episciences.org_7430_1653158788
1653158788
episciences.org
raphael.tournoy+crossrefapi@ccsd.cnrs.fr
episciences.org
HardyRamanujan Journal
28047370
10.46298/journals/hrj
https://hrj.episciences.org
05
06
2021
Volume 43  Special...
Bounds for ddistinct partitions
SoonYi
Kang
Young
Kim
Euler's identity and the RogersRamanujan identities are perhaps the most famous results in the theory of partitions. According to them, 1distinct and 2distinct 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 ddistinct partitions. Instead, there are partition inequalities and mock modularity related with ddistinct partitions. For example, the AlderAndrews Theorem states that the number of ddistinct 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 AlderAndrews Theorem and establish asymptotic lower and upper bounds for the ddistinct partitions. Using the asymptotic relations and data obtained from computation, we propose a conjecture on a partition inequality that gives an upper bound for ddistinct partitions. Specifically, for d ≥ 4, the number of ddistinct 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] .
05
06
2021
7430
https://hal.archivesouvertes.fr/hal03208434v1
10.46298/hrj.2021.7430
https://hrj.episciences.org/7430

https://hrj.episciences.org/7430/pdf