Chain Partitions of Normalized Matching...

Chain Partitions of Normalized Matching Posets

Shahriar Shahriari

Department of MathematicsPomona College

Claremont, CaliforniaUSA

IPM 20—Reza 70Tehran, IranMay 16, 2009

Chains & Posets

Chain Partitions in Posets

Definition

P = finite partially ordered set (poset)

A chain in P = a linearly ordered subset of P.i.e., a0, a1, . . ., an is a chain of size n + 1 (and length n) in P if

a0 < a1 < . . . < an.

Question

P a poset.µ = (µ1, µ2, . . . , µk) an integer partition of |P|.i.e., µ1 ≥ µ2 ≥ · · · ≥ µk positive integers with

µ1 + µ2 + · · ·+ µk = |P| .

Can we partition the poset P into k chains with sizes µ1, . . ., µk?

Chains & Posets

Definition

P = finite partially ordered set (poset)A chain in P = a linearly ordered subset of P.

i.e., a0, a1, . . ., an is a chain of size n + 1 (and length n) in P if

a0 < a1 < . . . < an.

Question

µ1 + µ2 + · · ·+ µk = |P| .

Chains & Posets

Definition

P = finite partially ordered set (poset)A chain in P = a linearly ordered subset of P.i.e., a0, a1, . . ., an is a chain of size n + 1 (and length n) in P if

a0 < a1 < . . . < an.

Question

µ1 + µ2 + · · ·+ µk = |P| .

Chains & Posets

Definition

a0 < a1 < . . . < an.

Question

P a poset.

µ = (µ1, µ2, . . . , µk) an integer partition of |P|.i.e., µ1 ≥ µ2 ≥ · · · ≥ µk positive integers with

µ1 + µ2 + · · ·+ µk = |P| .

Chains & Posets

Definition

a0 < a1 < . . . < an.

Question

P a poset.µ = (µ1, µ2, . . . , µk) an integer partition of |P|.

i.e., µ1 ≥ µ2 ≥ · · · ≥ µk positive integers with

µ1 + µ2 + · · ·+ µk = |P| .

Chains & Posets

Definition

a0 < a1 < . . . < an.

Question

µ1 + µ2 + · · ·+ µk = |P| .

Chains & Posets

Definition

a0 < a1 < . . . < an.

Question

µ1 + µ2 + · · ·+ µk = |P| .

Chains & Posets

Example (Boolean Lattice of order 3)

{1, 2}

{1, 3}

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Chains & Posets

The question is too general.

We want to consider a class of posets that is a generalizationof the Boolean Lattices.

To focus on the so-called “Normalized Matching” or LYMposets.

Chains & Posets

rank & rank numbers in graded posets

Definition

P is a graded poset of rank n if all maximal chains in P havelength n.If P graded, then rank of x ∈ P is the length of any maximal chainfrom a minimal element of P to x .In a graded poset, each element has a well-defined rank, and theposet is partitioned into levels with each level consisting ofelements of the same rank.The set of elements of rank k in a graded poset P = the k-th levelof P.The size of the k-th level of P = the kth rank number of P.

Chains & Posets

Definition

P is a graded poset of rank n if all maximal chains in P havelength n.

If P graded, then rank of x ∈ P is the length of any maximal chainfrom a minimal element of P to x .In a graded poset, each element has a well-defined rank, and theposet is partitioned into levels with each level consisting ofelements of the same rank.The set of elements of rank k in a graded poset P = the k-th levelof P.The size of the k-th level of P = the kth rank number of P.

Chains & Posets

Definition

P is a graded poset of rank n if all maximal chains in P havelength n.If P graded, then rank of x ∈ P is the length of any maximal chainfrom a minimal element of P to x .

In a graded poset, each element has a well-defined rank, and theposet is partitioned into levels with each level consisting ofelements of the same rank.The set of elements of rank k in a graded poset P = the k-th levelof P.The size of the k-th level of P = the kth rank number of P.

Chains & Posets

Definition

P is a graded poset of rank n if all maximal chains in P havelength n.If P graded, then rank of x ∈ P is the length of any maximal chainfrom a minimal element of P to x .In a graded poset, each element has a well-defined rank, and theposet is partitioned into levels with each level consisting ofelements of the same rank.

The set of elements of rank k in a graded poset P = the k-th levelof P.The size of the k-th level of P = the kth rank number of P.

Chains & Posets

Definition

P is a graded poset of rank n if all maximal chains in P havelength n.If P graded, then rank of x ∈ P is the length of any maximal chainfrom a minimal element of P to x .In a graded poset, each element has a well-defined rank, and theposet is partitioned into levels with each level consisting ofelements of the same rank.The set of elements of rank k in a graded poset P = the k-th levelof P.

The size of the k-th level of P = the kth rank number of P.

Chains & Posets

Definition

P is a graded poset of rank n if all maximal chains in P havelength n.If P graded, then rank of x ∈ P is the length of any maximal chainfrom a minimal element of P to x .In a graded poset, each element has a well-defined rank, and theposet is partitioned into levels with each level consisting ofelements of the same rank.The set of elements of rank k in a graded poset P = the k-th levelof P.The size of the k-th level of P = the kth rank number of P.

Chains & Posets

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Normalized Matching Posets

Matchings and Marriage

Definition

Let G = (X ,E ,Y ) be a bipartite graph with X ∪ Y = the set ofvertices and E = the set of edges.

For Z ⊆ X , define Γ(Z ) = theset of neighbors of X in Y . i.e.,

Γ(Z ) = {y ∈ Y | ∃ z ∈ Z such that (z , y) ∈ E}.

Theorem (Marriage Theorem)

Let G = (X ,E ,Y ) be a bipartite graph, with |X | ≤ |Y |. Anecessary and sufficient condition for the existence of a matchingfrom X to Y is:

|Γ(Z )| ≥ |Z | , ∀ Z ⊆ X .

Definition

Let G = (X ,E ,Y ) be a bipartite graph with X ∪ Y = the set ofvertices and E = the set of edges. For Z ⊆ X , define Γ(Z ) = theset of neighbors of X in Y .

|Γ(Z )| ≥ |Z | , ∀ Z ⊆ X .

Definition

Let G = (X ,E ,Y ) be a bipartite graph with X ∪ Y = the set ofvertices and E = the set of edges. For Z ⊆ X , define Γ(Z ) = theset of neighbors of X in Y . i.e.,

|Γ(Z )| ≥ |Z | , ∀ Z ⊆ X .

Definition

Let G = (X ,E ,Y ) be a bipartite graph, with |X | ≤ |Y |.

Anecessary and sufficient condition for the existence of a matchingfrom X to Y is:

|Γ(Z )| ≥ |Z | , ∀ Z ⊆ X .

Definition

|Γ(Z )| ≥ |Z | , ∀ Z ⊆ X .

Definition

|Γ(Z )| ≥ |Z | , ∀ Z ⊆ X .

The normalized matching condition

Question

Given a bipartite graph with parts X and Y , what is the maximumpossible constant α such that

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

The inequality should work for Z = X . Assuming no isolatedpoints, we have Γ(X ) = Y , and so:

|Y | ≥ α |X | ⇒ α ≤ |Y | / |X | .

Thus the best we can hope for is:

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Question

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

The inequality should work for Z = X .

Assuming no isolatedpoints, we have Γ(X ) = Y , and so:

|Y | ≥ α |X | ⇒ α ≤ |Y | / |X | .

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Question

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

|Y | ≥ α |X | ⇒ α ≤ |Y | / |X | .

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Question

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

|Y | ≥ α |X |

⇒ α ≤ |Y | / |X | .

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Question

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

|Y | ≥ α |X | ⇒ α ≤ |Y | / |X | .

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Question

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

|Y | ≥ α |X | ⇒ α ≤ |Y | / |X | .

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Question

|Γ(Z )| ≥ α |Z | , ∀ Z ⊆ X .

|Y | ≥ α |X | ⇒ α ≤ |Y | / |X | .

|Γ(Z )||Y |

≥ |Z ||X |

∀Z ⊆ X .

Definition

Definition (Graham & Harper 1969)

P a finite graded poset of rank n.

P is normalized matching(NM) (or has the LYM property) if, for any consecutive levels Xand Y in P and Z ⊆ X , we have

|Z ||X |

≤ |Γ(Z )||Y |

where Γ(Z ) is the set of neighbors of Z in Y .

Notation

r0, r1, . . ., rn positive integers.Then NM(r0, . . . , rn) denotes theset of normalized matching posets with rank #s r0, . . ., rn.

Definition

P a finite graded poset of rank n. P is normalized matching(NM)

(or has the LYM property) if, for any consecutive levels Xand Y in P and Z ⊆ X , we have

|Z ||X |

≤ |Γ(Z )||Y |

Notation

Definition

P a finite graded poset of rank n. P is normalized matching(NM) (or has the LYM property)

if, for any consecutive levels Xand Y in P and Z ⊆ X , we have

|Z ||X |

≤ |Γ(Z )||Y |

Notation

Definition

P a finite graded poset of rank n. P is normalized matching(NM) (or has the LYM property) if, for any consecutive levels Xand Y in P and Z ⊆ X , we have

|Z ||X |

≤ |Γ(Z )||Y |

Notation

Definition

|Z ||X |

≤ |Γ(Z )||Y |

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Definition

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Definition

|Z ||X |

≤ |Γ(Z )||Y |

Notation

Definition

|Z ||X |

≤ |Γ(Z )||Y |

Notation

r0, r1, . . ., rn positive integers.

Then NM(r0, . . . , rn) denotes theset of normalized matching posets with rank #s r0, . . ., rn.

Definition

|Z ||X |

≤ |Γ(Z )||Y |

Notation

r0, r1, . . ., rn positive integers.Then NM(r0, . . . , rn)

denotes theset of normalized matching posets with rank #s r0, . . ., rn.

Definition

|Z ||X |

≤ |Γ(Z )||Y |

Notation

Relation to matching

Remark

Let P ∈ NM(r0, r1).

If r0 = r1, then P is NM iff there is a perfect matchingbetween level 0 and level 1 of P.

If r0 < r1, then P NM is more than the existence of amatching from level 0 into level 1:

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Normalized Matching

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P is NM iff you can make such an assignment for each twoconsecutive levels.

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Theorem

Any regular poset (a poset where all the elements at a given levelhave the same up-degree, and all the elements at the same levelhave the same down-degree) is a normalized matching poset.

Corollary

The Subset Lattices, the Subspace Lattices, and the DivisorLattices are all normalized matching posets.

Examples

Theorem

Corollary

The Subset Lattices,

the Subspace Lattices, and the DivisorLattices are all normalized matching posets.

Examples

Theorem

Corollary

The Subset Lattices, the Subspace Lattices,

and the DivisorLattices are all normalized matching posets.

Examples

Theorem

Corollary

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Theorem

Corollary

anti-chains and k-families

Definition

P a poset

An anti-chain in P is a set of pairwise incomparable elements.The width of P is the size of the largest anti-chain in P.A k-family in P is a subset of P that contains no chains of lengthk.

1-family = anti-chain.

In a graded poset, the union of any k levels is a k-family.

[Dilworth’s Theorem] The width of P is the minimumnumber of chains needed to cover P.

Definition

P a posetAn anti-chain in P is a set of pairwise incomparable elements.

The width of P is the size of the largest anti-chain in P.A k-family in P is a subset of P that contains no chains of lengthk.

Definition

P a posetAn anti-chain in P is a set of pairwise incomparable elements.The width of P is the size of the largest anti-chain in P.

A k-family in P is a subset of P that contains no chains of lengthk.

Definition

P a posetAn anti-chain in P is a set of pairwise incomparable elements.The width of P is the size of the largest anti-chain in P.A k-family in P is a subset of P that contains no chains of lengthk.

Definition

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Equivalent Formulations

Theorem (Kleitman)

P a graded poset with rank numbers r0, . . ., rn.

The Following AreEquivalent:

P ∈ NM(r0, . . . , rn).

For any anti-chain (a set of pair-wise incomparable elements)A ⊆ P, let ai = number of elements of A of rank i . Then

n∑i=0

ri≤ 1 LYM inequality

P has a regular chain cover, i.e., can find a number of (notnecessarily disjoint) maximal chains that cover P and all theelements in the same level are on the same number of chains.

Theorem (Kleitman)

P a graded poset with rank numbers r0, . . ., rn. The Following AreEquivalent:

P ∈ NM(r0, . . . , rn).

n∑i=0

Theorem (Kleitman)

P ∈ NM(r0, . . . , rn).

n∑i=0

Theorem (Kleitman)

P ∈ NM(r0, . . . , rn).

For any anti-chain (a set of pair-wise incomparable elements)A ⊆ P,

let ai = number of elements of A of rank i . Then

n∑i=0

Theorem (Kleitman)

P ∈ NM(r0, . . . , rn).

For any anti-chain (a set of pair-wise incomparable elements)A ⊆ P, let ai = number of elements of A of rank i .

n∑i=0

Theorem (Kleitman)

P ∈ NM(r0, . . . , rn).

n∑i=0

ri≤ 1

LYM inequality

Theorem (Kleitman)

P ∈ NM(r0, . . . , rn).

n∑i=0

Theorem (Kleitman)

P ∈ NM(r0, . . . , rn).

n∑i=0

Nested Chain Decomposition

Definition

Definition (Gansner 1982)

P a finite graded poset. A nested chain decomposition ornesting of P is a partition of P into chains such that for any twochains in the partition, the ranks of elements of one of the chainsis a subset of the ranks of the elements of the other.

P is nested if it has a nesting.

Definition

P a finite graded poset.

A nested chain decomposition ornesting of P is a partition of P into chains such that for any twochains in the partition, the ranks of elements of one of the chainsis a subset of the ranks of the elements of the other.

Definition

P a finite graded poset. A nested chain decomposition

ornesting of P is a partition of P into chains such that for any twochains in the partition, the ranks of elements of one of the chainsis a subset of the ranks of the elements of the other.

Definition

P a finite graded poset. A nested chain decomposition ornesting of P

is a partition of P into chains such that for any twochains in the partition, the ranks of elements of one of the chainsis a subset of the ranks of the elements of the other.

Definition

P a finite graded poset. A nested chain decomposition ornesting of P is a partition of P into chains

such that for any twochains in the partition, the ranks of elements of one of the chainsis a subset of the ranks of the elements of the other.

Definition

P a finite graded poset. A nested chain decomposition ornesting of P is a partition of P into chains such that for any twochains in the partition,

the ranks of elements of one of the chainsis a subset of the ranks of the elements of the other.

Definition

P is nested

if it has a nesting.

Definition

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The NCD partition of |P|

A nested chain decomposition

may or may not exist.

may or may not be unique.

However, if a nested chain decomposition exists, then the numberof chains and the sizes of the chains in such a chain partition areunique, and only depend on the rank numbers of the poset.

Definition

P a finite graded poset of rank n.σ the partition of the integer |P| given by the sizes of thehypothetical nested chain decomposition of P.σ is called the NCD partition of |P|.

Definition

However, if a nested chain decomposition exists, then the numberof chains and the sizes of the chains in such a chain partition areunique,

and only depend on the rank numbers of the poset.

Definition

P a finite graded poset of rank n.

σ the partition of the integer |P| given by the sizes of thehypothetical nested chain decomposition of P.σ is called the NCD partition of |P|.

Definition

P a finite graded poset of rank n.σ the partition of the integer |P| given by the sizes of thehypothetical nested chain decomposition of P.

σ is called the NCD partition of |P|.

Definition

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Majorization or Dominance

Definition

µ = (µi ) and ν = (νi ) two partitions of a positive integer m.Then µ ≤ ν in the dominance (or majorization) order if and onlyif, for all j ,

j∑i=1

µi ≤j∑

Example

5, 5, 4, 3, 3, 3, 3, 2, 2, 2 < 6, 4, 4, 4, 4, 2, 2, 2, 2, 2

4, 3, 1, 1 and 5, 1, 1, 1, 1 are incomparable.

Definition

µ = (µi ) and ν = (νi ) two partitions of a positive integer m.

Then µ ≤ ν in the dominance (or majorization) order if and onlyif, for all j ,

j∑i=1

µi ≤j∑

Example

5, 5, 4, 3, 3, 3, 3, 2, 2, 2 < 6, 4, 4, 4, 4, 2, 2, 2, 2, 2

Definition

µ = (µi ) and ν = (νi ) two partitions of a positive integer m.Then µ ≤ ν in the dominance (or majorization) order

if and onlyif, for all j ,

j∑i=1

µi ≤j∑

Example

5, 5, 4, 3, 3, 3, 3, 2, 2, 2 < 6, 4, 4, 4, 4, 2, 2, 2, 2, 2

Definition

j∑i=1

µi ≤j∑

Example

5, 5, 4, 3, 3, 3, 3, 2, 2, 2 < 6, 4, 4, 4, 4, 2, 2, 2, 2, 2

Definition

j∑i=1

µi ≤j∑

Example

5, 5, 4, 3, 3, 3, 3, 2, 2, 2 < 6, 4, 4, 4, 4, 2, 2, 2, 2, 2

Definition

j∑i=1

µi ≤j∑

Example

5, 5, 4, 3, 3, 3, 3, 2, 2, 2 < 6, 4, 4, 4, 4, 2, 2, 2, 2, 2

Generalized Griggs Conjecture

Generalized Griggs Conjecture (SS)

P be a finite normalized matching poset,σ the NCD partition of |P|,µ = (µ1, . . . , µm) an arbitrary partition of |P|.A partition of the poset P into m chains with sizes µ1, . . . , µm

exists if and only if µ ≤ σ in the dominance order.

The only if direction is straightforward.

P be a finite normalized matching poset,

σ the NCD partition of |P|,µ = (µ1, . . . , µm) an arbitrary partition of |P|.A partition of the poset P into m chains with sizes µ1, . . . , µm

P be a finite normalized matching poset,σ the NCD partition of |P|,

µ = (µ1, . . . , µm) an arbitrary partition of |P|.A partition of the poset P into m chains with sizes µ1, . . . , µm

P be a finite normalized matching poset,σ the NCD partition of |P|,µ = (µ1, . . . , µm) an arbitrary partition of |P|.

A partition of the poset P into m chains with sizes µ1, . . . , µm

exists

if and only if µ ≤ σ in the dominance order.

Example

{1} {2} {3}

{1, 2} {1, 3} {2, 3}

{1, 2, 3}

��

�� PPPP

��

�� PPPP

22, 14

23, 12

3, 22, 1

32, 12

4, 2, 12

3, 2, 13

��

Example

{1} {2} {3}

{1, 2} {1, 3} {2, 3}

{1, 2, 3}

��

�� PPPP

��

�� PPPP

22, 14

23, 12

3, 22, 1

32, 12

4, 2, 12

3, 2, 13

��

The Furedi Partition

The top partition is the NCD partition, the bottom partition isalways 1|P|.

The number of chains in the NCD partition is the minimumnumber needed for a partition of P(size of the largest anti-chain ofP = the width of P).

Definition

Among the partitions dominated by NCD, the minimal partitionwith the minimum number of chains is called the Furedi partition.

The Furedi partition consists of only two chain sizes (twoconsecutive integers). It is an attempt to partition P into as fewchains as possible and with uniform size.

The top partition is the NCD partition, the bottom partition isalways 1|P|.The number of chains in the NCD partition is the minimumnumber needed for a partition of P

(size of the largest anti-chain ofP = the width of P).

Definition

The top partition is the NCD partition, the bottom partition isalways 1|P|.The number of chains in the NCD partition is the minimumnumber needed for a partition of P(size of the largest anti-chain ofP = the width of P).

Definition

Among the partitions dominated by NCD,

the minimal partitionwith the minimum number of chains is called the Furedi partition.

Definition

The Furedi partition consists of only two chain sizes

(twoconsecutive integers). It is an attempt to partition P into as fewchains as possible and with uniform size.

Definition

Example

t t tt t t

t t t t��

��

PPPPPP

��

PPPPPP t��

��

3, 3, 2, 2

4, 2, 2, 2 3, 3, 3, 1

4, 3, 2, 1

��

4, 3, 2, 1

3, 3, 2, 2

3, 3, 3, 14, 2, 2, 2

Example

t t tt t t

t t t t��

��

PPPPPP

��

PPPPPP t��

��

3, 3, 2, 2

4, 2, 2, 2 3, 3, 3, 1

4, 3, 2, 1

��

4, 3, 2, 1

3, 3, 2, 2

3, 3, 3, 14, 2, 2, 2

Example

t t tt t t

t t t t��

��

PPPPPP

��

PPPPPP t

��

3, 3, 2, 2

4, 2, 2, 2 3, 3, 3, 1

4, 3, 2, 1

��

4, 3, 2, 1

3, 3, 2, 2

3, 3, 3, 14, 2, 2, 2

Example

t t tt t t

t t t t��

��

PPPPPP

��

PPPPPP

t��

��

3, 3, 2, 2

4, 2, 2, 2 3, 3, 3, 1

4, 3, 2, 1

��

4, 3, 2, 1

3, 3, 2, 2

3, 3, 3, 1

4, 2, 2, 2

Example

t t tt t t

t t t t��

��

PPPPPP

��

PPPPPP

t��

��

3, 3, 2, 2

4, 2, 2, 2 3, 3, 3, 1

4, 3, 2, 1

��

4, 3, 2, 1

3, 3, 2, 2

3, 3, 3, 1

4, 2, 2, 2

Special Cases: Griggs Nesting Conjecture, Furedi’s Conjecture, Griggs Dominance Conjecture

Special cases of the Generalized Griggs Conjecture are:

Conjecture (Griggs’ Nesting Conjecture 1975)

Every finite normalized matching poset is nested.

Conjecture (Furedi’s Question 1985)

The Boolean lattices can be partitioned into chains according tothe Furedi partition.

Conjecture (Sands 1985 for c = 4, Griggs 1988 for general c)

Fix c ≥ 1. For n sufficiently large, the Boolean Lattices can bepartitioned into chains of length c, except for at most c − 1elements, which also belong to a single chain.

Conjecture (Griggs’ Dominance Conjecture 1988)

The generalized Griggs conjecture is true for the Boolean Lattices.

34 year old Griggs Nesting Conjecture open even for rank 3posets.

Furedi’s Question is open for the Boolean Lattices.

Griggs Dominance conjecture is open for the BooleanLattices.

On Sands and Griggs Conjecture

On Sands and Griggs’ Conjecture:

Problem (Sands, 1985)

Can 2[n] be partitioned into chains of size 4 for sufficiently large n?Given k, can 2[n] be partitioned into chains of size 2k , for n largeenough?

For k = 0, 1, the problem is easy. For k = 1, just take the chainsX ,X ∪ {n} for all X ⊆ [n − 1].

Theorem (Griggs, Grinstead, Yeh, 1987)

2[n] can be partitioned into chains of size 4 if and only if n ≥ 9.

Conjecture (Griggs’ c-conjecture 1988)

Fix c ≥ 1. For n sufficiently large, 2[n] can be partitioned intochains of length c, except for at most c − 1 elements, which alsobelong to a single chain.

Theorem (Lonc 1991)

The Griggs’ c-conjecture is true.

For a given c , Lonc’s proof needs n to be very large. Elzobi &Lonc: for n sufficiently large, 2[n] can be partitioned into chains ofsize c = b1

√log log nc, except possibly c − 1 elements which also

form a chain.According to the Griggs’ dominating conjecture, we should be ableto partition 2[n] as above for c an appropriate constant multiple of√

Theorem (Lonc 1991)

form a chain.

According to the Griggs’ dominating conjecture, we should be ableto partition 2[n] as above for c an appropriate constant multiple of√

Theorem (Lonc 1991)

On Furedi’s Question

On Furedi’s Question:

In the (hypothetical) Furedi partition of 2[n], the number of chainsis

( nbn/2c

), and the sizes of the chains are a(n) and a(n) + 1, where

a(n) = b2n/( nb n

2c)c ∼

√π/2

Theorem (Hsu, Logan, SS., Towse 02 & 03)

2[n] has a partition into( nbn/2c

)chains such that the size of the

shortest chains are at least 12

√n, and, for n > 16, the size of the

longest chains is at most 32

√n log n.

( nbn/2c

a(n) = b2n/( nb n

2c)c ∼

√π/2

√n log n.

( nbn/2c

a(n) = b2n/( nb n

2c)c ∼

√π/2

)chains

such that the size of the

√n log n.

( nbn/2c

a(n) = b2n/( nb n

2c)c ∼

√π/2

and, for n > 16, the size of thelongest chains is at most 3

√n log n.

( nbn/2c

a(n) = b2n/( nb n

2c)c ∼

√π/2

√n log n.

Theorem (Hsu, Logan, S. 06—DM’s IPM Special Issue)

Let P be a rank-unimodal, rank-symmetric normalized matchingposet.

The Generalized Furedi conjecture is true if

P is of rank 2, or

P has rapidly decreasing rank numbers, i.e., if the ratiobetween consecutive rank sizes is at least 2 when they are notboth equal to the maximum possible rank size.

Corollary (Hsu, Logan, SS 06—DM’s IPM Special Issue)

Let P be the poset of subspaces of a finite dimensional vectorspace over a finite field ordered by inclusion. Then there exists apartition of P into chains whose sizes are given by the Furedipartition.

Let P be a rank-unimodal, rank-symmetric normalized matchingposet. The Generalized Furedi conjecture is true if

P is of rank 2, or

P is of rank 2,

P is of rank 2, or

P has rapidly decreasing rank numbers,

i.e., if the ratiobetween consecutive rank sizes is at least 2 when they are notboth equal to the maximum possible rank size.

P is of rank 2, or

The full generalized Griggs Conjecture is known only for two cases:

Theorem (Lonc, Elzobi 99)

The generalized Griggs Conjecture is true for the product of twochains.

Theorem (Pearsall, SS)

The generalized Griggs Conjecture is true for all normalizedmatching posets of rank 2.

Griggs’ Nesting Conjecture

Rank-Unimodal and Rank-Symmetric Posets

On Griggs’ Nesting Conjecture:

Definition

Let r0, r1, . . ., rn be a sequence of positive integers.This sequence is unimodal if there is an index 0 ≤ k ≤ n such that

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

The sequence is symmetric if, for 0 ≤ k ≤ n, rk = rn−k .P a graded poset with rank numbers r0, . . ., rn.If r0, . . . , rn is unimodal then P is called rank-unimodal.If r0, . . . , rn is symmetric then P is called rank-symmetric.

Remark

P rank-unimodal, rank-symmetric finite graded poset.Then a Nested Chain Decomposition for P is called a SymmetricChain Decomposition.

Definition

Let r0, r1, . . ., rn be a sequence of positive integers.This sequence is unimodal

if there is an index 0 ≤ k ≤ n such that

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

The sequence is symmetric if

, for 0 ≤ k ≤ n, rk = rn−k .P a graded poset with rank numbers r0, . . ., rn.If r0, . . . , rn is unimodal then P is called rank-unimodal.If r0, . . . , rn is symmetric then P is called rank-symmetric.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

The sequence is symmetric if, for 0 ≤ k ≤ n, rk = rn−k .

P a graded poset with rank numbers r0, . . ., rn.If r0, . . . , rn is unimodal then P is called rank-unimodal.If r0, . . . , rn is symmetric then P is called rank-symmetric.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

The sequence is symmetric if, for 0 ≤ k ≤ n, rk = rn−k .P a graded poset with rank numbers r0, . . ., rn.

If r0, . . . , rn is unimodal then P is called rank-unimodal.If r0, . . . , rn is symmetric then P is called rank-symmetric.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

The sequence is symmetric if, for 0 ≤ k ≤ n, rk = rn−k .P a graded poset with rank numbers r0, . . ., rn.If r0, . . . , rn is unimodal then P is called rank-unimodal.

If r0, . . . , rn is symmetric then P is called rank-symmetric.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

Remark

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

Remark

P rank-unimodal, rank-symmetric finite graded poset.

Then a Nested Chain Decomposition for P is called a SymmetricChain Decomposition.

Definition

r0 ≤ r1 ≤ · · · ≤ rk and rk ≥ rk+1 ≥ · · · ≥ rn.

Remark

Anderson-Griggs Theorem

Theorem (Anderson 1967, Griggs 1977)

Let r0, r1, . . ., rn be a symmetric and unimodal sequence ofpositive integers.

Then every P ∈ NM(r0, . . . , rn) is nested.i.e., every rank-unimodal, rank-symmetric, normalized matchingposet has a symmetric chain decomposition.

Let r0, r1, . . ., rn be a symmetric and unimodal sequence ofpositive integers. Then every P ∈ NM(r0, . . . , rn) is nested.

i.e., every rank-unimodal, rank-symmetric, normalized matchingposet has a symmetric chain decomposition.

Let r0, r1, . . ., rn be a symmetric and unimodal sequence ofpositive integers. Then every P ∈ NM(r0, . . . , rn) is nested.i.e., every rank-unimodal, rank-symmetric, normalized matchingposet has a symmetric chain decomposition.

k-saturated partitions

Definition

If a partition of a poset P into chains has the following property:

Choose K a maximum size k-family. Then every element in P\K isin a chain that intersects K in exactly k elements.Then this partition is called a k-saturated partition of P.

Theorem

1 If P is NM of rank n, then P is nested if and only if P has achain partition that is k-saturated for all k with 1 ≤ k ≤ n.

2 [Greene & Kleitman 1976] For every positive integer k, everyposet P has a chain partition that is simultaneously k andk + 1 saturated.

Corollary

Every rank 2 normalized matching poset is nested.

Definition

If a partition of a poset P into chains has the following property:Choose K a maximum size k-family. Then every element in P\K isin a chain that intersects K in exactly k elements.

Then this partition is called a k-saturated partition of P.

Theorem

Corollary

Definition

If a partition of a poset P into chains has the following property:Choose K a maximum size k-family. Then every element in P\K isin a chain that intersects K in exactly k elements.Then this partition is called a k-saturated partition of P.

Theorem

Corollary

Definition

Theorem

Corollary

Definition

Theorem

Corollary

Definition

Theorem

Corollary

Griggs’ Nesting Conjecture—rank 3 posets

Standard case

On Griggs’ Nesting Conjecture for rank 3 posets:

Standard case

To prove Griggs Nesting conjecture for rank 3 posets it is enoughto prove that every P ∈ NM(r0, r1, r2, r3) is nested where r0, r1,r2, and r3 are arbitrary positive integers with

r0 = r3 < r1 ≤ r2.

Standard case

r0 = r3 < r1 ≤ r2.

Standard case

r0 = r3 < r1 ≤ r2.

NM(5, 16, r, 5)

Consider P ∈ NM(5, 16, r , 5) with r ≥ 16.

s s s s ss s s s s s s s s s s s s s s ss s s s s s s s s s s s s s s s s s ss s s s s. . .

A nested chain decomposition of P would consist of five chains ofsize 4, 11 chains of size 2, and r − 16 singletons. How hard can itbe?

NM(5, 16, r, 5)

A nested chain decomposition of P would consist of five chains ofsize 4, 11 chains of size 2, and r − 16 singletons.

How hard can itbe?

NM(5, 16, r, 5)

Before our work the only case (from among NM(5, 16, r , 5)) donewas the case r = 16.

Every rank-unimodal, rank-symmetric, normalized matching posethas a symmetric chain decomposition.

Idea of Proof.

Growing chains from the middle

Question

When can we use the Anderson-Griggs strategy

and extend anymatching of the middle two levels to a nesting?

For NM(5, 16, r , 5), we showed that we can do so also forr = 17, 18, and 19.

Theorem (Hsu, Logan, SS 08—DM’s 2nd IPM Special Issue)

Let r0 = r3 < r1 < r2 be positive integers. Then the following areequivalent.

1 For any P ∈ NM(r0, r1, r2, r3), any matching of level 1 intolevel 2 can be extended to a nesting of P.

r2 − r1 ≤ d r2r0e − 1.

Question

When can we use the Anderson-Griggs strategy and extend anymatching of the middle two levels to a nesting?

r2 − r1 ≤ d r2r0e − 1.

Question

r2 − r1 ≤ d r2r0e − 1.

Question

r2 − r1 ≤ d r2r0e − 1.

Question

r2 − r1 ≤ d r2r0e − 1.

Question

r2 − r1 ≤ d r2r0e − 1.

Growing good chains from the middle

For NM(5, 16, r , 5), the above nesting method does not work forr ≥ 20.

But for r = 20 and 21, we have an alternative.

Start with a partition of the poset that is simultaneously 1 and2-saturated(guaranteed by Greene and Kleitman).

We have shown that for r = 20 and 21, the matching between themiddle two levels, given by this partition, can be extended outwardto get a nesting!

For NM(5, 16, r , 5), the above nesting method does not work forr ≥ 20. But for r = 20 and 21, we have an alternative.

Start with a partition of the poset that is simultaneously 1 and2-saturated

(guaranteed by Greene and Kleitman).

Theorem (Escamilla, Nicolae, Salerno, SS, Tirrell, 09+)

Let r0 = r3 < r1 < r2 be positive integers.For 0 ≤ i ≤ r1 − r0 − 1, define

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

Let i0 be the largest integer (with 0 ≤ i0 ≤ r1 − r0 − 1) such thatf (i0) 6= 0.If f (i0) > 0, then every P ∈ NM(r0, r1, r2, r3) is nested.

Corollary (Escamilla, Nicolae, Salerno, SS, Tirrell, 09+)

Let r0 = r3 < r1 < r2 ≤ 11 be positive integers. Then everyP ∈ NM(r0, r1, r2, r3) is nested.

Let r0 = r3 < r1 < r2 be positive integers.

For 0 ≤ i ≤ r1 − r0 − 1, define

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

Let i0 be the largest integer (with 0 ≤ i0 ≤ r1 − r0 − 1) such thatf (i0) 6= 0.

If f (i0) > 0, then every P ∈ NM(r0, r1, r2, r3) is nested.

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

Let r0 = r3 < r1 < r2 ≤ 11 be positive integers.

Then everyP ∈ NM(r0, r1, r2, r3) is nested.

f (i) =

⌈r0(1 + i)

r2 − r0

⌉−

⌊r0i

r1 − r0

Starting with long chains

Question

For an arbitrary poset P ∈ NM(5, 16, r , 5), can we start with anarbitrary set of 5 long chains from level 0 to level 3,

and then findadditional 11 chains from level 1 to level 2 to get a nesting?

We showed that for r ≥ 91, the answer is yes.

1 For any P ∈ NM(r0, r1, r2, r3), every collection of r0 chainsof size 4 from level 0 to level 3 can be completed to a nesting.

r0r1 < r2.

Question

For an arbitrary poset P ∈ NM(5, 16, r , 5), can we start with anarbitrary set of 5 long chains from level 0 to level 3, and then findadditional 11 chains from level 1 to level 2 to get a nesting?

r0r1 < r2.

Question

r0r1 < r2.

Question

r0r1 < r2.

Question

r0r1 < r2.

Question

r0r1 < r2.

Starting with good long chains

We have improved this last result that starting with a particularlygood set of long chains allows you to find a nesting for r ≥ 74.

Let r0 = r3 < r1 < r2 be positive integers. Assumer2 > r0r1 − r0 gcd(r1, r2), Then every P ∈ NM(r0, r1, r2, r3) isnested.

Assumer2 > r0r1 − r0 gcd(r1, r2), Then every P ∈ NM(r0, r1, r2, r3) isnested.

Let r0 = r3 < r1 < r2 be positive integers. Assumer2 > r0r1 − r0 gcd(r1, r2), Then every P ∈ NM(r0, r1, r2, r3) isnested.

So, for P ∈ NM(5, 16, r , 5), we are guaranteed a nesting if r ≤ 21or r ≥ 74.

What about any other cases?We can also produce nestings for r = 32, 48, 64, and 72! In fact,

Theorem

Let r0 = r3 < r1 < r2 be positive integers. Assume one of thefollowing holds

1 [Hsu, Logan, SS 08] r1 | r2,2 [Escamilla, Nicolae, Salerno, SS, Tirrell, 09+] r0 | r1,3 [Escamilla, Nicolae, Salerno, SS, Tirrell, 09+] (r0 + 1) | r1,

Then every P ∈ NM(r0, r1, r2, r3) is nested.

So, for P ∈ NM(5, 16, r , 5), we are guaranteed a nesting if r ≤ 21or r ≥ 74. What about any other cases?

We can also produce nestings for r = 32, 48, 64, and 72! In fact,

Theorem

So, for P ∈ NM(5, 16, r , 5), we are guaranteed a nesting if r ≤ 21or r ≥ 74. What about any other cases?We can also produce nestings for r = 32, 48, 64, and 72! In fact,

Theorem

Assume one of thefollowing holds

Theorem

1 [Hsu, Logan, SS 08] r1 | r2,

2 [Escamilla, Nicolae, Salerno, SS, Tirrell, 09+] r0 | r1,3 [Escamilla, Nicolae, Salerno, SS, Tirrell, 09+] (r0 + 1) | r1,

Theorem

1 [Hsu, Logan, SS 08] r1 | r2,2 [Escamilla, Nicolae, Salerno, SS, Tirrell, 09+] r0 | r1,

3 [Escamilla, Nicolae, Salerno, SS, Tirrell, 09+] (r0 + 1) | r1,Then every P ∈ NM(r0, r1, r2, r3) is nested.

Theorem

Example

Consider P ∈ NM(5, r1, r2, 5).

If r1 = 15, then P is nested for all r2 ≥ 15.If r1 = 16, then P is nested if

16 ≤ r2 ≤ 21, r2 = 32, 48, 64, 72, or r2 ≥ 74.

If r1 = 17, then P is nested if

17 ≤ r2 ≤ 23, r2 = 34, 51, 68, or r2 ≥ 81.

If r1 = 18, then P is nested for all r2 ≥ 18.

Example

Consider P ∈ NM(5, r1, r2, 5).If r1 = 15, then P is nested for all r2 ≥ 15.

16 ≤ r2 ≤ 21, r2 = 32, 48, 64, 72, or r2 ≥ 74.

17 ≤ r2 ≤ 23, r2 = 34, 51, 68, or r2 ≥ 81.

Example

Consider P ∈ NM(5, r1, r2, 5).If r1 = 15, then P is nested for all r2 ≥ 15.If r1 = 16, then P is nested if

16 ≤ r2 ≤ 21, r2 = 32, 48, 64, 72, or r2 ≥ 74.

17 ≤ r2 ≤ 23, r2 = 34, 51, 68, or r2 ≥ 81.

Example

16 ≤ r2 ≤ 21, r2 = 32, 48, 64, 72, or r2 ≥ 74.

17 ≤ r2 ≤ 23, r2 = 34, 51, 68, or r2 ≥ 81.

Example

16 ≤ r2 ≤ 21, r2 = 32, 48, 64, 72, or r2 ≥ 74.

17 ≤ r2 ≤ 23, r2 = 34, 51, 68, or r2 ≥ 81.

Example

Some of the results can be used for posets of larger rank.

For example, we can prove that

P ∈ NM(3, 5, 8, 32, 29, 10, 9, 5, 2) ⇒ P is nested.

Example

Some of the results can be used for posets of larger rank.For example, we can prove that

P ∈ NM(3, 5, 8, 32, 29, 10, 9, 5, 2) ⇒ P is nested.

Example

Some of the results can be used for posets of larger rank.For example, we can prove that

P ∈ NM(3, 5, 8, 32, 29, 10, 9, 5, 2) ⇒ P is nested.

References

A. Pearsall and S. Shahriari. Chain decompositions of normalizedmatching posets of rank 2. Submitted.

E. Escamilla, A. Nicolae, P. Salerno, S. Shahriari, and J. Tirrell. Onnested chain decompositions of normalized matching posets of rank3. To appear in Order.

T. Hsu, M. Logan, and S. Shahriari. Methods for nesting rank 3normalized matching rank-unimodal posets. Discrete Math.,309(3):521–531, February 2009.

T. Hsu, M. Logan, and S. Shahriari. The generalized Furediconjecture holds for finite linear lattices. Discrete Math.,306(23):3140–3144, 2006.

T. Hsu, M. Logan, S. Shahriari, and C. Towse. Partitioning theBoolean lattice into a minimal number of chains of relatively uniformsize. European Journal of Combinatorics, 24:219–228, 2003.

T. Hsu, M. Logan, S. Shahriari, and C. Towse. Partitioning the

Boolean lattice into chains of large minimum size. Journal of

Combinatorial Theory A, 97(1):62–84, 2002.

References

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PARTITIONS OF UNITY Contents 1. Introduction. 1 2. Partitions of