Related papers: Generating functions for fixed points of the Mulli…
For any prime p, we construct, and simultaneously count, all of the complex Specht modules in a given p-block of the symmetric group which remain irreducible when reduced modulo p. We call the Specht modules with this property p-irreducible…
We study infinitesimal generators of one-parameter semigroups in the unit disk $\mathbb D$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein-Milman…
In this short note we prove that, if $p$ is an odd prime dividing the order of a sporadic simple group, then with the exception of four groups for $p=3$, all sporadic simple groups are generated by an involution and an element of order $p$.
In materials science, the challenge of rapid prototyping materials with desired properties often involves extensive experimentation to find suitable microstructures. Additionally, finding microstructures for given properties is typically an…
Let $p$ be an odd prime, and define $$G_p(x)=\prod_{k=1}^{(p-1)/2}\left(x-e^{2\pi i k^2/p}\right).$$ In this paper we study values of $G_p(x)$ at roots of unity via Galois theory, and confirm some previous conjectures. For example, for any…
The analogue of the Riesz-Dunford functional calculus has been introduced and studied recently as well as the theory of semigroups and groups of linear quaternionic operators. In this paper we suppose that $T$ is the infinitesimal generator…
In a seminal 2007 paper, Andrews introduced a class of combinatorial objects that generalize partitions called $k$-marked Durfee symbols. Multivariate rank generating functions for these objects have been shown by many to have interesting…
We study normal reflection subgroups of complex reflection groups. Our approach leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a…
Feit and Fine derived a generating function for the number of ordered pairs of commuting n by n matrices over the finite field F_q. This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas…
Given a smooth toric variety $X$, the action of the torus $T$ lifts to the moduli space $\mathcal{M}$ of stable sheaves on $X$. Using the pioneering work of Klyacho, a fairly explicit combinatorial description of the fixed point locus…
Given a barrier $0 \leq b_0 \leq b_1 \leq ...$, let $f(n)$ be the number of nondecreasing integer sequences $0 \leq a_0 \leq a_1 \leq ... \leq a_n$ for which $a_j \leq b_j$ for all $0 \leq j \leq n$. Known formul\ae for $f(n)$ include an $n…
Recently, Capparelli, Meurman, A. Primc and M. Primc introduced a class of colored partitions which has since been called CMPP partitions. This generalized earlier work by M. Primc and \v{S}iki\'{c}, and by Trup\v{c}evi\'{c}. One main…
The generating function for $p_N(n)$, the number of partitions of $n$ into at most $N$ parts, may be written as a product of $N$ factors. In part I, we studied the behavior of coefficients in the partial fraction decomposition of this…
This is a note for constructing fundamental invariants and computing the Hilbert series of the invariant subalgebras of tensor products of polynomial rings under the action by a direct product of symmetric groups. Our computation relies on…
We study constructible invariants of the moduli space $\overline{\mathcal{M}}(\boldsymbol{x})$ of stable maps from genus zero curves to $\mathbb{P}^1$, relative to $0$ and $\infty$, with ramification profiles specified by…
For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…
We verify a recent conjecture of Kenyon/Szendroi, arXiv:0705.3419, by computing the generating function for pyramid partitions. Pyramid partitions are closely related to Aztec Diamonds; their generating function turns out to be the…
We describe fixed points of an infinite dimensional non-linear operator related to a hard core (HC) model with a countable set $\mathbb{N}$ of spin values on the Cayley tree. This operator is defined by a countable set of parameters…
We study normal reflection subgroups of complex reflection groups. Our point of view leads to a refinement of a theorem of Orlik and Solomon to the effect that the generating function for fixed-space dimension over a reflection group is a…
We extend the Andrews-Olsson identity to two-colored partitions. Regarding the sets of proper Young walls of quantum affine algebras $\g_n=A^{(2)}_{2n}$, $A^{(2)}_{2n-1}$, $B^{(1)}_{n}$, $D^{(1)}_{n}$ and $D^{(2)}_{n+1}$ as the sets of…