New equalities and inequalities for the ranks and cranks of partitions

Let p(n), N(r, m; n) and C(r, m; n) denote the number of partitions of n, the number of partitions of n with rank congruent to r modulo m and the number of partitions of n with crank congruent to r modulo m, respectively. Applying some properties of Appell-Lerch sums and a universal mock theta function g(x, q), we establish the generating functions for N(a,12;n) and C(a,12;n) with 0 < a < 11. With those generating functions, we obtain some new equalities and inequalities involving p(n), N(a, 12; n) and C(a, 12; n). In particular, we confirm several conjectures due to Aygin and Chan and prove that for n >= 0, N(2, 12; 2n) + N(5, 12; 2n) + N(6, 12; 2n) = C(1, 12; 2n) + C(3, 12; 2n) + C(5, 12; 2n).(c) 2023 Elsevier Inc. All rights reserved.