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36ch3_p36.txt Mon May 30 09:42:09 2005 1

EXAMPLE (6)===========Maple calculations for vector partitions of 4 and their crank mod 5

==============================================================================> read "FUNCS.txt":> V:=vecptns(4);> m:=matrix(nops(V),3):> for i from 1 to nops(V) do> m[i,1]:=V[i]:> m[i,2]:=vpw(V[i]):> m[i,3]:=vpcrank(V[i]):> od:> #Table of vector partitions of 4 with weight and crank> print(m); [[[], [], [1, 1, 1, 1]] 1 -4] [ ] [ [[], [], [1, 1, 2]] 1 -3] [ ] [ [[], [], [2, 2]] 1 -2] [ ] [ [[], [], [1, 3]] 1 -2] [ ] [ [[], [], [4]] 1 -1] [ ] [ [[], [1], [1, 1, 1]] 1 -2] [ ] [ [[], [1], [1, 2]] 1 -1] [ ] [ [[], [1], [3]] 1 0] [ ] [ [[], [1, 1], [1, 1]] 1 0] [ ] [ [[], [1, 1], [2]] 1 1] [ ] [ [[], [2], [1, 1]] 1 -1] [ ] [ [[], [2], [2]] 1 0] [ ] [ [[], [1, 1, 1], [1]] 1 2] [ ] [ [[], [1, 2], [1]] 1 1] [ ] [ [[], [3], [1]] 1 0] [ ] [[[], [1, 1, 1, 1], []] 1 4] [ ] [ [[], [1, 1, 2], []] 1 3] [ ] [ [[], [2, 2], []] 1 2] [ ] [ [[], [1, 3], []] 1 2] [ ] [ [[], [4], []] 1 1] [ ] [ [[1], [], [1, 1, 1]] -1 -3] [ ] [ [[1], [], [1, 2]] -1 -2] [ ] [ [[1], [], [3]] -1 -1] [ ] [ [[1], [1], [1, 1]] -1 -1] [ ] [ [[1], [1], [2]] -1 0] [ ]

37ch3_p37.txt Mon May 30 09:42:09 2005 1

[ [[1], [1, 1], [1]] -1 1] [ ] [ [[1], [2], [1]] -1 0] [ ] [ [[1], [1, 1, 1], []] -1 3] [ ] [ [[1], [1, 2], []] -1 2] [ ] [ [[1], [3], []] -1 1] [ ] [ [[2], [], [1, 1]] -1 -2] [ ] [ [[2], [], [2]] -1 -1] [ ] [ [[2], [1], [1]] -1 0] [ ] [ [[2], [1, 1], []] -1 2] [ ] [ [[2], [2], []] -1 1] [ ] [ [[1, 2], [], [1]] 1 -1] [ ] [ [[3], [], [1]] -1 -1] [ ] [ [[1, 2], [1], []] 1 1] [ ] [ [[3], [1], []] -1 1] [ ] [ [[1, 3], [], []] 1 0] [ ] [ [[4], [], []] -1 0]

> > #Total weight of vector partitions of 4> > add(vpw(V[i]),i=1..nops(V)); 5> nops(V); 41

> #There are 41 vector partitions of 4 with total weight 5 #which is p(4).

> > > # vector partitions of 4 with crank = 0 mod 5> > V0:=vecptnsC(4,0,5);V0 := [[[], [1], [3]], [[], [1, 1], [1, 1]], [[], [2], [2]], [[], [3], [1]],

[[1], [1], [2]], [[1], [2], [1]], [[2], [1], [1]], [[1, 3], [], []],

[[4], [], []]]

> nops(V0); 9

> add(vpw(V0[i]),i=1..nops(V0)); 1

> > # There are 9 vector partitions of 4 with crank = 0 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 1 mod 5

38ch3_p38.txt Mon May 30 09:42:09 2005 1

> > V1:=vecptnsC(4,1,5);V1 := [[[], [], [1, 1, 1, 1]], [[], [1, 1], [2]], [[], [1, 2], [1]],

[[], [4], []], [[1], [1, 1], [1]], [[1], [3], []], [[2], [2], []],

[[1, 2], [1], []], [[3], [1], []]]

> nops(V1); 9

> add(vpw(V1[i]),i=1..nops(V1)); 1

> > # There are 9 vector partitions of 4 with crank = 1 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 2 mod 5> > V2:=vecptnsC(4,2,5);V2 := [[[], [], [1, 1, 2]], [[], [1, 1, 1], [1]], [[], [2, 2], []],

[[], [1, 3], []], [[1], [], [1, 1, 1]], [[1], [1, 2], []],

[[2], [1, 1], []]]

> nops(V2); 7

> add(vpw(V2[i]),i=1..nops(V2)); 1

> > # There are 7 vector partitions of 4 with crank = 0 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 3 mod 5> > V3:=vecptnsC(4,3,5);V3 := [[[], [], [2, 2]], [[], [], [1, 3]], [[], [1], [1, 1, 1]],

[[], [1, 1, 2], []], [[1], [], [1, 2]], [[1], [1, 1, 1], []],

[[2], [], [1, 1]]]

> nops(V3); 7

> add(vpw(V3[i]),i=1..nops(V3)); 1

> > # There are 7 vector partitions of 4 with crank = 3 mod 5> # and total weight = 1.> > # vector partitions of 4 with crank = 4 mod 5> > V4:=vecptnsC(4,4,5);V4 := [[[], [], [4]], [[], [1], [1, 2]], [[], [2], [1, 1]],

[[], [1, 1, 1, 1], []], [[1], [], [3]], [[1], [1], [1, 1]], [[2], [], [2]],

[[1, 2], [], [1]], [[3], [], [1]]]

39ch3_p39.txt Mon May 30 09:42:09 2005 1

> nops(V4); 9

> add(vpw(V4[i]),i=1..nops(V4)); 1

> > # There are 9 vector partitions of 4 with crank = 4 mod 5> # and total weight = 1.>

==============================================================================

We see that

NV(0,5,4)=NV(1,5,4)=NV(2,5,4)=NV(3,5,4)=NV(4,5,4)=1=p(4)/5

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