Untrodden pathways in the theory of the restricted partition function p(n, N)

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:180

Article

Dixit, Atul¹  Eyyunni, Pramod^2,3  Maji, Bibekananda^1,4  Sood, Garima^1,5 

[1] Indian Inst Technol Gandhinagar, Discipline Math, Gandhinagar 382355, Gujarat, India

[2] HBNI, Harish Chandra Res Inst, Chhatnag Rd, Allahabad 211019, Uttar Pradesh, India

[3] Indian Inst Sci Educ & Res Berhampur, Ind Training Inst Berhampur, Engn Sch Rd, Berhampur 760010, Odisha, India

[4] Indian Inst Technol Indore, Dept Math, Indore 453552, Madhya Pradesh, India

[5] Univ Wien, Fak Math, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

关键词: Partitions;    Restricted partition function;    q-series;    Finite analogues;    Smallest parts function;    Divisor function;   

DOI  :  10.1016/j.jcta.2021.105423

来源: Elsevier

【 摘 要 】

We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for spt(n). The latter motivates us to extend the theory of the restricted partition function p(n, N), namely, the number of partitions of nwith largest parts less than or equal to N, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for our finite analogue of the spt-function, namely spt(n, N), we prove an inequality between the finite second rank and crank moments. The other results obtained include finite analogues of a recent identity of Garvan, an identity relating d(n, N) and lpt(n, N), namely the finite analogues of the divisor and largest parts functions respectively, and a finite analogue of the Beck-Chern theorem. (C) 2021 Elsevier Inc. All rights reserved.

【 授权许可】

Free