Related papers: A Partition Function Connected with the G\"ollnitz…
In this paper we refine a weighted partition identity of Alladi. We write explicit formulas of generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results…
The Rogers-Ramanujan-Gordon identities generalize the classical partition identities discovered independently by L. J. Rogers and S. Ramanujan. In 2021, Afsharijoo provided a commutative algebra proof of the Rogers-Ramanujan-Gordon…
We utilize false theta function results of Nathan Fine to discover three new partition identities involving weights. These relations connect G\"ollnitz--Gordon type partitions and partitions with distinct odd parts, partitions into distinct…
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…
This paper will primarily present a method of proving generating function identities for partitions from linked partition ideals. The method we introduce is built on a conjecture by George Andrews and that those generating functions satisfy…
The celebrated Rogers-Ramanujan identities equate the number of integer partitions of $n$ ($n\in\mathbb N_0$) with parts congruent to $\pm 1 \pmod{5}$ (respectively $\pm 2 \pmod{5}$) and the number of partitions of $n$ with super-distinct…
We study the generating function of the excess number of Rogers-Ramanujan partitions with odd rank over those with even rank, and, using combinatorial and analytical techniques, show that this generating function is closely connected with…
In this work we introduce new combinatorial objects called $d$--fold partition diamonds, which generalize both the classical partition function and the partition diamonds of Andrews, Paule and Riese, and we set $r_d(n)$ to be their counting…
The partition function $p(n)$ has been a testing ground for applications of analytic number theory to combinatorics. In particular, Hardy and Ramanujan invented the "circle method" to estimate the size of $p(n)$, which was later perfected…
We generalise Euler's partition theorem involving odd parts and different parts for all moduli and provide new companions to Rogers-Ramanujan- Andrews-Gordon identities related to this theorem.
We derive a combinatorial multisum expression for the number $D(n,k)$ of partitions of $n$ with Durfee square of order $k$. An immediate corollary is therefore a combinatorial formula for $p(n)$, the number of partitions of $n$. We then…
Sometimes we need the approximate value of the partition number in a simple and efficient way. There are already several formulae to calculate the partition number p(n). But they are either inconvenient for most people (not majored in math)…
Resorting to the recursions satisfied by the polynomials which converge to the right hand sides of the Rogers-Ramanujan type identities given by Sills and a determinant method presented in a paper by Ismail-Prodinger-Stanton, we obtain many…
We develop a new closed-form arithmetic and recursive formula for the partition function and a generalization of Andrews' smallest parts (spt) function. Using the inclusion-exclusion principle, we additionally develop a formula for the…
We extend partition-theoretic work of Andrews, Bressoud, and Burge to overpartitions, defining the notions of successive ranks, generalized Durfee squares, and generalized lattice paths, and then relating these to overpartitions defined by…
We give a series of recursive identities for the number of partitions with exactly $k$ parts and with constraints on both the minimal difference among the parts and the minimal part. Using these results we demonstrate that the number of…
For a subset $\mathcal A\subset \mathbb N$, let $p_{\mathcal A}(n)$ denote the restricted partition function which counts partitions of $n$ with all parts lying in $\mathcal A$. In this paper, we use a variation of the Hardy-Littlewood…
In this paper we develop a method to calculate the overpartition function efficiently using a Hardy-Rademacher-Ramanujan type formula, and we use this method to find many new Ramanujan-style congruences, whose existence is predicted by…
Recently Corteel and Welsh outlined a technique for finding new sum-product identities by using functional relations between generating functions for cylindric partitions and a theorem of Borodin. Here, we extend this framework to include…
Bressoud introduced the partition function $B(\alpha_1,\ldots,\alpha_\lambda;\eta,k,r;n)$, which counts the number of partitions with certain difference conditions. Bressoud posed a conjecture on the generating function for the partition…