Related papers: Glaisher combinatorics of regular partitions
Recently, Hirschhorn and Sellers defined the partition function $a_r(n)$, which counts the number of partitions of $n$ wherein even parts come in only one color, while the odd parts may appear in one of $r$-colors for fixed $r\ge1$. The aim…
In a recent paper, Carrell and Goulden found a combinatorial identity of the Bernstein operators that they then used to prove Bernstein's Theorem. We show that this identity is a straightforward consequence of the classical result. We also…
Motivated by Andrews' recent work related to Euler's partition theorem, we consider the set of partitions of an integer $n$ where the set of even parts has exactly $j$ elements, versus the set of partitions of $n$ where the set of repeated…
Over the past twenty years, lecture hall partitions have emerged as fundamental combinatorial structures, leading to new generalizations and interpretations of classical theorems and new results. In recent years, geometric approaches to…
We give a purely combinatorial proof of the Glaisher-Crofton identity which derives from the analysis of discrete structures generated by iterated second derivative. The argument illustrates utility of symbolic and generating function…
We show how certain suitably modified N-modular diagrams of integer partitions provide a nice combinatorial interpretation for the general term of Zeilberger's KOH identity. This identity is the reformulation of O'Hara's famous proof of the…
Consideration of a question of E. R. Berlekamp led Carlitz, Roselle, and Scoville to give a combinatorial interpretation of the entries of certain matrices of determinant~1 in terms of lattice paths. Here we generalize this result by…
The partitioning of space by hyperplanes in the context of discrete classification problem is considered. We obtain some relations for the number of partitions and establish a recurrence relation for the maximal number of partitions of R^n…
A cubic partition is an integer partition wherein the even parts can appear in two colors. In this paper, we introduce the notion of generalized cubic partitions and prove a number of new congruences akin to the classical Ramanujan-type. We…
Basis partitions are minimal partitions corresponding to successive rank vectors. We show combinatorially how basis partitions can be generated from primary partitions which are equivalent to the Rogers-Ramanujan partitions. This leads to…
Ramanujan gave a recurrence relation for the partition function in terms of the sum of the divisor function $\sigma(n)$. In 1885, J.W. Glaisher considered seven divisor sums closely related to the sum of the divisors function. We develop a…
We give short elementary expositions of combinatorial proofs of some variants of Euler's partitition problem that were first addressed analytically by George Andrews, and later combinatorially by others. Our methods, based on ideas from a…
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a…
We demonstrate that statistics of certain classes of set partitions is described by generating functions related to the Burgers, Ibragimov--Shabat and Korteweg--de Vries integrable hierarchies.
In this article, we first investigate the partitions whose parts are congruent to $a$ or $b$ modulo $k$ with the aid of separable integer partition classes with modulus $k$ introduced by Andrews. Then, we introduce the…
We provide a new foundation for combinatorial commutative algebra and Stanley-Reisner theory using the partition complex introduced in [Adi18]. One of the main advantages is that it is entirely self-contained, using only a minimal knowledge…
We use the celebrated circle method of Hardy and Ramanujan to develop convergent formulae for counting a restricted class of partitions that arise from the G\"ollnitz--Gordon identities.
In the first part of this article, we consider a Groebner basis of the differential ideal {x_1^2} with respect to "the" weighted lexicographical monomial order and show that its computation is related with an identity involving the…
We give a definition of partition C*-algebras: To any partition of a finite set, we assign algebraic relations for a matrix of generators of a universal C*-algebra. We then prove how certain relations may be deduced from others and we…
In our previous paper, we determined a unified combinatorial framework to look at a large number of colored partition identities, and studied the five identities corresponding to the exceptional modular equations of prime degree of the…