FUNCTION: partitions[PSCHUR] - number of Schur partitions
CALLING SEQUENCE: PSCHUR(n)
PARAMETERS: n - positive integer
-
SYNOPSIS:
Computes the number of ptns of n in which difference between parts
is at least 3 and such that no two consecutive multiples of 3
occur as parts
EXAMPLES:
> with(qseries):
> with(partitions):
> PSCHURC(12);
6
> GENFUNC:=1+add(PSCHURC(n)*q^n,n=1..40);
40 39 38 37 36 35 34
GENFUNC := 169 q + 153 q + 139 q + 126 q + 114 q + 102 q + 91 q
33 32 31 30 29 28 27 26
+ 82 q + 74 q + 67 q + 60 q + 53 q + 47 q + 42 q + 38 q
25 24 23 22 21 20 19 18
+ 34 q + 30 q + 26 q + 23 q + 20 q + 18 q + 16 q + 14 q
17 16 15 14 13 12 11 10 9
+ 12 q + 10 q + 9 q + 8 q + 7 q + 6 q + 5 q + 4 q + 3 q
8 7 6 5 4 3 2
+ 3 q + 3 q + 2 q + 2 q + q + q + q + q + 1
> prodmake(GENFUNC,q,40);
5 7 11 13 17 19
1/((1 - q) (-q + 1) (-q + 1) (-q + 1) (-q + 1) (-q + 1) (-q + 1)
23 25 29 31 35 37
(-q + 1) (-q + 1) (-q + 1) (-q + 1) (-q + 1) (-q + 1))
> prodmake(GENFUNC,q,40,list);
[-1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0,
-1, 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0]
> seq(PSCHURC(n),n=1..20);
1, 1, 1, 1, 2, 2, 3, 3, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18
DISCUSSION: Number of Schur partitions of 12 is 6. The generating
function looks like a nice product.
SEE ALSO: ptnSCHUR