Bressoud style identities for regular partitions and overpartitions

Abstract

Abstract. We construct a family of partition identities which contain the following identities: Rogers-Ramanujan-Gordon identities, Bressoud's even moduli generalization of them, and their counterparts for overpartitions due to Lovejoy et al. and Chen et al. We obtain unusual companion identities to known theorems as well as to the new ones in the process. The proof is, against tradition, constructive and open to automation.

2-colored Rogers-Ramanujan partition identities

“…Using steps described on [3] and [4], with straightforward but long computations which we skipped here, by…”

“…As an example, all 2-colored Rogers-Ramanujan partitions of 6 are 6, 6, 5 + 1, 5 + 1, 5 + 1, 5 + 1, 4 + 2, 4 + 2, 4 + 2, 4 + 2, 3 + 2 + 1, 3 + 2 + 1 A constructive way from Kurşungöz's papers [3] and [4] applied on the generating functions for these types of partitions to find identities on 2-colored Rogers-Ramanujan partitions.…”

Overpartitions and Bressoud's conjecture, I

“…The conjecture for j = 1 has been recently proved by Kim [27]. The overpartition analogues of classical partition theorems have been received great attention, see for example Chen, Sang and Shi [13][14][15], Choi, Kim and Lovejoy [16], Corteel and Lovejoy [17], Corteel, Lovejoy and Mallet [18], Corteel and Mallet [19], Dousse [20,21], Goyal [24], He, Ji, Wang and Zhao [25], He, Wang and Zhao [26], Kurşungöz [31], Lovejoy [32,33,[35][36][37], Lovejoy and Mallet [38], Padmavathamma and Raghavendra [39], and Sang and Shi [41]. The main objective of this paper is to give an overpartition analogue of Bressoud's conjectured combinatorial theorem, which provides overpartition analogues of many classical partition theorems including Euler's partition theorem, the Rogers-Ramanujan-Gordon theorem and the Andrews-Göllnitz-Gordon theorem.…”