Soon-Yi Kang ; Young Kim - Bounds for d-distinct partitions

hrj:7430 - Hardy-Ramanujan Journal, May 6, 2021, Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020 - https://doi.org/10.46298/hrj.2021.7430
Bounds for d-distinct partitionsArticle

Authors: Soon-Yi Kang 1; Young Kim

• 1 Kangwon National University

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

Volume: Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020
Published on: May 6, 2021
Accepted on: May 6, 2021
Submitted on: April 30, 2021
Keywords: asymptotic formulas,partition inequalities,partition identities,d-distinct partitions,partitions,2010 Mathematics Subject Classification. 11P82, 11P84,[MATH]Mathematics [math]