10.46298/hrj.2021.7430
https://hrj.episciences.org/7430
Kang, Soon-Yi
Soon-Yi
Kang
Kim, Young,
Young
Kim
Bounds for d-distinct partitions
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] .
episciences.org
asymptotic formulas
partition inequalities
partition identities
d-distinct partitions
partitions
2010 Mathematics Subject Classification. 11P82, 11P84
[MATH]Mathematics [math]
2021-05-06
2021-05-06
2021-05-06
en
journal article
https://hal.archives-ouvertes.fr/hal-03208434v1
2804-7370
https://hrj.episciences.org/7430/pdf
VoR
application/pdf
Hardy-Ramanujan Journal
Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020
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