Alder-type partition inequality at the general level

A known Alder-type partition inequality of level a, which involves the second Rogers–Ramanujan identity when the level a is 2, states that the number of partitions of n into parts differing by at least d with the smallest part being at least a is greater than or equal to that of partitions of n into parts congruent to ±a(modd+3), excluding the part d+3−a. In this paper, we prove that for all values of d with a finite number of exceptions, an arbitrary level a Alder-type partition inequality holds without requiring the exclusion of the part d+3−a in the latter partition.