arXiv subjects

Gaurab Bardhan

Publications and source records attributed to Gaurab Bardhan.

5 recordsLinked to original sources

arXiv2026

Some new results for Andrews' Kimberling partitions

George E. Andrews (2016) introduced the Kimberling index, $K(\pi)$, of a partition $\pi$ of a positive integer $n$, which is defined as $$K(\pi) = (\text{largest part of } \pi) - (\text{least part of } \pi) - (\text{number of parts of } \pi).$$ 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(\pi) $ 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.

math.CO↗

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$SOME(\lambda)\equiv0\pmod{5^\alpha}$, $\alpha\ge 1$ and $\lambda\ge0$ are integers such that $24\lambda\equiv1\pmod{5^\alpha}$ 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 $DSOME(n)$.

math.NT↗

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.

math.NT↗

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.

math.NT↗

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.

math.NT↗