A four parameter generalization of Gollnitz's (BIG) partition theorem

Using $P(n,m)$, the number of integer partitions of $n$ into exactly $m$ parts, which was the subject of an earlier paper, $P(n,m,p)$, the number of integer partitions of $n$ into exactly $m$ parts with each part at most $p$, can be computed in $O(n^2)$, and the q-binomial coefficient can be computed in $O(n^3)$. Using the definition of the q-binomial coefficient, some properties of the q-binomial coefficient and $P(n,m,p)$ are derived. The q-multinomial coefficient can be co...

We present what we call a "motivated proof" of the Bressoud-G\"ollnitz-Gordon partition identities. Similar "motivated proofs" have been given by Andrews and Baxter for the Rogers-Ramanujan identities and by Lepowsky and Zhu for Gordon's identities. Additionally, "motivated proofs" have also been given for the Andrews-Bressoud partition identities by Kanade, Lepowsky, Russell, and Sills and for the G\"ollnitz-Gordon-Andrews identities by Coulson, Kanade, Lepowsky, McRae, Qi, ...

Schur's partition theorem states that the number of partitions of n into distinct parts congruent 1, 2 (mod 3) equals the number of partitions of n into parts which differ by >= 3, where the inequality is strict if a part is a multiple of 3. We establish a double bounded refined version of this theorem by imposing one bound on the parts congruent 0,1 (mod 3) and another on the parts congruent 2 (mod 3), and by keeping track of the number of parts in each of the residue classe...

In 1967, Andrews found a combinatorial generalization of the G\"ollnitz-Gordon theorem, which can be called the Andrews-G\"ollnitz-Gordon theorem. In 1980, Bressoud derived a multisum Rogers-Ramanujan-type identity, which can be considered as the generating function counterpart of the Andrews-G\"ollnitz-Gordon theorem. Lovejoy gave an overpartition analogue of the Andrews-G\"ollnitz-Gordon theorem for $i=k$. In this paper, we give an overpartition analogue of this theorem in ...

Many asymptotic formulas exist for unrestricted integer partitions as well as for distinct partitions of integers into a finite number of parts. Szekeres and Canfield have derived an asymptotic formula for the number of partitions that is valid for any value of the number of parts. We obtain general asymptotic formulas for distinct partitions that are valid in a wider range of parameters than the existing asymptotic formulas, and we recover the known asymptotic results as spe...

Berkovich-Uncu have recently proved a companion of the well-known Capparelli's identities as well as refinements of Savage-Sills' new little G\"ollnitz identities. Noticing the connection between their results and Boulet's earlier four-parameter partition generating functions, we discover a new class of partitions, called $k$-strict partitions, to generalize their results. By applying both horizontal and vertical dissections of Ferrers' diagrams with appropriate labellings, w...

Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for four types of partitions using elementary series manipulations. In this paper, we generalize their congruences using arithmetic properties of certain quadratic forms.

We provide new formulas for the coefficients in the partial fraction decomposition of the restricted partition generating function. These techniques allow us to partially resolve a recent conjecture of Sills and Zeilberger. We also describe upcoming work, giving a resolution to Rademacher's conjecture on the asymptotics of these coefficients.

In a previous paper of the second author with K. Ono, surprising multiplicative properties of the partition function were presented. Here, we deal with $k$-regular partitions. Extending the generating function for $k$-regular partitions multiplicatively to a function on $k$-regular partitions, we show that it takes its maximum at an explicitly described small set of partitions, and can thus easily be computed. The basis for this is an extension of a classical result of Lehmer...

In a work of 1995, Alladi, Andrews, and Gordon provided a generalization of the two Capparelli identities involving certain classes of integer partitions. Inspired by that contribution, in particular as regards the general setting and the tools the authors employed, we obtain new partition identities by identifying further sets of partitions that can be explicitly put into a one-to-one correspondence by the method described in the 1995 paper. As a further result, although of ...