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Nipen Saikia

Publications and source records attributed to Nipen Saikia.

17 recordsLinked to original sources
arXiv2026

Some new results for Andrews' Kimberling partitions

George E. Andrews (2016) introduced the Kimberling index, $K(π)$, of a partition $π$ of a positive integer $n$, which is defined as $$K(π) = (\text{largest part of } π) - (\text{least part of } π) - (\text{number of parts of } π).$$ Based on Kimberling index, Andrews defined five partition functions, $K_>(n),$ $K_<(n),$ $K_\leq(n),$ $K_=(n),$ and $K_\geq(n)$, called Kimberling partition functions, which count the numbers of partitions of a positive integer $n$ for which the Kimberling index $K(π) $ is $>0$, $<0$, $\leq0$, $=0$ and $\geq 0$, respectively. He also gave the generating functions for $K_\le(n),$ $K_<(n),$ and $K_>(n)$ and established some relations connecting Kimberling partitions and other partition functions. Since then, the Kimberling partition functions and their generating functions remained unexplored. In this paper, we derive generating functions for $K_=(n),$ and $K_\geq(n)$, and establish some congruence relations of the five Kimberling partition functions by using the method of $q$-series identities.

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arXiv2026

Proofs of the Conjectures on $SOME(n)$ and $DSOME(n)$ Functions Related to Integer Partitions

Andrews and Dastidar (2026) introduced $SOME(n)$ and $DSOME(n)$ functions related to partitions of a positive integer $n$, where $SOME(n)$ is the sum of all the odd parts in the partitions of $n$ minus the sum of all the even parts and $DSOME(n)$ is the sum of all the odd parts in the partitions of $n$ into distinct parts minusthe sum of all the even parts in the same partitions. The purpose of this paper is to establish the conjecture$SOME(λ)\equiv0\pmod{5^α}$, $α\ge 1$ and $λ\ge0$ are integers such that $24λ\equiv1\pmod{5^α}$ due to Andrews and Dastidar, and the conjecture $DSOME(50n+21)\equiv0\pmod{8}$ due to Baruah and Gogoi (2026). In the process, we establish some new infinite families of congruences modulo 2, 4, and 8 for $DSOME(n)$.

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arXiv2026

Analytic proofs of Andrews-Bachraoui identities related to two-color partitions with evens in one color

Andrews and Bachraoui (\textit{Int. J. Number Theory} (2026)) studied the two-color partition function $F(n)$ of a non-negative integer $n$ wherein odd parts may appear in two colors (red and blue) and even parts appear in one color (blue). For any non-negative integer $n,$ they also considered some restricted versions of $F(n)$: $F_0(n)$: the number of partitions of $n$ counted by $F(n)$ such that the number of odd parts in red color is even; $F_1(n)$: the number of partitions counted by $F(n)$ such that the number of odd parts in red color is odd; $H(n)$: the number of partitions of $n$ counted by $F(n)$ such that the parts of the same color do not repeat. The main purpose of this paper is to present the analytic proofs of the $q$-series identities connected with $F(n)$ and $H(n),$ which appeared as open problems in the original paper. We also prove some congruences of $F_0(n)$ and $F_1(n)$ modulo $2,$ $4,$ and $8$ by using $ q $-series.

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arXiv2026

Some new congruences and identities for $SOME(n)$, $DSOME(n)$, $\overline{SOME}(n)$ functions and analogues

Andrews and Dastidar (\textit{Ramanujan J. 69, Article Number 26, (2026)} ) introduced the $SOME(n)$ and $DSOME(n)$ functions that calculate the sum of all odd parts minus the sum of all even parts of ordinary partitions and distinct partitions, respectively of a positive integer $n$, and proved their generating functions and some congruences modulo 4 and 5. Recently, Gireesh and Hemanthkumar introduced an overpartition analogue of $SOME(n)$ function, denoted by $\overline{SOME}(n)$ and proved some congruences modulo 3, 5 and powers of 2. In this paper, we prove some new identities and congruences for $SOME(n)$, $DSOME(n)$, and $\overline{SOME}(n)$ functions, including monotonicity results. We also define a general analogue of $SOME(n)$ function, denoted by $S_{\mathcal P}(n)$, which calculates the sum of all odd parts minus the sum of all even parts in any arbitrary family of partitions $\mathcal P(n)$ of a positive integer $n$, and prove some divisibility properties. Additionally, we define a colour partition analogue of $SOME(n)$ function and prove divisibility properties.

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arXiv2026

Relations for partitions with distinct even parts except the largest part which is even

In this paper, we prove some new \(q\)-series identities connecting \(4\)-regular partitions and partitions with distinct even parts with largest part being odd. We also define three new partition functions with distinct even parts except the largest part which is even, and prove identities connecting the three partitions with \(4\)-regular partitions. Moreover, we also offer some congruence for the three newly defined partitions.

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arXiv2025

Integer Partitions With Restricted Distinct Parts

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for $Q_{t}^{s}(n)$ for some particular values of $s$ and $t$ by employing $q$-series and theta function identities.

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arXiv2025

Some New Congruences and Partition-Theoretic Interpretations for the Coefficients of Some Rogers-Ramanujan Type Identities

Ramanujan listed several q-series identities in his lost notebook. The most well known q-series identities are the Rogers-Ramanujan type identities which are first discovered by Rogers and then rediscovered by Ramanujan. In this paper, we give partition-theoretic interpretations of some of the Rogers-Ramanujan type identities using overpartition and colour partition of positive integers, and prove infinite families of congruences modulo powers of 2.

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arXiv2025

Some New Congruences For Overpartition Function With $\ell$-Regular Non-Overlined Parts

Alanzi et al. (2022) investigated overpartition of a positive integer $n$ with $\ell$-regular non-overlined parts denoted by $\overline R_\ell^\ast (n)$, and proved some results for the case $\ell=3$. As extension to the results of Alanzi et al., Sellers (2024) proved some new congruences for $\overline R_3^\ast (n)$. In this paper, we prove some new infinite families and particular congruences for $\overline R_\ell^\ast (n)$ for $\ell=4, 5k, 6$, and 8, where $k$ is any positive integer. We also offer some congruences connecting $\overline R_\ell^\ast (n)$ with some other partition functions.

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arXiv2025

Some new congruences for generalized overcubic partition function

Amdeberhan et al. (2024) introduced the notion of a generalized overcubic partition function $\overline a_c (n)$ and proved an infinite family of congruences modulo a prime $p\ge 3$ and some Ramanujan type congruences. In this paper, we show that $\overline a_{2^λm+t}(n) \equiv \overline a_t (n) \pmod {2^{λ+1}}$, where $λ\geq1, m\geq0,$ and $t\geq1$ are integers. We also prove some new congruences modulo $8$ and $16$ for $\overline a_{2m+1}(n), \overline a_{2m+2}(n), \overline a_{8m+3}(n)$, where $m$ is any non-negative integer.

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arXiv2023

Some Identities of Ramanujan's q-Continued Fractions of Order Fourteen and Twenty-Eight, and Vanishing Coefficients

We deduce $q$-continued fractions $S_{1}(q)$, $S_{2}(q)$ and $S_{3}(q)$ of order fourteen, and continued fractions $V_{1}(q)$, $V_{2}(q)$ and $V_{3}(q)$ of order twenty-eight from a general continued fraction identity of Ramanujan. We establish some theta-function identities for the continued fractions and derive some colour partition identities as applications. Some vanishing coefficients results arising from the continued fractions are also offered.

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arXiv2021

Some New Congruences Modulo Powers of 2 For $(j,k)$-Regular Overpartition

Let $\overline{p}_{j,k}(n)$ denotes the number of $(j,k)$-regular overpartitions of a positive integer $n$ such that none of the parts is congruent to $j$ modulo $k$. Naika et. al. (2021) proved infinite families of congruences modulo powers of 2 for $\overline{p}_{3,6}(n)$, $\overline{p}_{5,10}(n)$ and $\overline{p}_{9,18}(n)$. In this paper, we obtain infinite families of congruences modulo power of 2 for $\overline{p}_{4,8}(n)$, $\overline{p}_{6,12}(n)$ and $\overline{p}_{8,16}(n)$. For example, we prove that, for all integers $n\geq 0$ and $α\geq 0$, $$ \overline{p}_{4,8}\left( 5^{2α+1}\left( 16(5n+j)+14\right) \right) q^n\equiv 0\pmod{64}; \qquad j=1,2,3,4.$$

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arXiv2021

Arithmetic Properties For $(r,s)$-Regular Partition Functions With Distinct Parts

For any relatively prime integers $r$ and $s$, let $a_{r,s}(n)$ denote the number of $(r,s)$-regular partitions of a positive integer of $n$ into distinct parts. Prasad and Prasad (2018) proved many infinite families of congruences modulo 2 for $a_{3,5}(n)$. In this paper, we establish families of congruences modulo 2 and 4 for $a_{r,s}(n)$ with $(r,s)\in$ \{(2,5), (2,7), (4,5), (4,9)\}. For example, we show that for all $β\geq 0$ and $n \geq 0,$ we have $$a_{2,5}\Big(4\cdot 5^{2β+1}n+\dfrac{37\cdot5^{2β}-1}{6}\Big)\equiv0 \pmod4. $$

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arXiv2020

General Congruences Modulo 5 and 7 for Colour Partitions

For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove some general congruences modulo $5$ and $7$ for the colour partition function $p_r(n)$ by considering some general values of $r$. To prove the congruences we employ some $q$-series identities which is also in the spirit of Ramanujan.

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arXiv2016

Some Congruences of a Restricted Bipartition Function

Let $c_N(n)$ denotes the number of bipartitions $(λ, μ)$ of a positive integer $n$ subject to the restriction that each part of $μ$ is divisible by $N$. In this paper, we prove some congruence properties of the function $c_N(n)$ for $N=7$, 11, and $5l$, for any integer $l\ge 1$, by employing Ramanujan's theta-function identities.

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