The Heisenberg–Weyl algebra, which underlies virtually all physical representations of Quantum Theory, is considered from the combinatorial point of view. We provide a concrete model of the algebra in terms of paths on a lattice with some decomposition rules. We also discuss the rook problem on the associated Ferrers board; this is related to the calculus in the normally ordered basis. From thi...

In this manuscript we study inclusion posets of Borel orbit closures on (symmetric) matrices. In particular, we show that the Bruhat poset of partial involutions is a lexicographically shellable poset. We determine which subintervals of the Bruhat posets are Eulerian, and moreover, by studying certain embeddings of the symmetric groups and their involutions into rook matrices and partial involu...

We present guesses, based on intensive computer algebra calculations, for recurrence equations of the sequences enumerating rook walks in up to twelve dimensions ending on the main diagonal. Computer proofs can in principle be constructed for all of them. For the moment, however, these computations are feasible only for low dimensions. We pose it as a challenge to develop algorithms which can a...

In this paper, we 2nd an expression of the rook vector of a matrix A (not necessarily square) in terms of permanents of some matrices associated with A, and obtain some simple exact formulas for the permanents of all n × n Toeplitz band matrices of zeros and ones whose bands are of width not less than n− 1. c © 2002 Elsevier Science B.V. All rights reserved.

Our purpose in this article is to investigate the order complex of inclusion poset PFn of Borel orbit closures in skew-symmetric matrices. We prove that PFn is an EL-shellable poset and furthermore its order complex triangulates a ball. We investigate (rook-theoretic) combinatorial properties of the rank-generating function of PFn in contrast with the zeta function of the variety of skew-symmet...

1. Yeatman GW, Shaw C, Barlow Md, et al: Pseudohattering in Vietnamese children. Pediatrics 1976;58:616-618 2. Waskerwitz 5, Christoffel KK, Hauger 5: Hypersensitivity vasculitis presenting as suspected child abuse: Case report and literature review. Pediatrics 1981;67:283-284 3. Rook A J (ed): Textbook ofDermatologv, ed 2. Oxford, Blackwell Scientific Publications, 1972 4. Feigin RD, Cherry JD...

Inspired by the results of [APR], we propose combinatorial Gelfand models for semigroup algebras of some finite semigroups, which include the symmetric inverse semigroup, the dual symmetric inverse semigroup, the maximal factorizable subsemigroup in the dual symmetric inverse semigroup, and the factor power of the symmetric group. Furthermore we extend the Gelfand model for the semigroup algebr...

We consider a tower of generalized rook monoid algebras over the field $\mathbb{C}$ complex numbers and observe that Bratteli diagram associated to this is simple graph. construct modules describe Jucys-Murphy elements for algebras. Over an algebraically closed $\Bbbk$ positive characteristic $p$, utilizing algebras, $0\leq i\leq p-1$ we define corresponding $i$-restriction $i$-induction functo...