Some partition and analytical identities arising from the Alladi, Andrews, Gordon bijection
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Some partition and analytical identities arising from the Alladi, Andrews, Gordon bijection S. Capparelli1 · A. Del Fra1 · P. Mercuri2 · A. Vietri1 Received: 6 April 2020 / Accepted: 19 August 2020 © The Author(s) 2020
Abstract In the work of Alladi et al. (J Algebra 174:636–658, 1995) the authors provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of a different nature, we obtain an analytical identity of Rogers–Ramanujan type, involving generating functions, for a class of partition identities already found in that paper and that generalize the first Capparelli identity and include it as a particular case. To achieve this, we apply the same strategy as Kanade and Russell did in a recent paper. This method relies on the use of jagged partitions that can be seen as a more general kind of integer partitions. Keywords Partition identity · Rogers–Ramanujan identity · Jagged partition · Analytical identity Mathematics Subject Classification Primary 11P84; Secondary 05A17 · 11P82 · 11P83
1 Introduction In a 1969 paper, [2], Andrews characterized the type of partition sets that could be set into a bijection using Euler’s classical trick to show that partitions of n into distinct
P. Mercuri is supported by the research grant “Ing. Giorgio Schirillo” of the Istituto Nazionale di Alta Matematica “F. Severi”, Rome.
B
A. Vietri [email protected]
1
Università degli Studi di Roma La Sapienza, Rome, Italy
2
Università degli Studi di Roma “Tor Vergata”, Rome, Italy
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parts are as many as partitions of n into odd parts. In particular, Andrews proved that an identity of Schur ([12]) and one of Göllnitz ([10]) provide examples of “Euler-pairs”. Inspired by that paper, here we look at one of the identities given in [6], see also [8]. We study the bijection provided by Alladi et al. in [1] and we find new sets of partitions that can be set into a bijection using the same approach. For further details on this subject and some generalizations we refer the reader to [3–5,7,9]. As the starting point of our research, we consider the partition identity which was proved in [1] and in particular the Concluding Remarks (Section 7), according to which the first Capparelli’s identity (see [7]) can be generalized from modulo 3 to modulo t by means of suitable dilations. In Sect. 2 we find an analytical identity for the partition identity modulo t. This is done using the same method as in [11] to compute the generating functions of the sum side. In Sect. 3 we again look back at [1], this time by generalizing the machinery which led the authors to build up the partition identity. In particular, we study a different class of partition
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