Related papers: Demystifying a divisibility property of the Kostan…

In this paper, first we introduce a quantity called a partition function for a quiver mutation sequence. The partition function is a generating function whose weight is a $q$-binomial associated with each mutation. Then, we show that the…

The paper contains an exposition of part of topology using partitions of unity. The main idea is to create variants of the Tietze Extension Theorem and use them to derive classical theorems. This idea leads to a new result generalizing…

The partition function of composite bosons ("cobosons" for short) is calculated in the canonical ensemble, with the Pauli exclusion principle between their fermionic components included in an exact way through the finite temperature…

In this paper, we present a generalization of one of the theorems in [G. E. Andrews, Partitions with parts separated by parity, \textit{Annals of Combinatorics} \textbf{23}(2019), 241 - 248], and give its bijective proof. Further variations…

We employ computer algebra algorithms to prove a collection of identities involving Bessel functions with half-integer orders and other special functions. These identities appear in the famous Handbook of Mathematical Functions, as well as…

A generalization of a beautiful $q$-series identity found in the unorganized portion of Ramanujan's second and third notebooks is obtained. As a consequence, we derive a new three-parameter identity which is a rich source of…

Almost nothing is known about the parity of the partition function $p(n)$, which is conjectured to be random. Despite this expectation, Ono surprisingly proved the existence of infinitely many linear dependence congruence relations modulo 4…

Recently the second named author discovered a combinatorial identity in the context of vertex representations of quantum Kac-Moody algebras. We give a direct and elementary proof of this identity. Our method is to show a related identity of…

We study the finite dimensional partition properties of the countable homogeneous dense local order. Some of our results use ideas borrowed from the partition calculus of the rationals and are obtained thanks to a strengthening of…

New identities and congruences involving the ranks and cranks of partitions are proved. The proof depends on a new partial differential equation connecting their generating functions.

The first article in this series presented a thorough discussion of particle weights and their characteristic properties. In this part a disintegration theory for particle weights is developed which yields pure components linked to…

By investigating a recurrence relation about functions, we first give alternative proofs of various identities on Fibonacci numbers and Lucas numbers, and then, make certain well known identities visible via certain trivalent graph…

In 2002, Andrews, Lewis, and Lovejoy introduced the combinatorial objects which they called \emph{partitions with designated summands}. These are built by taking unrestricted integer partitions and designating exactly one of each occurrence…

We give bijective proofs using Fomin's growth diagrams for identities involving numbers of vacillating tableaux that arose in the representation theory of partition algebras or are inspired by such identities.

In this paper, we gave some properties of binomial coefficient.

We study a large class of stochastic $p$-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions.…

We prove analytic and combinatorial identities reminiscent of Schur's classical partition theorem. Specifically, we show that certain families of overpartitions whose parts satisfy gap conditions are equinumerous with partitions whose parts…

Partition functions for non-interacting particles are known to be symmetric functions. It is shown that powerful group-theoretical techniques can be used not only to derive these relationships, but also to significantly simplify calculation…

It was shown recently that entanglement of identical particles has a feature called dualism [Phys. Rev. Lett. 110, 140404 (2013)], which is fundamentally connected with quantum indistinguishability. Here we report an experiment that…

Indistinguishability of particles is normally considered to be an inherently quantum property which cannot be possessed by a classical theory. However, Saunders has argued that this is incorrect, and that classically indistinguishable…