arXiv subjects
Publications and source records attributed to Rinchin Drema.
For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for $Q_{t}^{s}(n)$ for some particular values of $s$ and $t$ by employing $q$-series and theta function identities.
For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2 for $a_{3,5}(n)$. In this paper, we establish families of congruences modulo 2 and 4 for $a_{r,s}(n)$ with $(r,s)\in$ \{(2,5), (2,7), (4,5), (4,9)\}. For example, we show that for all $β\geq 0$ and $n \geq 0,$ we have $$a_{2,5}\Big(4\cdot 5^{2β+1}n+\dfrac{37\cdot5^{2β}-1}{6}\Big)\equiv0 \pmod4. $$