Chiranjit Ray

Indian Institute of Technology Delhi | IIT Delhi · Department of Mathematics

Andrews introduced the partition function \({\overline{C}}_{k, i}(n)\), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts \(\equiv \pm i\pmod {k}\) may be overlined. We study the parity and distribution results for \({\overline{C}}_{k,i}(n),\) where \(k>3\) and \(1\le i \...

Let $pod_{\ell}(n)$ be the number of $\ell$-regular partitions of $n$ with distinct odd parts. In this article, we prove that for any positive integer $k$, the set of non-negative integers $n$ for which $pod_{\ell}(n)\equiv 0 \pmod{p^{k}}$ has density one. We also exhibit several multiplicative identities for $pod_{3}(n)$, $pod_{5}(n)$ and $pod_{7}...

Suppose $j_N(\tau)$ and $j_N^{*}(\tau)$ are the Hauptmoduln of the congruence subgroup $\Gamma_0(N)$ and the Fricke group $\Gamma^{*}_0(N)$, respectively. In [7], the authors predicted that, like Klein's $j$-function, the Fourier coefficients of $j_N(\tau)$ and $j_{N}^{*}(\tau)$ in some arithmetic progression are both even and odd with density $\fr...

Andrews and Newman introduced the minimal excludant or ``$mex$'' function for an integer partition $\pi$ of a positive integer $n$, $mex(\pi)$, as the smallest positive integer that is not a part of $\pi$. They defined $\sigma mex(n)$ to be the sum of $mex(\pi)$ taken over all partitions $\pi$ of $n$. We prove infinite families of congruence and mu...

Using the relationship between Siegel cusp forms of degree 2 and cuspidal automorphic representations of GSp(4,AQ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathr...

Andrews and Newman introduced the minimal excludant or “[Formula: see text]” function for an integer partition [Formula: see text] of a positive integer [Formula: see text], [Formula: see text], as the smallest positive integer that is not a part of [Formula: see text]. They defined [Formula: see text] to be the sum of [Formula: see text] taken ove...

Let \({\overline{A}}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In this paper, we prove that \({\overline{A}}_{\ell }(n)\) is almost always divisible by \(p_i^j\) if \(p_i^{2a_i}\ge \ell \), where j is a fixed positive integer and \(\ell =p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}\) with primes \(p_i>3\). We ob...

Let $pod_{\ell}(n)$ be the number of $\ell$-regular partitions of $n$ with distinct odd parts. In this article, prove that for any positive integer $k$, the set of non-negative integers $n$ for which $pod_{\ell}(n)\equiv 0 \pmod{p^{k}}$ has density one under certain conditions on $\ell$ and $p$. For $p \in \{3,5,7\}$, we also exhibit multiplicative...

International audience The minimal excludant or "mex" function for an integer partition π of a positive integer n, mex(π), is the smallest positive integer that is not a part of π. Andrews and Newman introduced σmex(n) to be the sum of mex(π) taken over all partitions π of n. Ballantine and Merca generalized this combinatorial interpretation to σrm...

Using the relationship between Siegel cusp forms of degree $2$ and cuspidal automorphic representations of $\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}})$, we derive some congruences involving dimensions of spaces of Siegel cusp forms of degree $2$ and the class number of $\mathbb{Q}(\sqrt{-p})$. We also obtain some congruences between the $4$-core parti...

The minimal excludant or ``$mex$'' function for an integer partition $\pi$ of a positive integer $n$, $mex(\pi)$, is the smallest positive integer that is not a part of $\pi$. Andrews and Newman introduced $\sigma mex(n)$ to be the sum of $mex(\pi)$ taken over all partitions $\pi$ of $n$. Ballantine and Merca generalized this combinatorial interpre...

Let $\overline{A}_{\ell}(n)$ be the number of overpartitions of $n$ into parts not divisible by $\ell$. In this paper, we prove that $\overline{A}_{\ell}(n)$ is almost always divisible by $p_i^j$ if $p_i^{2a_i}\geq \ell$, where $j$ is a fixed positive integer and $\ell=p_1^{a_1}p_2^{a_2} \dots p_m^{a_m}$ with primes $p_i>3$. We obtain a Ramanujan-t...

Let b(n) denote the number of cubic partition pairs of n. We affirm a conjecture of Lin by proving that $$\begin{aligned} b(49n+37)\equiv 0 \pmod {49} \end{aligned}$$for all \(n\ge 0\). We also prove two congruences modulo 256 satisfied by \(\overline{b}(n)\), the number of overcubic partition pairs of n. Let \(\overline{a}(n)\) denote the number o...

Recently, Andrews defined a partition function EO(n) which counts the number of partitions of n in which every even part is less than each odd part. He also defined a partition function EO‾(n) which counts the number of partitions of n enumerated by EO(n) in which only the largest even part appears an odd number of times. Andrews proposed to undert...

Andrews introduced the partition function C¯k,i(n), called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. He also proved that C¯3,1(9n+3) and C¯3,1(9n+6) are divisible by 3 for n≥0. Recently Aricheta proved that for an infinite family of k, C¯3k,k(...

Andrews introduced the partition function $\overline{C}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of $n$ in which no part is divisible by $k$ and only parts $\equiv \pm i\pmod{k}$ may be overlined. He also proved that $\overline{C}_{3, 1}(9n+3)$ and $\overline{C}_{3, 1}(9n+6)$ are divisible by $3$ for $n\g...

Recently, Andrews defined a partition function $\mathcal{EO}(n)$ which counts the number of partitions of $n$ in which every even part is less than each odd part. He also defined a partition function $\overline{\mathcal{EO}}(n)$ which counts the number of partitions of $n$ enumerated by $\mathcal{EO}(n)$ in which only the largest even part appears...

Let $b(n)$ denote the number of cubic partition pairs of $n$. We give affirmative answer to a conjecture of Lin, namely, we prove that $$b(49n+37)\equiv 0 \pmod{49}.$$ We also prove two congruences modulo $256$ satisfied by $\overline{b}(n)$, the number of overcubic partition pairs of $n$. Let $\overline{a}(n)$ denote the number of overcubic partit...

Let (Formula presented.) be the number of overpartitions of (Formula presented.) into parts not divisible by (Formula presented.). In this paper, we find infinite families of congruences modulo 4, 8 and 16 for (Formula presented.) and (Formula presented.) for any (Formula presented.). Along the way, we obtain several Ramanujan type congruences for...

Let \(\overline{A}_{\ell }(n)\) be the number of overpartitions of n into parts not divisible by \(\ell \). In a recent paper, Shen calls the overpartitions enumerated by the function \(\overline{A}_{\ell }(n)\) as \(\ell \)-regular overpartitions. In this paper, we find certain congruences for \(\overline{A}_{\ell }(n)\), when \(\ell =4, 8\), and...