# Ajit SinghUniversity of Virginia | UVa · Department of Mathematics

Ajit Singh

Doctor of Philosophy

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28

Publications

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Introduction

**Skills and Expertise**

## Publications

Publications (28)

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other...

We study "partition Eisenstein series", extensions of the Eisenstein series $G_{2k}(\tau),$ defined by $$\lambda=(1^{m_1}, 2^{m_2},\dots, k^{m_k}) \vdash k \ \ \ \ \ \longmapsto \ \ \ \ \ G_{\lambda}(\tau):= G_2(\tau)^{m_1} G_4(\tau)^{m_2}\cdots G_{2k}(\tau)^{m_k}. $$ For functions $\phi: \mathcal{P}\rightarrow \mathbb{C}$ on partitions, the weight...

In his ``lost notebook'', Ramanujan used iterated derivatives of two theta functions to define sequences of $q$-series $\{U_{2t}(q)\}$ and $\{V_{2t}(q)\}$ that he claimed to be quasimodular. We give the first explicit proof of this claim by expressing them in terms of ``partition Eisenstein series'', extensions of the classical Eisenstein series $E...

In 2010, G.-N. Han obtained the generating function for the number of size t t hooks among integer partitions. Here we obtain these generating functions for self-conjugate partitions, which are particularly elegant for even t t . If n t ( λ ) n_t(\lambda ) is the number of size t t hooks in a partition λ \lambda and S C \mathcal {SC} denotes the se...

A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We emphasize two methods of proofs, one elementary (relying significantly on functional equations) and the other...

We confirm the speculation that the distribution of $t$-hooks among unrestricted integer partitions essentially descends to self-conjugate partitions. Namely, we prove that the number of hooks of length $t$ among the size $n$ self-conjugate partitions is asymptotically normally distributed with mean $\mu_t(n) \sim \frac{\sqrt{6n}}{\pi} + \frac{3}{\...

Andrews and Newman introduced the mex-function \(\text {mex}_{A,a}(\lambda )\) for an integer partition \(\lambda \) of a positive integer n as the smallest positive integer congruent to a modulo A that is not a part of \(\lambda \). They then defined \(p_{A,a}(n)\) to be the number of partitions \(\lambda \) of n satisfying \(\text {mex}_{A,a}(\la...

For a positive integer t≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\ge 2$$\end{document}, let bt(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackag...

Lin introduced the partition function $\text {PDO}_t(n)$ , which counts the total number of tagged parts over all the partitions of n with designated summands in which all parts are odd. Lin also proved some congruences modulo 3 and 9 for $\text {PDO}_t(n)$ , and conjectured certain congruences modulo $3^{k+2}$ for $k\geq 0$ . He proved the conject...

Andrews and Newman introduced the mex-function $\text{mex}_{A,a}(\lambda)$ for an integer partition $\lambda$ of a positive integer $n$ as the smallest positive integer congruent to $a$ modulo $A$ that is not a part of $\lambda$. They then defined $p_{A,a}(n)$ to be the number of partitions $\lambda$ of $n$ satisfying $\text{mex}_{A,a}(\lambda)\equ...

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered new infinite families of Ramanujan type congruences modulo 2 for $b_{21}(n)$ involving every prime $p$ with $p\equiv 13...

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-simi...

For a positive integer \(\ell \), let \(b_{\ell }(n)\) denote the number of \(\ell \)-regular partitions of a nonnegative integer n. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo 2 for \(b_3(n)\) and \(b_{21}(n)\). We prove a specific case of a conjecture of Keith and Zanello on self...

For a positive integer $t$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. In a recent paper, Keith and Zanello established infinite families of congruences and self-similarity results modulo $2$ for $b_{t}(n)$ for certain values of $t$. Further, they proposed some conjectures on self-similarities of $b_t(n...

Let $p_{\{3, 3\}}(n)$ denote the number of $3$ -regular partitions in three colours. Da Silva and Sellers [‘Arithmetic properties of 3-regular partitions in three colours’, Bull. Aust. Math. Soc. 104 (3) (2021), 415–423] conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$ . We confirm these conjectural congruences...

For a positive integer $\ell$, let $b_{\ell}(n)$ denote the number of $\ell$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_3(n)$ and $b_{21}(n)$. We prove a specific case of a conjecture of Keith and Zanello on self-similari...

Let $p_{\{3, 3\}}(n)$ denote the number of $3$-regular partitions in three colours. In a very recent paper, da Silva and Sellers studied certain arithmetic properties of $p_{\{3, 3\}}(n)$. They further conjectured four Ramanujan-like congruences modulo $5$ satisfied by $p_{\{3, 3\}}(n)$. In this article, we confirm the conjectural congruences of da...

The minimal excludant, or “mex” function, on a set S of positive integers is the least positive integer not in S. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity considerations o...

Andrews introduced the singular overpartition function C‾k,i(n) which counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. In this article, we study the divisibility properties of C‾4k,k(n) and C‾6k,k(n) by arbitrary powers of 2 and 3 for infinite families of k. For an infinite famil...

In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely p t,t (n) and p 2t,t (n). We establish identities connecting the ordinary partition function p(n) to p t,t (n) and p...

Andrews introduced the partition function $\overline {C}_{k, i}(n)$ , called the singular overpartition function, 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 prove that $\overline {C}_{6, 2}(n)$ is almost always divisible by $2^k$ for any positive inte...

Andrews' $(k, i)$-singular overpartition function $\overline{C}_{k, i}(n)$ 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. In recent times, divisibility of $\overline{C}_{3\ell, \ell}(n)$, $\overline{C}_{4\ell, \ell}(n)$ and $\overline{C}_{6\ell, \ell}(n)$ by $2...

In order to give overpartition analogues of Rogers-Ramanujan type theorems for the ordinary partition function, Andrews defined the so-called singular overpartitions. Singular overpartition function C‾k,i(n) counts the number of overpartitions of n in which no part is divisible by k and only parts ≡±i(modk) may be overlined. Andrews also proved two...

The minimal excludant, or "mex" function, on a set $S$ of positive integers is the least positive integer not in $S$. In a recent paper, Andrews and Newman extended the mex-function to integer partitions and found numerous surprising partition identities connected with these functions. Very recently, da Silva and Sellers present parity consideratio...

In a recent paper, Andrews and Newman introduced certain families of partition functions using the minimal excludant or "mex" function. In this article, we study two of the families of functions Andrews and Newman introduced, namely $p_{t,t}(n)$ and $p_{2t,t}(n)$. We establish identities connecting the ordinary partition function $p(n)$ to $p_{t,t}...