E�� (X) +

n�2X�=2

c�(k)E�(X) ; (4.73)

where

a0(k) =

�k

n

�+

�k

n

�(4.74)

a1(k) =

�k

n� 1

�+

�k

n� 1

�(4.75)

b�(k) =

�k

n � �

�(4.76)

c�(k) =

�k

n� �

��

�k

n � �

�(4.77)

(4.78)

31

with �k

n

�=

k(k + 1) � � �(k + n � 1)

n!(4.79)�

k

n

�=

k(k � 1) � � �(k � n + 1)

n!: (4.80)

ProofWe start with the following version of addition formula (3.12) with k+1 variables X0; X1; : : : ; Xk

E�n (X0+X1+ � � �+Xk) =

Xn0+n1+���+nk=n

(8 i) ni�1

2Xn00 Xn1

1 � � �Xnkk +

Xn0+n1+���+nk=n

(9 i) ni�2

Xn00 Xn1

1 � � �Xnkk

An0;n1 ;:::;nk: (4.81)

Making the substitution X0 := X and Xi := 1, for i = 1; : : : ; k, we have, by Theorem 4.7,

E�n (k+X) =

Xn0+n1+���+nk=n

(8 i) ni�1

2Xn0 +X

n0+n1+���+nk=n(9 i) ni�2

Xn0

�n0An0 ;n1;:::;nk: (4.82)

In the right-hand side of (4.82), the �rst sum splits into two sums and the second sum splits intofour sums as follows:

E�n (k +X) =

XP1

2 +XP2

2X +XQ1

1 +XQ2

X +XQ3

Xn0

�n0An0;n1;:::;nk+XQ4

Xn0

�n0An0;n1;:::;nk; (4.83)

where

P1 = f(n0; n1; : : : ; nk) j n0 + n1 + � � �+ nk = n; n0 = 0; ni � 1; i = 1; : : : ; kg ;

P2 = f(n0; n1; : : : ; nk) j n0 + n1 + � � �+ nk = n; n0 = 1; ni � 1; i = 1; : : : ; kg ;

Q1 = f(n0; n1; : : : ; nk) j n0 + n1 + � � �+ nk = n; n0 = 0; ni = 0 for some i, 1 � i � kg ;

Q2 = f(n0; n1; : : : ; nk) j n0 + n1 + � � �+ nk = n; n0 = 1; ni = 0 for some i, 1 � i � kg ;

Q3 = f(n0; n1; : : : ; nk) j n0 + n1 + � � �+ nk = n; n0 � 2; ni � 1; i = 1; : : : ; kg ;

Q4 = f(n0; n1; : : : ; nk) j n0 + n1 + � � �+ nk = n; n0 � 2; ni � 2 for some i, 1 � i � kg :

The result follows by collecting similar terms in (4.83), making use of the facts that

(n0; n1; : : : ; nk) 2 Q3 ) �n0An0;n1;:::;nk = An0 ; (4.84)

(n0; n1; : : : ; nk) 2 Q4 ) �n0An0;n1;:::;nk = Sn0 ; (4.85)

and by replacing n0 by the simpler summation index �. 2

Formula (4.73) shows that for M = E�n , the generalized binomial coe�cients

�E�nN

�kde�ned

by (4.72) are equal to 0, except possibly when N is of the form E�� (X) and E�(X), 0 � � � n. For

n = 3 and n = 4, formula (4.73) reduces to

E�3 (k +X) = C3(k +X) =

k3 + 2k

2+ k2X + kX2 + C3(X) ; (4.86)

32

and

E�4 (k +X) =

k4 + 11k

12+k3 + 2k

3X +

k(k � 1)

2X2 + kE2(X) + kC3(X) +E�

4 (X) : (4.87)

Formulas such as (4.73), (4.86) and (4.87) can be used to analyse the alternating nth-roots � of Xintroduced by the authors in [Auger et al. (2000)]. These are C -species, � = �(X), satisfying theequation

E�n (�) = X : (4.88)

We conclude with formulas for the two cases of interest Cn(k + X) and Pn(k + X), follo-wing (4.66){(4.68). Hence we introduce the alphabet X1;k = f0; 1; : : : ; kg. We say that a word �in X�1;k has distribution (i; j) if there are i occurences of the letter 0 and j occurences of the otherletters 1; : : : ; k, where i+ j is the length of �. We introduce the following notation:

Xi;j1;k is the set of words in X

�1;k with distribution (i; j);

pali;j(k) is the number of palindromes in Xi;j1;k;

�i;j(k) is the number of Lyndon words in Xi;j1;k;

�i;j(k) is the number of skew primitive classes [[�]] with � 2 Xi;j1;k;

�i;j(k) is the number of palindromic primitive classes [[�]] with � 2 Xi;j1;k;

�i;j(k) is the number of dexterpalindromic primitive classes [[�]] with � 2 Xi;j1;k;

�(1)i;j (k) is the number of dexterpalindromic primitive classes [[�]] with � 2 Xi;j1;k where the letter 0

is a pivot;

�(2)i;j (k) is the number of dexterpalindromic primitive classes [[�]] with � 2 Xi;j1;k where one of the

letters 1; : : : ; k is a pivot.

There are explicit formulas for these numbers. We set n = i + j and write dj(i; j) to denotethat d divides both i and j. Note �rst that �i;j(k) = �i;j(1; k) is a special case of (4.51):

�i;j(k) =1

i+ j

Xdj(i;j)

�(d)

n=d

i=d

!kj=d : (4.89)

Moreover

pali;j(k) =

�bn=2c

bi=2c

�kdj=2e if ij is even, and 0 otherwise; (4.90)

�i;j(k) =1

1 + �(n even)

Xdj(i;j)

�(d) pali=d;j=d(k); (4.91)

�(1)i;j (k) =

1

1 + �(i even)

Xdj(i;j)

�(d) pali=d�1;j=d(k); (4.92)

�(2)i;j (k) =

1

1 + �(j even)

Xdj(i;j)

k�(d) pali=d;j=d�1(k); (4.93)

�i;j(k) =1

1 + �2(i; j)

��(1)i;j (k) + �

(2)i;j (k)

�; (4.94)

33

where �2(i; j) = �(i and j are both odd), and

�i;j(k) =1

2(�i;j(k)� �i;j(k)� �(n even)�i;j(k)) : (4.95)

As an immediate consequence of Theorem 4.5, we obtain the following formula for Cn(k+X) =Cn(X + k).

Theorem 4.9 For n � 2, we have

Cn(k +X) = Cn(k) +Xdjn

Xi+j=di>0

�i;j(k)Cn=d(Xi) ; (4.96)

where

Cn(k) = ZCn(k; k; k; : : :) =1

n

Xdjn

�(d)kn=d : (4.97)

This theorem can be applied to analyse the cyclic nth-roots � of the species X . These areC -species � = �(X) satisfying the equation

Cn(�) = X : (4.98)

It has been conjectured in [Auger et al. (2000)] that, for n � 2, the number of cyclic nth-roots ofX is exactly n� 1.

In order to state the addition formula for Pn(k + X) we introduce the two quotient speciesCm(X

i)�Z2 and Cm(X

i)��Z2 as special cases of (3.23) where there is only one color:

Cm(Xi)�Z2 =

�Cn(X

i1)�1Z2

� ���X1 :=X

(4.99)

andCm(X

i)��Z2 =

�Cn(X

i1)��1Z2

� ���X1 :=X

: (4.100)

We also use the notation �m(i; j) introduced in (3.26), for m = 0, 1 or 2.

Theorem 4.10 for n � 3, we have

Pn(k +X) = Pn(k) +Xdjn

dXi=1

5X�=1

(i;j)�;k (X) (4.101)

where

Pn(k) = ZPn(k; k; k; : : :)

=1

2Cn(k) +

( 12k

(n+1)=2 if n is odd,14

�kn=2 + k(n+2)=2

�if n is even

(4.102)

34

and where, for � = 1; : : : ; 5, the species (i;j)�;k are de�ned as follows, with d = i+ j:

(i;j)1;k = �i;j(k)Cn=d(X

i) (4.103)

(i;j)2;k = �1(i; j)�i;j(k)Cn=d(X

i)�Z2 (4.104)

(i;j)3;k = �2(i; j)�

(1)i;j (k)Cn=d(X

i)�Z2 (4.105)

(i;j)4;k = �0(i; j)

��(2)i;j (k) + �i;j(k)

�Cn=d(X

i)�Z2 (4.106)

(i;j)5;k = �0(i; j)�

(1)i;j (k)Cn=d(X

i)��Z2 : (4.107)

Proof

The result follows immediately from Theorem 3.4. Details are left to the reader. 2

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35