Related papers: Linked partition ideals and Kanade--Russell conjec…
We construct Andrews-Gordon type evidently positive series as generating functions of partitions satisfying certain difference conditions in six conjectures by Kanade and Russell. We construct generating functions for missing partition…
Finding an Andrews--Gordon type generating function identity for a linked partition ideal is difficult in most cases. In this paper, we will handle this problem in the setting of graph theory. With the generating function of directed graphs…
We give an alternative construction for a family of partition generating functions due to Kanade and Russell. In our alternative construction, we use ordinary partitions instead of jagged partitions. We also present new generating functions…
We propose a method to construct a variety of partition identities at once. The main application is an all-moduli generalization of some of Andrews' results in [5]. The novelty is that the method constructs solutions to functional equations…
We consider differences of one- and two-variable finite products and provide combinatorial proofs of the nonnegativity of certain coefficients. Since the products may be interpreted as generating functions for certain integer partitions,…
We present a new partition identity and give a combinatorial proof of our result. This generalizes a result of Andrew's in which he considers the generation function for partitions with respect to size, number of odd parts, and number of…
Inspired by a number of recent papers by Corteel, Dousse, Foda, Uncu and Welsh on cylindric partitions and Rogers-Ramanujan-type identities, we obtain the $\mathrm{A}_2$ (or $\mathrm{A}_2^{(1)}$) analogues of the celebrated Andrews-Gordon…
Partitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition…
In this paper, we discuss a few recent conjectures made by George Beck related to the ranks and cranks of partitions. The conjectures for the rank of a partition were proved by Andrews by using results due to Atkin and Swinnerton-Dyer on a…
In his paper, "On a Partition Function of Richard Stanley," George Andrews proves a certain partition identity analytically and asks for a combinatorial proof. This paper provides the requested combinatorial proof.
Let $\mathscr{S}$ denote the set of integer partitions into parts that differ by at least $3$, with the added constraint that no two consecutive multiples of $3$ occur as parts. We derive trivariate generating functions of Andrews--Gordon…
We present proofs of two new families of sum-product identities arising from the cylindric partitions paradigm. Most of the presented expressions, the related sum-product identities, and the ingredients for the proofs were first conjectured…
We construct Andrews-Gordon type evidently positive series as generating functions for the partitions satisfying the difference conditions imposed by Capparelli's identities and G\"{o}llnitz-Gordon identities. The construction involves…
Noting a curious link between Andrews' even-odd crank and the Stanley rank, we adopt a combinatorial approach building on the map of conjugation and continue the study of integer partitions with parts separated by parity. Our motivation is…
Recently, George Andrews has given a Glaisher style proof of a finite version of Euler's partition identity. We generalise this result by giving a finite version of Glaisher's partition identity. Both the generating function and bijective…
Kanade and Russell conjectured several Rogers-Ramanujan-type partition identities, some of which are related to level $2$ characters of the affine Lie algebra $A_9^{(2)}$. Many of these conjectures have been proved by Bringmann,…
In this paper, we give a conjecture, which generalises Euler's partition theorem involving odd parts and different parts for all moduli. We prove this conjecture for two family partitions. We give $q$-difference equations for the related…
We prove a family of partition identities which is "dual" to the family of Andrews-Gordon's identities. These identities are inspired by a correspondence between a special type of partitions and "hypergraphs" and their proof uses…
We construct an evidently positive multiple series as a generating function for partitions satisfying the multiplicity condition in Schur's partition theorem. Refinements of the series when parts in the said partitions are classified…
We study cylindric partitions with two-element profiles using MacMahon's partition analysis. We find explicit formulas for the generating functions of the number of cylindric partitions by first finding the recurrences using partition…