Explanipedia

Hook length inequalities for <i>t</i>-regular partitions in the <i>t</i>-aspect Open

Gurinder Singh, Rupam Barman · 2025

Let $t\geq 2$ and $k\geq 1$ be integers. A t -regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t . Let $b_{t,k}(n)$ denote the number of hooks of length k in all the t -regular parti…

Generalized twisted Edwards curves over finite fields and hypergeometric functions Open

Rupam Barman, Sipra Mairty, Sulakashna · 2024

Let $\mathbb{F}_q$ be a finite field with $q$ elements. For $a,b,c,d,e,f \in \mathbb{F}_q^{\times}$, denote by $C_{a,b,c,d,e,f}$ the family of algebraic curves over $\mathbb{F}_q$ given by the affine equation \begin{align*} C_{a,b,c,d,e,f}…

Recursive Formulas for MacMahon and Ramanujan $q$-series Open

Tewodros Amdeberhan, Rupam Barman, Ajit Singh · 2024

In the present work, we extend current research in a nearly-forgotten but newly revived topic, initiated by P. A. MacMahon, on a generalized notion which relates the divisor sums to the theory of integer partitions and two infinite familie…

Hook length inequalities for $t$-regular partitions in the $t$-aspect Open

Gurinder Singh, Rupam Barman · 2024

Let $t\geq2$ and $k\geq1$ be integers. A $t$-regular partition of a positive integer $n$ is a partition of $n$ such that none of its parts is divisible by $t$. Let $b_{t,k}(n)$ denote the number of hooks of length $k$ in all the $t$-regula…

$p$-Adic quotient sets: linear recurrence sequences with reducible characteristic polynomials Open

Deepa Antony, Rupam Barman · 2024

Let $(x_n)_{n\geq0}$ be a linear recurrence sequence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In 2017, …

Hook length biases in ordinary and $t$-regular partitions Open

Rupam Barman, Gurinder Singh · 2024

In this article, we study hook lengths of ordinary partitions and $t$-regular partitions. We establish hook length biases for the ordinary partitions and motivated by them we find a few interesting hook length biases in $2$-regular partiti…

$p$-Adic hypergeometric functions and certain weight three newforms Open

Sulakashna, Rupam Barman · 2024

For an odd prime $p$ and a positive integer $n$, let ${_n}G_n[\cdots]_p$ denote McCarthy's $p$-adic hypergeometric function. In this article, we prove $p$-adic analogue of certain classical hypergeometric identities and using these identit…

On the $p$-adic valuation of third order linear recurrence sequences Open

Deepa Antony, Rupam Barman · 2024

In a recent paper, Bilu et al. studied a conjecture of Marques and Lengyel on the $p$-adic valuation of the Tribonacci sequence. In this article, we study the $p$-adic valuation of third order linear recurrence sequences by considering a g…

Hypergeometric Functions for Dirichlet Characters and Peisert-Like Graphs on $$\mathbb {Z}_n$$ Open

Anwita Bhowmik, Rupam Barman · 2023

$p$-Adic hypergeometric functions and the trace of Frobenius of elliptic curves Open

Sulakashna, Rupam Barman · 2023

Let $p$ be an odd prime and $q=p^r$, $r\geq 1$. For positive integers $n$, let ${_n}G_n[\cdots]_q$ denote McCarthy's $p$-adic hypergeometric functions. In this article, we prove an identity expressing a ${_4}G_4[\cdots]_q$ hypergeometric f…

Arithmetic properties and asymptotic formulae for $σ_o\text{mex}(n)$ and $σ_e\text{mex}(n)$ Open

Gurinder Singh, Rupam Barman · 2023

The minimal excludant of an integer partition is the least positive integer missing from the partition. Let $σ_o\text{mex}(n)$ (resp., $σ_e\text{mex}(n)$) denote the sum of odd (resp., even) minimal excludants over all the partitions of $n…

On $p$-adic denseness of quotients of values of integral forms Open

Deepa Antony, Rupam Barman · 2023

Given $A\subseteq \mathbb{Z}$, the ratio set or the quotient set of $A$ is defined by $R(A):=\{a/b: a, b\in A, b\neq 0\}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attaine…

Diagonal hypersurfaces and elliptic curves over finite fields and hypergeometric functions Open

Sulakashna, Rupam Barman · 2023

Let $D_λ^{d,k}$ denote the family of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^{d,k}:X_1^d+X_2^d=λdX_1^kx_2^{d-k}, \end{align*} where $d\geq2$, $1\leq k\leq d-1$, and $\gcd(d,k)=1$. Let $\#D^{d,k}…

Congruences for the partition function $\text{PDO}_t(n)$ modulo powers of $2$ and $3$ Open

Gurinder Singh, Rupam Barman · 2023

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. For $k\geq0$, Lin conjectured congruences for $\text…

DIVISIBILITY OF THE PARTITION FUNCTION BY POWERS OF AND Open

Rupam Barman, Gurinder Singh, Ajit Singh · 2023

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 …

Parity distribution and divisibility of Mex-related partition functions Open

Subhrajyoti Bhattacharyya, Rupam Barman, Ajit Singh, Apu Kumar Saha · 2023

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

Certain Diophantine equations and new parity results for $21$-regular partitions Open

Ajit Singh, Gurinder Singh, Rupam Barman · 2023

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 infi…

Cliques of orders three and four in the Paley-type graphs Open

Anwita Bhowmik, Rupam Barman · 2023

Let $n=2^s p_{1}^{α_{1}}\cdots p_{k}^{α_{k}}$, where $s=0$ or $1$, $α_i\geq 1$, and the distinct primes $p_i$ satisfy $p_i\equiv 1\pmod{4}$ for all $i=1, \ldots, k$. Let $\mathbb{Z}_n^\ast$ denote the group of units in the commutative ring…

-ADIC QUOTIENT SETS: LINEAR RECURRENCE SEQUENCES Open

Deepa Antony, Rupam Barman · 2023

Let $(x_n)_{n\geq 0}$ be a linear recurrence of order $k\geq 2$ satisfying $x_n=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$ , where $a_1,\ldots ,a_k,x_0,\ldots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . Sanna [‘…

Certain Diophantine equations and new parity results for 21-regular partitions Open

Ajit Singh, Gurinder Singh, Rupam Barman · 2023

For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a non-negative integer $n$. In a recent paper, Keith and Zanello (2022) investigated the parity of $b_{t}(n)$ when $t\leq 28$. They discovered …

Hypergeometric functions for Dirichlet characters and Peisert-like graphs on $\mathbb{Z}_n$ Open

Anwita Bhowmik, Rupam Barman · 2022

For a prime $p\equiv 3\pmod 4$ and a positive integer $t$, let $q=p^{2t}$. The Peisert graph of order $q$ is the graph with vertex set $\mathbb{F}_q$ such that $ab$ is an edge if $a-b\in\langle g^4\rangle\cup g\langle g^4\rangle$, where $g…

Number of $\mathbb{F}_q$-points on Diagonal hypersurfaces and hypergeometric function Open

Sulakashna, Rupam Barman · 2022

Let $D_λ^d$ denote the family of monomial deformations of diagonal hypersurface over a finite field $\mathbb{F}_q$ given by \begin{align*} D_λ^d: X_1^d+X_2^d+\cdots+X_n^d=λd X_1^{h_1}X_2^{h_2}\cdots X_n^{h_n}, \end{align*} where $d,n\geq2$…

p-adic quotient sets: diagonal forms Open

Deepa Antony, Rupam Barman, Piotr Miska · 2022

Arithmetic properties of certain $t$-regular partitions Open

Rupam Barman, Ajit Singh, Gurinder Singh · 2022

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 $2$ …

$p$-Adic quotient sets: linear recurrence sequences Open

Deepa Antony, Rupam Barman · 2022

Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient …

On denseness of certain direction and generalized direction sets Open

Deepa Antony, Rupam Barman, Jaitra Chattopadhyay · 2022

Direction sets, recently introduced by Leonetti and Sanna, are generalization of ratio sets of subsets of positive integers. In this article, we generalize the notion of direction sets and define {\it $k$-generalized direction sets} and {\…

Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions Open

Anwita Bhowmik, Rupam Barman · 2022

For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ …

Certain transformations and values of p-adic hypergeometric functions Open

Sulakashna, Rupam Barman · 2022

We prove two transformations for the $p$-adic hypergeometric functions which can be described as $p$-adic analogues of a Euler's transformation and a transformation of Clausen. We first evaluate certain character sums, and then relate them…

On a Paley-Type Graph on $${\mathbb {Z}}_n$$ Open

Anwita Bhowmik, Rupam Barman · 2022

p-adic quotient sets: cubic forms Open

Deepa Antony, Rupam Barman · 2022

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