An analogue of Alder-Andrews Conjecture generalizing the 2nd Rogers-Ramanujan identity

초록

In 1956, Alder conjectured that q(d)(n) - Q(d)(n) >= 0, where q(d)(n) and Q(d)(n) are the number of partitions of n into parts differing by at least d and the number of partitions of n into parts which are congruent to +/- 1 (mod d+3), respectively. It took more than 50 years to complete the proof and the first breakthrough was made by Andrews in 1971, who proved that the conjecture holds for d = 2(r) - 1 (r >= 4). In this paper, we prove two analogous partition inequalities following Andrew's method. One of them generalizes the second Rogers-Ramanujan identity, which is the number of partitions of n into parts differing by at least d with the smallest part at least 2 is greater than or equal to that of partitions of n into parts congruent to +/- 2 (mod d + 3) excluding d + 1 when d = 2(r )- 2 (r >= 2, r not equal 3,4). (C) 2020 Elsevier B.V. All rights reserved.

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