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Arithmetic Properties of 9-Regular Partitions with Distinct Odd Parts
B. Hemanthkumar
,
H. S. Sumanth Bharadwaj
,
M. S. Mahadeva Naika
Abstract
Let $\text{pod}_{9}(n)$, $\text{ped}_{9}(n)$, and $\overline{A}_{9}(n)$ 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 $\text{pod}_{9}(n)$ 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 $n\geq 0$, $ 6 \text{pod}_{9}(2n + 1) = 2 \text{ped}_{9}(2n + 3) = 3 \overline{A}_{9}(n + 1).$