Inequalities for the M2-rank modulo 12 of partitions without repeated odd parts

Let N 2 ( a , M ; n ) denote the number of partitions of n without repeated odd parts whose M 2 -rank is congruent to a modulo M . Lovejoy, Osburn and Mao have found formulas for M 2 -rank differences modulo 3, 5, 6, and 10. Recently, Xia and Zhao established generating functions for N 2 ( a , 8 ; n...

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Bibliographic Details
Published in:The Ramanujan journal 2024, Vol.63 (1), p.105-130
Main Authors: Fan, Yan, Liu, Eric H., Xia, Ernest X. W.
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Identities
Inequalities
Mathematics
Mathematics and Statistics
Number Theory
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Summary:Let N 2 ( a , M ; n ) denote the number of partitions of n without repeated odd parts whose M 2 -rank is congruent to a modulo M . Lovejoy, Osburn and Mao have found formulas for M 2 -rank differences modulo 3, 5, 6, and 10. Recently, Xia and Zhao established generating functions for N 2 ( a , 8 ; n ) with 0 ≤ a ≤ 7 . Motivated by their works, we establish generating functions for N 2 ( a , 12 ; n ) with 0 ≤ a ≤ 11 by using some identities for Appell–Lerch sums and theta functions. Based on these generating functions, we prove some inequalities for certain linear combinations of N 2 ( a , 12 ; n ) by utilizing asymptotic formulas of eta quotients due to Chern.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-023-00783-5