A generalization of Sylvester's identity

✍ Scribed by Igor Pak; Alexander Postnikov

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 218 KB
• Volume: 178
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(97)81838-9

✦ Synopsis

We consider a new generalization of Euler's and Sylvester's identities for partitions. Our proof is based on an explicit bijection.

1. Main results

A partition 2 of n is a sequence (21,22 ..... 2/) of positive integers such that )-1 >t 22 >~ ... ~> 2/> 0 and y~ 2/= n. The numbers 2i are called parts of 2. Denote by l(2) the number l of parts in 2.

One of the well-known facts in the theory of partitions is Euler's identity.

Theorem (Euler, 1748). The number of partitions of n with odd parts is equal to the number of partitions of n with distinct parts.

There exist several generalizations of Euler's identity (e.g. see [2,5]). One of them is Sylvester's identity.

By sO(n, k) denote the set of partitions of n into odd parts (repetitions allowed) with exactly k different parts. By ~(n, k) denote the set of partitions 2 = (21 > 22 > ... > 2t) of n such that the sequence (21 -l, 22 -l + 1,..., 2t --1) has exactly k different elements. Let A(n, k) = # sO(n, k) and B(n, k) = # ~(n, k).

Theorem (Sylvester, 1882). A(n, k) = B(n, k).