Inequalities for odd ranks of odd Durfee symbols

• Published in: The Ramanujan journal Vol. 56; no. 3; pp. 911 - 929
• Main Author: Liu, Edward Y. S.
• Format: Journal Article
• Language: English
• Published: New York Springer US 2021; Springer Nature B.V
• Subjects: Asymptotic methods; Combinatorial analysis; Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Inequalities; Mathematics; Mathematics and Statistics; Number Theory; Symbols; 11A82; Odd Durfee symbols; Odd ranks; Inequalities; 05A20; Asymptotic formula; 05A17
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-020-00329-z

Andrews introduced odd Durfee symbols to give an interesting combinatorial interpretation of ω ( q ) invoked by MacMahon’s modular partitions, where ω ( q ) is one of the mock theta functions defined by Watson. In analogy with Dyson’s rank, Andrews defined the odd rank of an odd Durfee symbol as the number of entries in the top row minus the number of entries in the bottom row. Let N 0 ( m , n ) be the number of odd Durfee symbols of n with odd rank m . In this paper, we employ Wright’s circle method to give an asymptotic formula for N 0 ( m , n ) which implies that the inequalities N 0 ( m , n ) ≥ N 0 ( m + 2 , n ) and N 0 ( m , n ) ≤ N 0 ( m , n + 2 ) hold for sufficient large n . Motivated by the work of Chan and Mao, we proved that the above inequalities hold for all nonnegative integers m and n .