Dissecting the Stanley Partition Function
| dc.creator | Berkovich, Alexander | |
| dc.creator | Garvan, Frank G. | |
| dc.date | 2004-09-24 | |
| dc.date | 2005-03-02 | |
| dc.date.accessioned | 2026-07-07T05:12:34Z | |
| dc.date.available | 2026-07-07T05:12:34Z | |
| dc.description | Let p(n) denote the number of unrestricted partitions of n. For i=0, 2, let p[i](n) denote the number of partitions pi of n such that O(pi) - O(pi') = i mod 4. Here O(pi) denotes the number of odd parts of the partition pi and pi' is the conjugate of pi. R. Stanley [13], [14] derived an infinite product representation for the generating function of p[0](n)-p[2](n). Recently, Holly Swisher[15] employed the circle method to show that limit[n->oo] p[0](n)/p(n) = 1/2 (i) and that for sufficiently large n 2 p[0](n) > p(n), if n=0,1 mod 4, 2 p[0](n) < p(n), otherwise. (ii) In this paper we study even/odd dissection of the Stanley product, and show how to use it to prove (i) and (ii) with no restriction on n. Moreover, we establish the following new result |p[0](2n) - p[2](2n)| > |p[0](2n+1) - p[2](2n+1)|, n>0. Two proofs of this surprising inequality are given. The first one uses the Gollnitz-Gordon partition theorem. The second one is an immediate corollary of a new partition inequality, which we prove in a combinatorial manner. Our methods are elementary. We use only Jacobi's triple product identity and some naive upper bound estimates. | |
| dc.description | 13 pages, new theorems, examples and Note added, to appear in JCT(A) | |
| dc.identifier | https://arxiv.org/abs/math/0409480 | |
| dc.identifier | http://arxiv.org/abs/math/0409480 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/72618 | |
| dc.subject | Combinatorics | |
| dc.subject | Number Theory | |
| dc.subject | 11P81, 11P82, 11P83, 05A17, 05A19 | |
| dc.title | Dissecting the Stanley Partition Function | |
| dc.type | text |