Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts

Abstract

AbstractLet pod9(n)$\\text {pod}_{9}(n)$, ped9(n)$\\text {ped}_{9}(n)$, and A¯9(n)$\\overline {A}_{9}(n)$ denote the number of 9-regular partitions of n wherein odd parts are distinct, even parts are distinct, and the number of 9-regular overpartitions of n, respectively. By considering pod9(n)$\\text {pod}_{9}(n)$ from an arithmetic point of view, we establish a number of infinite families of congruences modulo 16 and 32, and some internal congruences modulo small powers of 3. A relation connecting above partition functions in arithmetic progressions is obtained as follows. For any n≥0$n\\geq 0$,\n6pod9(2n+1)=2ped9(2n+3)=3A¯9(n+1).$ 6 \\text {pod}_{9}(2n + 1) =\u20092 \\text {ped}_{9}(2n + 3) =\u20093 \\overline {A}_{9}(n + 1).$