Related papers: Self-conjugate $t$-core partitions and application…
We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different…
Recently, Merca and Schmidt proved a number of identities relating partitions of an integer with two classic number-theoretic functions, namely the M\"obius function and Euler's totient function. Their demonstrations were mainly algebraic.…
We determine an infinite family of linear identities for the number $A_4(n)$ of partition pairs of $n$ with $4$-cores by employing elementary $q$-series techniques and certain $3$-dissection formulas. We then discover an infinite family of…
Using equidistribution criteria, we establish divisibility by cyclotomic polynomials of several partition polynomials of interest, including $spt$-crank, overpartition pairs, and $t$-core partitions. As corollaries, we obtain new proofs of…
The Fibonacci numbers are the prototypical example of a recursive sequence, but grow too quickly to enumerate sets of integer partitions. The same is true for the other classical sequences $a(n)$ defined by Fibonacci-like recursions: the…
Based on two involutions and a bijection, we completely determine the difference between the number of $\ell$-regular partitions of $n$ into an even number of parts and into an odd number of parts for all positive integers $n$ and $\ell>1$,…
This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of…
We consider the $t$-core of an $s$-core partition, when $s$ and $t$ are coprime positive integers. Olsson has shown that the $t$-core of an $s$-core is again an $s$-core, and we examine certain actions of the affine symmetric group on…
Strict partitions are enumerated with respect to the weight, the number of parts, and the number of sequences of odd length. We write this trivariate generating function as a double sum $q$-series. Equipped with such a combinatorial set-up,…
The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely the conjecture that the partition function $p(n)$ is equidistributed modulo 2. Our…
Zeckendorf proved that every positive integer has a unique partition as a sum of non-consecutive Fibonacci numbers. Similarly, every natural number can be partitioned into a sum of non-consecutive terms of the Lucas sequence, although such…
Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain…
This note reports on the number of s-partitions of a natural number n. In an s-partition each cell has the form $2^k-1$ for some integer k. Such partitions have potential applications in cryptography, specifically in distributed…
We show that, in many cases, there are infinitely many sets of partitions corresponding to a single analytical Rogers-Ramanujan type identity. This means that a single analytical Rogers-Ramanujan type identity implies the existence of…
For rational $\alpha$, the fractional partition functions $p_\alpha(n)$ are given by the coefficients of the generating function $(q;q)^\alpha_\infty$. When $\alpha=-1$, one obtains the usual partition function. Congruences of the form…
Recently, Pankaj Jyoti Mahanta and Manjil P. Saikika proved some identities relating certain restricted partitions into distinct odd parts with the partition whose odd parts are distinct combinatorially. They asked for the q-series proofs.…
Recently, a novel method based on coding partitions [1]-[4] has been used to derive power series expansions to previously intractable problems. In this method the coefficients at $k$ are determined by summing the contributions made by each…
In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and…
Let $\bar{a}_s(n)$ denote the number of partitions of $n$, wherein each odd part is multicolored (atmost $s\ge 1$ colors) and the first appearance of parts may be overlined. In this paper, we establish new families of congruences modulo…
In this paper, we investigate the combinatorial properties of three classes of integer partitions: (1) $s$-modular partitions, a class consisting of partitions into parts with a number of occurrences (i.e., multiplicity) congruent to $0$ or…