# [seqfan] On the Parity of Stanley's Partition Function -- should a comment go in A097567 or elsewhere?

Jonathan Post jvospost3 at gmail.com
Tue Jun 15 02:37:01 CEST 2010

http://arxiv.org/abs/1006.2450

On the Parity of Stanley's Partition Function
Authors: William Y. C. Chen, Albert J. W. Zhu
(Submitted on 12 Jun 2010)

Abstract: Stanley defined a partition function t(n) as the number
of partitions $\lambda$ such that the number of odd parts of $\lambda$
is congruent to the number of odd parts of the conjugate of modulo 4.
We obtain a closed-form formula for the generating function for the
numbers p(n)-t(n). As a consequence, we see that t(n) has the same
parity as the ordinary partition function p(n) for any n.

Subjects: 	Combinatorics (math.CO); Number Theory (math.NT)
MSC classes: 	05A17
Cite as: 	arXiv:1006.2450v1 [math.CO]
Submission history
From: William Y. C. Chen [view email]
[v1] Sat, 12 Jun 2010 09:55:30 GMT (4kb)

A097567  	 	 T(n,k)= count of partitions p such that Abs(
Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts
the odd elements in p. Related to Stanley's 'f'.
1, 1, 0, 0, 0, 2, 1, 0, 2, 0, 3, 0, 0, 0, 2, 3, 0, 2, 0, 2, 0, 1, 0,
8, 0, 0, 0, 2, 3, 0, 8, 0, 2, 0, 2, 0, 10, 0, 2, 0, 8, 0, 0, 0, 2, 10,
0, 8, 0, 8, 0, 2, 0, 2, 0, 4, 0, 26, 0, 2, 0, 8, 0, 0, 0, 2, 10, 0,
26, 0, 8, 0, 8, 0, 2, 0, 2, 0, 27, 0, 10, 0, 28, 0, 2, 0, 8, 0, 0, 0,
2, 27, 0, 26, 0, 28, 0, 8, 0, 8 (list; graph; listen)

OFFSET
0,6

COMMENT
Table starts {1}, {1,0}, {0,0,2}, {1,0,2,0}, {3,0,0,0,2}, .. where the
odd columns are 0. Row sums are A000041 by definition.

George E. Andrews, On a Partition Function of Richard Stanley.

MATHEMATICA
Table[par=Partitions[n]; Table[Count[par, q_/; Abs[Count[q,
_?OddQ]-Count[TransposePartition[q], _?OddQ]]===k], {k, 0, n}], {n, 0,
16}]

CROSSREFS

Sequence in context: A127476 A140397 A120614 this_sequence A022881
A093201 A067613

Adjacent sequences: A097564 A097565 A097566 this_sequence A097568
A097569 A097570

KEYWORD
easy,nonn

AUTHOR
Wouter Meeussen (wouter.meeussen(AT)pandora.be), Aug 28 2004