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Nayandeep Deka Baruah

Publications and source records attributed to Nayandeep Deka Baruah.

At least 19 recordsLinked to original sources
arXiv2026

Congruences for the Andrews--Uncu partition function $\mathcal{EO}_u(n)$

In 2018, Andrews defined and studied the partition function $\mathcal{EO}(n)$, which counts the number of partitions of $n$ where each even part is less than each odd part. Thereafter, Uncu considered a different subset of such partitions, namely those partitions counted by $\mathcal{EO}(n)$ where there are no repeated even parts (we denote the number of such partitions by $\mathcal{EO}_u(n)$). In this paper, we prove several congruences modulo powers of $2$ for $\mathcal{EO}_u(n)$. We also prove some infinite families of congruences as well as a recurrence for the number of this class of partitions. Our methods involve elementary, algorithmic, and modular form techniques.

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arXiv2026

Difference of the Sum of All Odd and Even Overlined Parts of Overpartitions

Recently, Garvan and Sarma studied sums of non-overlined parts in overpartitions and partitions without repeated odd parts. In this paper, we study the corresponding statistic for overlined parts. We define $\mathrm{OSOME}(n)$ as the sum of all odd overlined parts minus the sum of all even overlined parts in the overpartitions of $n$. We obtain a closed form for its generating function and use it to prove congruences by elementary $q$-series methods, the theory of half-integral weight modular forms, and known congruences for the overpartition function.

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arXiv2026

Arithmetic properties of DSOME function

Recently, Andrews and Ghosh Dastidar (Ramanujan J. \textbf{69}, Art. No. 26, 2026) studied two interesting functions $SOME(n)$ and $DSOME(n)$, where $SOME(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all even parts and $DSOME(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minus sum of all the even parts. They expressed the generating functions of $SOME(n)$ and $DSOME(n)$ in terms of $q$-series and found several interesting congruences modulo 4 and 5. In this paper, we express the generating function of $DSOME(n)$ in a closed form, which allows us to find some new congruences and internal congruences modulo 4 and 8 for $DSOME(n)$.

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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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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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arXiv2025

A refinement of a result of Andrews and Newman on the sum of minimal excludants

In this article, we refine a result of Andrews and Newman, that is, the sum of minimal excludants over all the partitions of a number $n$ equals the number of partitions of $n$ into distinct parts with two colors. As a consequence, we find congruences modulo 4 and 8 for the functions appearing in this refinement. We also conjecture three further congruences for these functions. In addition, we also initiate the study of $k^{th}$ moments of minimal excludants. At the end, we also provide an alternate proof of a beautiful identity due to Hopkins, Sellers and Stanton.

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arXiv2025

Sign Patterns and Congruences of certain infinite products involving the Rogers-Ramanujan continued fraction

We study the behavior of the signs of the coefficients of certain infinite products involving the Rogers-Ramanujan continued fraction. For example, if $$\sum_{n=0}^{\infty}A(n)q^{n}:= \dfrac{(q^2;q^5)_\infty^5(q^3;q^5)_\infty^5}{(q;q^5)_\infty^5(q^4;q^5)_\infty^5},$$then $A(5n+1)>0$, $A(5n+2)>0$, $A(5n+3)>0$, and $A(5n+4)<0$. We also find a few congruences satisfied by some coefficients. For example, for all nonnegative integers $n$, $A(9n+4)\equiv 0 \pmod3$, $ A(16n+13)\equiv 0 \pmod4$, and $A(15n+r)\equiv0\pmod{15}$, where $r\in\{4, 8, 13, 14\}$.

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arXiv2024

Arithmetic properties of $5$-regular partitions into distinct parts

A partition is said to be $\ell$-regular if none of its parts is a multiple of $\ell$. Let $b^\prime_5(n)$ denote the number of 5-regular partitions into distinct parts (equivalently, into odd parts) of $n$. This function has also close connections to representation theory and combinatorics. In this paper, we study arithmetic properties of $b^\prime_5(n)$. We provide full characterization of the parity of $b^\prime_5(2n+1)$, present several congruences modulo 4, and prove that the generating function of the sequence $(b^\prime_5(5n+1))$ is lacunary modulo any arbitrary positive powers of 5.

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arXiv2024

Identities for the Rogers-Ramanujan Continued Fraction

We prove some new modular identities for the Rogers\textendash Ramanujan continued fraction. For example, if $R(q)$ denotes the Rogers\textendash Ramanujan continued fraction, then \begin{align*}&R(q)R(q^4)=\dfrac{R(q^5)+R(q^{20})-R(q^5)R(q^{20})}{1+R(q^{5})+R(q^{20})},\\ &\dfrac{1}{R(q^{2})R(q^{3})}+R(q^{2})R(q^{3})= 1+\dfrac{R(q)}{R(q^{6})}+\dfrac{R(q^{6})}{R(q)}, \end{align*}and\begin{align*}R(q^2)=\dfrac{R(q)R(q^3)}{R(q^6)}\cdot\dfrac{R(q) R^2(q^3) R(q^6)+2 R(q^6) R(q^{12})+ R(q) R(q^3) R^2(q^{12})}{R(q^3) R(q^6)+2 R(q) R^2(q^3) R(q^{12})+ R^2(q^{12})}.\end{align*} In the process, we also find some new relations for the Rogers-Ramanujan functions by using dissections of theta functions and the quintuple product identity.

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arXiv2024

The $n$-Color Partition Function and Some Counting Theorems

Recently, Merca and Schmidt found some decompositions for the partition function $p(n)$ in terms of the classical Möbius function as well as Euler's totient. In this paper, we define a counting function $T_k^r(m)$ on the set of $n$-color partitions of $m$ for given positive integers $k, r$ and relate the function with the $n$-color partition function and other well-known arithmetic functions like the Möbius function, Liouville function, etc. and their divisor sums. Furthermore, we use a counting method of Erdös to obtain some counting theorems for $n$-color partitions that are analogous to those found by Andrews and Deutsch for the partition function.

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arXiv2024

Arithmetic Identities for Some Analogs of $5$-core Partition Function

Recently, Gireesh, Ray, and Shivashankar studied an analog, $\overline{a}_t(n)$, of the $t$-core partition function, $c_t(n)$. In this paper, we study the function $\overline{a}_5(n)$ in conjunction with $c_5(n)$ as well as another analogous function $\overline{b}_5(n)$. We also find several arithmetic identities for $\overline{a}_5(n)$ and $\overline{b}_5(n)$.

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arXiv2023

Congruences for the difference of even and odd number of parts of the cubic and some analogous partition functions

Partitions wherein the even parts appear in two different colours are known as cubic partitions. Recently, Merca introduced and studied the function $A(n)$, which is defined as the difference between the number of cubic partitions of $n$ into an even number of parts and the number of cubic partitions of $n$ into an odd number of parts. In particular, using Smoot's \textsf{RaduRK} Mathematica package, Merca proved the following congruences by finding the exact generating functions of the respective functions. For all $n\ge0$, \begin{align*}A(9n+5)\equiv 0\pmod 3,\\ A(27n+26)\equiv 0\pmod 3. \end{align*} By using generating function manipulations and dissections, da Silva and Sellers proved these congruences and two infinite families of congruences modulo 3 arising from these congruences. In this paper, by employing Ramanujan's theta function identities, we present simplified formulas of the generating functions from which proofs of the congruences of Merca as well as those of da Silva and Sellers follow quite naturally. We also study analogous partition functions wherein multiples of $k$ appear in two different colours, where $k\in\{3,5,7,23\}$.

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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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