Letting n≥2k, the partition algebra CAk≥2(n) has two one-dimensional subrepresentations that correspond in a natural way to alternating and trivial characters of symmetric group Sk. In 2019, Benkart Halverson introduced proved evaluations distinguished bases CAk(n) for nonzero elements regular CAk(n)-submodule corresponds Young symmetrizer ∑σ∈Skσ; 2016, Xiao an explicit formula analogue sign re...

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes...

A Gelfand model for a semisimple algebra A over C is a complex linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of combinatorial diagram algebras including: the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid alg...

Stephen Richardson, MD; Jeanne B. Budgin, DVM; Jacqueline M. Junkins-Hopkins, MD; Carmela C. Vittorio, MD; Jason Lee, MD; Wallace T. Miller, Jr, MD; Alain H. Rook, MD; Ellen J. Kim, MD; Departments of Dermatology (Drs Richardson, Junkins-Hopkins, Vittorio, Rook, and Kim) and Radiology (Dr Miller), University of Pennsylvania Health System; Department of Dermatology, University of Pennsylvania Sc...

We define the Rook Jumping Maze, provide historical perspective, and describe a generation method for such mazes. When applying stochastic local search algorithms to maze design, most creative effort concerns the definition of an objective function that rates maze quality. We define and discuss several maze features to consider in such a function definition. Finally, we share our preferred desi...

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give ...