Bounds for d-distinct partitions

Euler's identity and the Rogers-Ramanujan identities are perhaps most famous results in theory of partitions. According to them, 1-distinct 2-distinct partitions n equinumerous with into parts congruent ±1 modulo 4 5, respectively. Furthermore, their generating functions modular up multiplication by rational powers q. For d ≥ 3, however, there is neither same type partition nor modularity for d-distinct Instead, inequalities mock related example, Alder-Andrews Theorem states that number greater than or equal which (mod d+3). In this note, we present recent developments generalizations analogs establish asymptotic lower upper bounds Using relations data obtained from computation, propose a conjecture on inequality gives an bound Specifically, 4, less m), where m ≤ 2dπ^2 / [3 log^2 (d)+6 log d] .