A new companion to Capparelli's identities

Abstract

Abstract. We discuss a new companion to Capparelli's identities. Capparelli's identities for m = 1, 2 state that the number of partitions of n into distinct parts not congruent to m, −m modulo 6 is equal to the number of partitions of n into distinct parts not equal to m, where the difference between parts is greater than or equal to 4, unless consecutive parts are either both consecutive multiples of 3 or add up to to a multiple of 6. In this paper we show that the set of partitions of n into distinct parts where the odd-indexed parts are not congruent to m modulo 3, the even-indexed parts are not congruent to 3 − m modulo 3, and 3l + 1 and 3l + 2 do not appear together as consecutive parts for any integer l has the same number of elements as the above mentioned Capparelli's partitions of n. In this study we also extend the work of Alladi, Andrews and Gordon by providing a complete set of generating functions for the refined Capparelli partitions, and conjecture some combinatorial inequalities.

Generalizations of Capparelli's identity

“…Note that the interpretation of k in the definition of the D(n; k; i, j) comes from the combinatorial interpretation of (5.1) above. First, the thirteen jagged overpartitions of 13 counted by C(13; k; i, j) are (13), (13), (11,2), (10, 3), (10, 3), (10, 1, 2), (10, 1, 2), (9,4), (9, 4), (7, 6), (7, 4, 2), (7,4,2), (4,5,4), while the thirteen partitions into distinct parts not congruent to 5 modulo 6 counted by D(13; k; i, j) are (13), (12, 1), (10, 3), (10, 2, 1), (9, 4), (9, 3, 1), (8, 4, 1), (8, 3, 2), (7, 6), (7, 4, 2), (7, 3, 2, 1), (6,4,3), (6, 4, 2, 1).…”

“…Let us now give an example for Corollary 1.12. First, the twelve jagged overpartitions of 11 counted by C (11; i, j) are (9, 2), (9, 2), (8, 1, 2), (8, 1, 2), (6, 5), (6, 2, 1, 2), (5,6), (5,6), (5, 1, 2, 1, 2), (4, 5, 2), (4, 4, 1, 2), (2, 1, 2, 1, 2, 1, 2), while the twelve overpartitions counted by D (11; i, j) are (11), (9, 2), (8, 3), (8, 3), (6, 5), (6, 3, 2), (5, 3, 3), (5,4,2), (5,4,2), (4,4,3), (4,4,3), (3,3,3,2).…”

“…It was first proved by Andrews in [2] using q-trinomial coefficients, then it was refined and generalized by Alladi, Andrews and Gordon in [1] using the method of 'weighted words', and then proved by Capparelli in [6] and by Tamba-Xie in [19] using Lie algebraic techniques. For other combinatorial studies related to the Capparelli identity, see [3,4,8,18].…”

Polynomial identities implying Capparelli's partition theorems

“…List of the related sizes and partitions, where 1 and 2 modulo 3 parts are colored differently. n π n π n π n π 16 (11, 5) 28 (17,8,3) 34 (17,12,5) 40 (18,14,8) 19 (14,5) (17, 11) (18,11,5) 43 (17,12,9,5) 22 (14,8) 31 (15,11,5) (17,11,5,1) 40 (17,11,7,5) 49 (18,15,11,5) 28 (14,9,5) (17, 11, 6) (17,12,8,3) 52 (19,17,11,5) One example of this corollary is presented in Table 1.…”

A unifying combinatorial approach to refined little Göllnitz and Capparelli's companion identities

“…As remarked by Berkovich and Uncu [8] the second condition of A m (n) can be replaced with the condition that all parts are distinct and 3l + 1 and 3l + 2 do not appear together as consecutive parts for any integer l ≥ 0. To prove the result, as for Theorem 1.2, they derived a finite version of the above identity using recurrence relations and proved a finite analogue [8, Theorem 2.5] of (1.5).…”