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

Abstract

In 1956, Alder conjectured that qd(n)−Qd(n)≧0, where qd(n) and Qd(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=2r−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(modd+3) excluding d+1 when d=2r−2 (r≧2, r≠3,4) .