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Pankaj Jyoti Mahanta

Publications and source records attributed to Pankaj Jyoti Mahanta.

14 recordsLinked to original sources
arXiv2026

A Proof of a Conjecture on Fixed Perimeter Partitions

Finding fixed perimeter analogues of various partition theoretic identities and inequalities has recently emerged as an active area of research. Gray, Payne, Swisher, and Watson [\textit{Discrete Math.}, 2026] established several fixed perimeter analogues of partition theoretic results inspired by Euler's celebrated partition identity. Very recently, in a separate work [\textit{ar{X}iv:2608.00421}, 2026], they explored fixed perimeter analogues of inequalities related to parity biases. Introducing the concept of parity bias, Kim, Kim, and Lovejoy [\textit{Eur. J. Comb.}, 2020] conjectured that $pd_o(n)>pd_e(n)$ for all $n\ge 20$, where $pd_o(n)$ (respectively, $pd_e(n)$) denote the number of partitions of $n$ into distinct parts having more odd parts (respectively, even parts) than even parts (respectively, odd parts). The author, together with Banerjee, Bhattacharjee, Dastidar, and Saikia [\textit{Eur. J. Comb.}, 2022], proved this conjecture. Gray, Payne, Swisher, and Watson conjectured that a fixed perimeter analogue of this inequality holds for all $n\ge 9$. In this paper, we confirm their conjecture.

math.CO↗
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.

math.CO↗
arXiv2026

Combinatorial proofs of some identities on overpartitions with repeated smallest non-overlined part

Let $\overline{\mathrm{spt}}k(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part, say $s(π)$, appears $k$ times and every overlined part is bigger than $s(π)$. Let $\overline{\mathrm{spt}}k_o(n)$ denote the number of overpartitions of $n$ where the smallest non-overlined part appears $k$ times, every overlined part is bigger than $s(π)$ and all parts other than $s(π)$ are incongruent modulo $2$ with $s(π)$. Also, let $b_e(k,n)$ (resp., $b_o(k,n)$) denote the number of overpartitions of $n$ counted by $\overline{\mathrm{spt}}k_o(n)$ where the number of parts greater than $s(π)$ is even (resp., odd), and let $$\overline{\mathrm{spt}}k_o'(n)=b_e(k,n)-b_o(k,n).$$ Recently, Malik and Sarma (arXiv:2601.15601v1) expressed the generating functions of these partition functions in terms of linear combinations of $q$-series with polynomials in $q$ as coefficients. As corollaries, they derived some partition identities involving the functions for $k=1$ and sought for combinatorial proofs of their results. In this paper, we present some desired proofs.

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arXiv2026

Biases in Non-Unitary Partitions

Recently, the concept of parity bias in integer partitions has been studied by several authors. We continue this study here, but for non-unitary partitions (namely, partitions with parts greater than $1$). We prove analogous results for these restricted partitions to those that have been obtained by Kim, Kim, and Lovejoy (2020) and Kim and Kim (2021). We also look at inequalities between two classes of partitions studied by Andrews (2019), where the parts are separated by parity (either all odd parts are smaller than all even parts or vice versa).

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arXiv2025

On hook length biases in $t$-regular partitions

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 $t$-regular partitions of $n$. Recently, the first and the third authors proved that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n\geq 4$, and conjectured that $b_{t+1,2}(n)\geq b_{t,2}(n)$ for all $t\geq 3$ and $n\geq 0$. In this paper, we prove that the conjecture is true for $t=3$.

math.CO↗
arXiv2024

Combinatorial Proofs of Some Results of Andrews and El Bachraoui

Recently, Andrews and El Bachraoui (2024) proved three very interesting $q$-series identities, from which three simple looking identities involving certain restricted partitions into distinct even parts and $4$-regular partitions follow. In this short note, we give combinatorial proofs of these identities. We also prove the counterpart identities for the restricted partitions into distinct odd parts.

math.CO↗
arXiv2024

Extensions of some results of Jovovic and Dhar

We look at extensions of formulas given by Jovovic and recently proved by Dhar on integer partitions where the smallest part occurs at least $m$ times and on integer partitions with fixed differences between the largest and smallest parts where the smallest part occurs at least $k$ times. Our results extend Dhar's results for the $m=2$ and $k=1$ cases to the general cases for arbitrary $m$ and $k$. We also look at analogous results for overpartitions and $\ell$-regular partitions.

math.CO↗
arXiv2022

Parity biases in partitions and restricted partitions

Let $p_{o}(n)$ (resp. $p_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts). Recently, Kim, Kim, and Lovejoy proved that $p_{o}(n)>p_{e}(n)$ for all $n>2$ and conjectured that $d_{o}(n)>d_{e}(n)$ for all $n>19$ where $d_{o}(n)$ (resp. $d_{e}(n)$) denote the number of partitions into distinct parts having more odd parts (resp. even parts) than even parts (resp. odd parts). In this paper we provide combinatorial proofs for both the result and the conjecture of Kim, Kim and Lovejoy. In addition, we show that if we restrict the smallest part of the partition to be $2$, then the parity bias is reversed. That is, if $q_{o}(n)$ (resp. $q_{e}(n)$) denote the number of partitions of $n$ with more odd parts (resp. even parts) than even parts (resp. odd parts) where the smallest part is at least $2$, then we have $q_o(n) 7$. We also look at some more parity biases in partitions with restricted parts.

math.CO↗
arXiv2021

Refinement of some partition identities of Merca and Yee

Recently, Merca and Yee proved some partition identities involving two new partition statistics. We refine these statistics and generalize the results of Merca and Yee. We also correct a small mistake in a result of Merca and Yee.

math.CO↗
arXiv2021

Partial sums of the Gibonacci sequence

Recently, Chu studied some properties of the partial sums of the sequence $P^k(F_n)$, where $P(F_n)=\big(\sum_{i=1}^nF_i\big)_{n\geq1}$ and $(F_n)_{n\geq1}$ is the Fibonacci sequence, and gave its combinatorial interpretation. We generalize those results, introduce colored Schreier sets, and give another equivalent combinatorial interpretation by means of lattice path.

math.CO↗
arXiv2020

Some New and Old Gibonacci Identities

We present a different combinatorial interpretations of Lucas and Gibonacci numbers. Using these interpretations we prove several new identities, and simplify the proofs of several known identities. Some open problems are discussed towards the end of the paper.

math.CO↗
arXiv2020

Some properties of Zumkeller numbers and $k$-layered numbers

Generalizing the concept of a perfect number is a Zumkeller or integer perfect number that was introduced by Zumkeller in 2003. The positive integer $n$ is a Zumkeller number if its divisors can be partitioned into two sets with the same sum, which will be $σ(n)/2$. Generalizing even further, we call $n$ a $k$-layered number if its divisors can be partitioned into $k$ sets with equal sum. In this paper, we completely characterize Zumkeller numbers with two distinct prime factors and give some bounds for prime factorization in case of Zumkeller numbers with more than two distinct prime factors. We also characterize $k$-layered numbers with two distinct prime factors and even $k$-layered numbers with more than two distinct odd prime factors. Some other results concerning these numbers and their relationship with practical numbers and Harmonic mean numbers are also discussed.

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