Related papers: ($\ell,0)$-Carter partitions, a generating functio…
Jantzen-Seitz partitions are those $p$-regular partitions of~$n$ which label $p$-modular irreducible representations of the symmetric group $S_n$ which remain irreducible when restricted to $S_{n-1}$; they have recently also been found to…
Let $\Gamma$ be a connected regular graph with an eigenvalue $\lambda$ and corresponding idempotent $E_{\lambda}$. Let ${\cal E}_{\lambda}=\langle J,E_{\lambda}\rangle^\circ$ be the algebra generated by $J$ and $E_\lambda$ with respect to…
Komj\'ath, Milner, and Polat investigated when a finitary matroid admits a partition into circuits. They defined the class of ``finite matching extendable'' matroids and showed in their compactness theorem that those matroids always admit…
The elementary divisors of the Gram matrices of Specht modules S^lambda over the symmetric group are determined for two-row partitions and for two-column partitions lambda. More precisely, the subquotients of the Jantzen filtration are…
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…
Stanley defined a partition function t(n) as the number of partitions $\lambda$ of n such that the number of odd parts of $\lambda$ is congruent to the number of odd parts of the conjugate partition $\lambda'$ modulo 4. We show that t(n)…
Ramanujan's celebrated partition congruences modulo $\ell\in \{5, 7, 11\}$ assert that $$ p(\ell n+\delta_{\ell})\equiv 0\pmod{\ell}, $$ where $0<\delta_{\ell}<\ell$ satisfies $24\delta_{\ell}\equiv 1\pmod{\ell}.$ By proving Subbarao's…
In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…
In this paper, we consider ordered set partitions obtained by imposing conditions on the size of the lists, and such that the first $r$ elements are in distinct blocks, respectively. We introduce a generalization of the Lah numbers. For…
The primary contribution of this thesis is to introduce and examine the planar modular partition monoid for parameters $m, k \in \mathbb{Z}_{>0}$, which has simultaneously and independently generated interest from other researchers as…
We consider a simple model of the dynamics of a single electron in a crystal lattice. Although this is a standard problem in condensed matter physics, alternative ways of evaluating a partition function for such a system lead to equalities,…
In this paper, we study arithmetic properties satisfied by the $k$-tuple $\ell$-regular partitions. A $k$-tuple of partitions $(\xi_1, \xi_2, \ldots, \xi_k)$ is said to be $\ell$-regular if all the $\xi_i$'s are $\ell$-regular. We study the…
We consider a partially asymmetric exclusion process (PASEP) on a finite number of sites with open and directed boundary conditions. Its partition function was calculated by Blythe, Evans, Colaiori, and Essler. It is known to be a…
We study the ideals of the partition, Brauer, and Jones monoid, establishing various combinatorial results on generating sets and idempotent generating sets via an analysis of their Graham--Houghton graphs. We show that each proper ideal of…
Buryak, Feigin and Nakajima computed a generating function for a family of partition statistics by using the geometry of the $Z/cZ$ fixed point sets in the Hilbert scheme of points on $C^2$. Loehr and Warrington had already shown how a…
The coefficients of the generating function $(q;q)^\alpha_\infty$ produce $p_\alpha(n)$ for $\alpha \in \mathbb{Q}$. In particular, when $\alpha = -1$, the partition function is obtained. Recently, Chan and Wang identified and proved…
A triangular partition is a partition whose Ferrers diagram can be separated from its complement (as a subset of $\mathbb{N}^2$) by a straight line. Having their origins in combinatorial number theory and computer vision, triangular…
In this paper we study the partial Brauer $\mathbb{C}$-algebras $\mathfrak{R}_n(\delta,\delta')$, where $n \in \mathbb{N}$ and $\delta,\delta'\in\mathbb{C}$. We show that these algebras are generically semisimple, construct the Specht…
For primes $\ell$ and nonnegative integers $a$, we study the partition functions $$p_\ell(a;n):= \#\{\lambda \vdash n : \text{ord}_\ell(H(\lambda))=a\},$$ where $H(\lambda)$ denotes the product of hook lengths of a partition $\lambda$.…
For a given prime $p$, we study the properties of the $p$-dissection identities of Ramanujan's theta functions $\psi(q)$ and $f(-q)$, respectively. Then as applications, we find many infinite family of congruences modulo 2 for some…