Staircases to Analytic Sum-Sides for Many New Integer Partition Identities of Rogers-Ramanujan Type

Abstract

A. We utilize the technique of staircases and jagged partitions to provide analytic sum-sides to some old and new partition identities of Rogers-Ramanujan type. Firstly, we conjecture a class of new partition identities related to the principally specialized characters of certain level 2 modules for the affine Lie algebra A (2) 9 . Secondly, we provide analytic sum-sides to some earlier conjectures of the authors. Next, we use these analytic sum-sides to discover a number of further generalizations. Lastly, we apply this technique to the well-known Capparelli identities and present analytic sum-sides which we believe to be new. All of the new conjectures presented in this article are supported by a strong mathematical evidence.

Proofs and reductions of various conjectured partition identities of Kanade and Russell

“…The product side generates the partitions where each part is congruent to 1, 4, 6, 8, or 11 modulo 12, and the conjectured identity is then equivalent to the statement that for all n there are an equal number of partitions of both types. The remaining conjectures are similar, and were discussed in detail in [15]. These interpretations are not immediately apparent from the series (1.5), but instead require a careful combinatorial analysis in which a partition satisfying difference conditions is decomposed into a staircase and jagged component.…”

“…In [15], Kanade and Russell gave more identities of this flavor. However, rather than searching based on partition gap conditions, they instead began their search from Bos' formulas for the level 2 principally specialized characters of A (2) 9 , namely χ(Ω(L(Λ 0 + Λ 1 ))) = 1 (q, q 4 , q 6 , q 8 , q 11 ; q 12 ) ∞ , all correspond to known partition identities (this is further explained in [3] and [15]).…”

“…In the present paper we consider the majority of the conjectures from [15]. We state these in an analytic form, as that is the most convenient formulation for our later calculations.…”

“…, and H 9 (1) are respectively identities 1, 2, 3, 4, 4a, 5, 5a, 6, and 6a of [15]. As alluded to above, the conjectures for H 1 (1), H 2 (1), and H 3 (1) arose from the principally specialized characters for A (2) 9 listed in (1.2), (1.3), and (1.4).…”

A Variant of IdentityFinder and Some New Identities of Rogers–Ramanujan–MacMahon Type

“…Some such identities may lie much deeper in the search space, or perhaps even more innovative searching parameters are required. As a testimony to the former, several ideas of this article along with [17] helped us identify conjectures related to certain level 2 modules for A (2) 9 which we presented in our article [19]. As will be clear from our discussion below, these identities lie quite deep in our current search space and hence had to be found by completely different methods.…”