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D. S. Gireesh

Publications and source records attributed to D. S. Gireesh.

7 recordsLinked to original sources
arXiv2026

On an Overpartition Analogue of $SOME(n)$

Recently, Andrews and Dastidar introduced the partition function $SOME(n)$, defined as the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts in the partitions of $n$. They derived its generating function and established some congruences satisfied by \(SOME(n)\). In this paper, we introduce an overpartition analogue of $SOME(n)$, denoted by $\overline{SOME}(n)$, the sum of all the odd parts in the overpartitions of \(n\) minus the sum of all the even parts in the overpartitions of \(n\). We derive the generating function for $\overline{SOME}(n)$ and obtain congruences modulo \(3, \ 5\) and powers of \(2\). Our method is based on classical $q$-series identities and manipulations of infinite products and sums.

math.CO↗
arXiv2025

On $\ell-$regular and $2-$color partition triples modulo powers of $3$

Let $T_\ell(n)$ denote the number of $\ell-$regular partition triples of $n$ and let $p_{\ell, 3}(n)$ enumerates the number of 2--color partition triples of $n$ where one of the colors appear only in parts that are multiples of $\ell$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $T_\ell(n)$ and $p_{\ell, 3}(n)$, where $\ell \geq 1$ and $\equiv 0\pmod{3^k}$, and $\equiv \pm 3^k \pmod{3^{k+1}}$.

math.CO↗
arXiv2025

On 2-color partitions where one of the colors is multiples of $7^k$

In this work, we investigate the arithmetic properties of $p_{1,7^k}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are multiples of $7^k$. By constructing generating functions for $p_{1,7^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type infinite family of congruences modulo powers of $7$.

math.NT↗
arXiv2025

On 2-color partitions where one of the color is multiples of $5^k$

In this work, we investigate the arithmetic properties of $p_{1,5^k}(n)$, which counts 2-color partitions of $n$ where one of the colors appears only in parts that are multiples of $5^k$. By constructing generating functions for $p_{1,5^k}(n)$ across specific arithmetic progressions, we establish a set of Ramanujan-type infinite family of congruences modulo powers of $5$.

math.NT↗
arXiv2025

On $7^k$-regular partitions modulo powers of $7$

In this study, we explore the arithmetic properties of $b_{7^k}(n)$ for any $k\geq1$, which enumerates the partitions of $n$ where no part is divisible by $7^k$. By constructing generating functions for $b_{7^k}(n)$ over specific arithmetic progressions, we establish a collection of Ramanujan-type congruences.

math.NT↗
arXiv2019

On 3 and 9-regular cubic partitions

Let $a_3(n)$ and $a_9(n)$ are 3 and 9-regular cubic partitions of $n$. In this paper, we find the infinite family of congruences modulo powers of 3 for $a_3(n)$ and $a_9(n)$ such as \[a_3\left (3^{2α}n+\frac{3^{2α}-1}{4}\right )\equiv 0 \pmod{3^α}\] and \[a_9\left (3^{α+1}n+3^{α+1}-1\right )\equiv 0 \pmod{3^{α+1}}.\]

math.NT↗