Related papers: Partition Statistics Equidistributed with the Numb…

Although symmetry methods and analysis are a necessary ingredient in every physicist's toolkit, rather less use has been made of combinatorial methods. One exception is in the realm of Statistical Physics, where the calculation of the…

We introduce the structure of vacillating Hecke tableaux, and establish a one-to-one correspondence between vacillating Hecke tableaux and linked partitions by using the Hecke insertion algorithm developed by Buch, Kresch, Shimozono,…

We give "hybrid" proofs of the $q$-binomial theorem and other identities. The proofs are "hybrid" in the sense that we use partition arguments to prove a restricted version of the theorem, and then use analytic methods (in the form of the…

Motivated by recent study on the number of $t$-hooks in partitions arising from Euler's partition identity, we investigate the number of $t$-hooks in the sets from the first Rogers-Ramanujan identity and the first little G\"ollitz identity.…

Recently, many authors have investigated how various partition statistics distribute as the size of the partition grows. In this work, we look at a particular statistic arising from the recent rejuvenation of MacMahon's partition analysis.…

Let \pi be a partition. In [2] we defined BG-rank(\pi) as an alternating sum of parities of parts. This statistic was employed to generalize and refine the famous Ramanujan modulo 5 partition congruence. Let p_j(n)(a_{t,j}(n)) denote a…

We introduce a new Partition of Unity Method for the numerical homogenization of elliptic partial differential equations with arbitrarily rough coefficients. We do not restrict to a particular ansatz space or the existence of a finite…

The number of standard Young tableaux of shape a partition $\lambda$ is called the dimension of the partition and is denoted by $f^{\lambda}$. Partitions with odd dimensions were enumerated by McKay and were further characterized by…

We look at the rank generating function $G_\lambda$ of partitions inside the Ferrers diagram of some partition $\lambda$, investigated by Stanton in 1990, as well as a closely related problem investigated by Stanley and Zanello in 2013. We…

Let $B$ be an infinite subset of $\mathbf{N}$. When we consider partitions of natural numbers into elements of $B$, a partition number without a restriction of the number of equal parts can be expressed by partition numbers with a…

The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a…

We find a new approach to computing the remainder of a polynomial modulo $x^n-1$; such a computation is called modular enumeration. Given a polynomial with coefficients from a commutative $\mathbb{Q}$-algebra, our first main result…

Let $X_1, \ldots, X_n$ be probability spaces, let $X$ be their direct product, let $\phi_1, \ldots, \phi_m: X \longrightarrow {\Bbb C}$ be random variables, each depending only on a few coordinates of a point $x=(x_1, \ldots, x_n)$, and let…

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our…

Recently, Blecher and Knopfmacher applied the notion of fixed points to integer partitions. This has already been generalized and refined in various ways such as $h$-fixed points for an integer parameter $h$ by Hopkins and Sellers. Here, we…

Let $M$ be a free module of rank $m$ over a commutative unital ring $R$ and let $N$ be its free submodule. We consider the problem when a given element of the exterior product $\Lambda^pM$ is divisible, in a sense, over elements of the…

We propound the thesis that there is a limitation to the number of possible structures which are axiomatically endowed with identities involving operations. In the case of algebras with a binary operation satisfying a formally reducible (to…

Stanley's theory of $(P,\omega)$-partitions is a standard tool in combinatorics. It can be extended to allow for the presence of a restriction, that is a given maximal value for partitions at each vertex of the poset, as was shown by Assaf…

The fundamentals of Statistical Mechanics require a fresh definition in the context of the developments in Classical Mechanics of integrable and chaotic systems. This is done with the introduction of Micro Partitions ; a union of disjoint…

Let $k$ be an algebraically closed field. Let $\Lambda$ be a noetherian commutative ring annihilated by an integer invertible in $k$ and let $\ell$ be a prime number different from the characteristic of $k$. We prove that if $X$ is a…