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Bounds for d-distinct partitions

Abstract : 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] .
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https://hal.archives-ouvertes.fr/hal-03208434
Contributor : Srinivas Kotyada Connect in order to contact the contributor
Submitted on : Monday, April 26, 2021 - 3:14:00 PM
Last modification on : Monday, March 28, 2022 - 8:14:08 AM
Long-term archiving on: : Tuesday, July 27, 2021 - 7:14:39 PM

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Soon-Yi Kang, Young Kim. Bounds for d-distinct partitions. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020, pp.83-90. ⟨10.46298/hrj.2021.7430⟩. ⟨hal-03208434⟩

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