General Colored Partition Identities
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Annals of Combinatorics
General Colored Partition Identities Sun Kim Abstract. Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7, and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees 5 and 11 were combinatorially proved by Sandon and Zanello, and it remains open to find a combinatorial proof of the partition identity of degree 23. In this paper, we prove general colored partition identities with a restriction on the number of parts, which are connected to the partition identities arising from those modular equations. We also provide bijective proofs of these partition identities. In particular, one of these proofs gives bijective proofs of the partition identity of degree 23 for some cases, which also work for the identities of degrees 5 and 11 for the same cases. Mathematics Subject Classification. Primary 05A17; Secondary 11P81. Keywords. Colored partitions, Modular equations, Theta functions.
1. Introduction Farkas and Kra [6,7] established the following identity using the theory of theta functions. (−q; q 2 )∞ (−q 7 ; q 14 )∞ − (q; q 2 )∞ (q 7 ; q 14 )∞ = 2q(−q 2 ; q 2 )∞ (−q 14 ; q 14 )∞ , (1.1)
The research of the author was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant agreement no. 335220 - AQSER This work was partially supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MSIP) (NRF-2016R1A5A1008055). 0123456789().: V,-vol
S. Kim
where |q| < 1 and (a; q)∞ =
∞
(1 − aq n ).
n=0
Throughout this paper, we assume that |q| < 1. From (1.1), they derived an elegant partition theorem as follows. Theorem 1.1. Consider the positive integers such that multiples of 7 occur in two copies. Let A(N ) denote the number of partitions of N into distinct odd parts and let B(N ) denote the number of partitions of N into distinct even parts. Then for N ≥ 1, A(2N + 1) = B(2N ). Here, we assume that multiples of 7 occur in two different colors, and we regard them as distinct parts. As in Theorem 1.1, if we can assign different colors to duplicate parts, then such a partition identity is called a colored partition identity. In this paper, we assume that duplicate parts are of different colors, and we regard them as distinct parts. Hirschhorn [10] gave an elementary q-series proof of (1.1), and in particular, he remarked that the referee of [10] pointed out that (1.1) is equivalent to a modular equation of degree 7 in Ramanujan’s Notebooks [2, Chapter 19, Entry 19 (i)], but actually due to Guetzlaff [9]. Farkas and Kra were the first to connect Ramanujan’s modular equations to partition identities. They actually proved two more identities of the same sort in [6,7], one of which is (−q; q 2 )2∞ (−q 3 ; q
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