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Hirakjyoti Das

Publications and source records attributed to Hirakjyoti Das.

10 recordsLinked to original sources
arXiv2026

Proof of a Conjecture of Cui, Gu and Tang on 18-Colored Generalized Frobenius Partitions

Recently, the study of the number of $k$-colored generalized Frobenius partitions, denoted by $cϕ_k(n)$, has witnessed renewed interest. In this paper, we investigate congruence properties of $cϕ_{16}(n)$ and $cϕ_{18}(n)$. Our main result is a proof of the conjecture of Cui, Gu, and Tang \cite{CGT25} that, for all $n\ge0$, $cϕ_{18}(3n+2)\equiv0\pmod{2187}$. The proof uses a $(p,k)$-parametrization together with $q$-series identities and dissections. We also establish congruences for $cϕ_{16}(n)$ modulo $1024$ and $2048$, and for $cϕ_{18}(n)$ modulo $8$ and $81$.

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arXiv2026

Arithmetic Properties modulo powers of $2$ and $3$ for Overpartition $k$-Tuples with Odd Parts

Recently, Drema and N. Saikia (2023) and M. P. Saikia, Sarma, and Sellers (2023) proved several congruences modulo powers of $2$ for overpartition triples with odd parts. In this paper, we study further divisibility properties of overpartition $k$-tuples with odd parts using elementary means as well as properties of modular forms. In particular, we prove several congruences modulo multiples of $3$, and an infinite family of congruences modulo powers of $3$; we also prove some cases of a conjecture of Saikia, Sarma, and Sellers.

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arXiv2026

Arithmetic Properties of Generalized Cubic and Overcubic Partitions

We prove several congruences satisfied by the generalized cubic and generalized overcubic partition functions, recently introduced by Amdeberhan, Sellers, and Singh. We also prove infinite families of congruences modulo powers of $2$ and modulo $12$ satisfied by the generalized overcubic partitions, as well as some density results that they satisfy. We use both elementary $q$-series techniques as well as the theory of modular forms to prove our results.

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arXiv2026

Hook Length Biases in $t$-Core Partitions

Recently, the theory of hook length biases has emerged as a prominent research topic. Led by Ballantine, Burson, Craig, Folsom, and Wen [\textit{Res. Math. Sci.}, 2023], hook length biases are being explored for ordinary partitions, odd versus distinct partitions, self-conjugate versus distinct odd partitions. Lately, Singh and Barman [\textit{J. Number Theory}, 2024] opened the door to hook length biases in $\ell$-regular partitions. In this work, we extend the theory of hook length biases to $t$-core partitions. For example, let $a_{t,k}(n)$ denote the number of hooks of length $k$ in all $t$-core partitions of $n$, then we find that $a_{3,1}(n)\ge a_{3,2}(n) \ge a_{3,4}(n)$ and $a_{4,1}(n)\ge a_{4,3}(n)$ for all $n$. The methods employed in this work are mainly combinatorial.

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arXiv2025

Infinite families of congruences for the second order mock theta function $\mathcal{B}(q)$

The arithmetic properties of the second order mock theta function $\mathcal{B}(q)$, introduced by McIntosh, defined by \begin{equation*} \mathcal{B}(q) := \sum_{n \geq 0} \frac{q^n (-q;q^2)_n}{(q;q^2)_{n+1}} = \sum_{n \geq 0}b(n)q^n, \end{equation*} have been extensively studied. For instance, for all $n\ge0$, Kaur and Rana established congruences such as for all $n\ge0$, \begin{align*} b(12n+10) &\equiv 0 \pmod{36}, \quad b(18n+16) \equiv 0 \pmod{72}, \end{align*} Chen and Mao proved that for all $n\ge0$, \begin{align*} b(4n+1) &\equiv 0 \pmod{2}, \quad b(4n+2) \equiv 0 \pmod{4}, \end{align*} while Mao also showed that for all $n\ge0$, \begin{align*} b(6n+2) &\equiv 0 \pmod{4}, \quad b(6n+4) \equiv 0 \pmod{9}. \end{align*} In this paper, we find new congruences and infinite families of congruences modulo $2, 4, 8, 36, 54, 72$ for the function $\mathcal{B}(q)$. For example, let $p \geq 5$ be a prime, if $\left(\frac{-3}{p}\right)_L = -1$, then for all $n, k \geq 0$ with $p \nmid n$, we have \begin{equation*} b\left( 3p^{2k+1}n + \frac{p^{2k+2}-1}{2} \right) \equiv 0 \pmod{2}. \end{equation*} Let $p \geq 5$ be a prime and $1 \leq \ell \leq p - 1$ such that $\left( \frac{12\ell + 9}{p} \right)_L = -1$. Then for all $n, k \geq 0$, we have \begin{equation*} b\left(6p^{2k+3}n + \frac{3p^{2k+2}(4\ell+3)-1}{2}\right) \equiv 0 \pmod{36}. \end{equation*} Our techniques involve elementary $q$-series and Maple.

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arXiv2022

Witness identities for three Ramanujan congruences

For the unrestricted partition function $p(n)$ for integers $n \geq 0$, it is known that $p(49n + 19) \equiv 0 \pmod{49}$, $p(49n + 33) \equiv 0 \pmod{49}$, and $p(49n + 40) \equiv 0 \pmod{49}$ for all $n \geq 0$. We find witness identities for these Ramanujan congruences.

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arXiv2022

Congruences for $k$-elongated plane partition diamonds

In the eleventh paper in the series on MacMahons partition analysis, Andrews and Paule [1] introduced the $k$ elongated partition diamonds. Recently, they [2] revisited the topic. Let $d_k(n)$ count the partitions obtained by adding the links of the $k$ elongated plane partition diamonds of length $n$. Andrews and Paule [2] obtained several generating functions and congruences for $d_1(n)$, $d_2(n)$, and $d_3(n)$. They also posed some conjectures, among which the most difficult one was recently proved by Smoot [11]. Da Silva, Hirschhorn, and Sellers [5] further found many congruences modulo certain primes for $d_k(n)$ whereas Li and Yee [8] studied the combinatorics of Schmidt type partitions, which can be viewed as partition diamonds. In this article, we give elementary proofs of the remaining conjectures of Andrews and Paule [2], extend some individual congruences found by Andrews and Paule [2] and da Silva, Hirschhorn, and Sellers [5] to their respective families as well as find new families of congruences for $d_k(n)$, present a refinement in an existence result for congruences of $d_k(n)$ found by da Silva, Hirschhorn, and Sellers [5], and prove some new individual as well as a few families of congruences modulo 5, 7, 8, 11, 13, 16, 17, 19, 23, 25, 32, 49, 64 and 128.

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arXiv2021

Matching coefficients in the series expansions of certain $q$-products and their reciprocals

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several of the results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote the Rogers-Ramanujan continued fraction having the well-known $q$-product repesentation $$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty}.$$ If \begin{align*} \sum_{n=0}^{\infty}α(n)q^n=\dfrac{1}{R^5\left(q\right)}=\left(\sum_{n=0}^{\infty}α^{\prime}(n)q^n\right)^{-1},\\ \sum_{n=0}^{\infty}β(n)q^n=\dfrac{R(q)}{R\left(q^{16}\right)}=\left(\sum_{n=0}^{\infty}β^{\prime}(n)q^n\right)^{-1}, \end{align*} then \begin{align*} α(5n+r)&=-α^{\prime}(5n+r-2) \quad r\in\{3,4\},\\ β(10n+r)&=-β^{\prime}(10n+r-6) \quad r\in\{7,9\}. \end{align*}

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arXiv2020

Families of Congruences for Fractional Partition Functions Modulo Powers of Primes

Recently, Chan and Wang (Fractional powers of the generating function for the partition function. Acta Arith. 187(1), 59--80 (2019)) studied the fractional powers of the generating function for the partition function and found several congruences satisfied by the corresponding coefficients. In this paper, we find some new families of congruences modulo powers of primes. We also find analogous results for the coefficients of the fractional powers of the generating function for the 2-color partition function.

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