Ramanujan’s Theta Functions and Parity of Parts and Cranks of Partitions

Abstract

In this paper, we explore intricate connections between Ramanujan’s theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.

Proof of a Conjecture of Banerjee and Dastidar on Odd Crank

“…(2.13) Taking all terms of the form q 2n in (2.13), after simplification, D. Tang [5] ∞ n=0 A(2n)q n ≡ 1 (q; q) 2 ∞ ((−q 3 , −q 5 , q 8 ; q 8 ) ∞ − (−q 14 , −q 18 , q 32 ; q 32 ) ∞ + 7q 3 (−q 2 , −q 30 , q 32 ; q 32 ) ∞ ) (mod 16).…”

“…Banerjee and Dastidar [5] verified that Conjecture 1.1 holds for any 1 ≤ n ≤ 2000. By using some q-series techniques, we not only confirm the above congruence modulo 4, but also establish another congruence modulo 8.…”

“…Recently, Banerjee and Dastidar [5] considered some arithmetic properties of c o (n). By means of q-series manipulations, Banerjee and Dastidar [5, (1.10)] proved that for any n ≥ 0, c o (5n + 4) ≡ 0 (mod 10).…”

PED and POD partitions: Combinatorial proofs of recurrence relations

“…These partitions were introduced by Andrews in [3,2]. Because the numbers of partitions of n with parts separated by parity (in different combinations) display modular properties, several authors have studied different aspects of their behavior, see for example [4,5,6,7,8,9,10,11,12,13,14,15,16]. Following the notation in [3], we denote by P wx yz (n) the set of partitions of n with parts of type w satisfying condition x or of type y satisfying condition z and all parts of type w satisfying condition x are larger than all parts of type y satisfying condition z.…”

“…Let n = 6. Then P ed ou (6) = {(6), (5, 1), (4, 2), (4, 1 2 ), (3 2 ), (3, 1 3 ), (2, 1 5 ), (1 7 )}. Thus, p ed ou (6) = 8.…”