{"docId":7430,"paperId":7430,"url":"https:\/\/hrj.episciences.org\/7430","doi":"10.46298\/hrj.2021.7430","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":438,"name":"Volume 43 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2020"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-03208434","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-03208434v1","dateSubmitted":"2021-04-30 14:29:02","dateAccepted":"2021-05-06 13:49:01","datePublished":"2021-05-06 13:50:24","titles":{"en":"Bounds for d-distinct partitions"},"authors":["Kang, Soon-Yi","Kim, Young, "],"abstracts":{"en":"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 \u00b11 modulo 4 and partitions of n into parts congruent to \u00b11 modulo 5, respectively. Furthermore, their generating functions are modular functions up to multiplication by rational powers of q. For d \u2265 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 \u00b11 (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 \u2265 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 \u00b11 (mod m), where m \u2264 2d\u03c0^2 \/ [3 log^2 (d)+6 log d] ."},"keywords":[{"en":"asymptotic formulas"},{"en":"partition inequalities"},{"en":"partition identities"},{"en":"d-distinct partitions"},{"en":"partitions"},"2010 Mathematics Subject Classification. 11P82, 11P84","[MATH]Mathematics [math]"]}