We study various versions of the univariate and multivariate matching and rook polynomials. We show that there is most general multivariate matching polynomial, which is, up the some simple substitutions and multiplication with a prefactor, the original multivariate matching polynomial introduced by C. Heilmann and E. Lieb. We follow here a line of investigation which was very successfully purs...

Connections between q-rook polynomials and matrices over nite elds are exploited to derive a new statistic for Garsia and Remmel's q-hit polynomial. Both this new statistic mat and another statistic for the q-hit polynomial recently introduced by Dworkin are shown to induce di erent multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are s...

Claudia Agnini, Jan Backman, Flavia Boscolo-Galazzo, Daniel J. Condon, Eliana Fornaciari, Simone Galeotti, Luca Giusberti, Paolo Grandesso, Lanci, Valeria Luciani, Simonetta Monechi, Giovanni Muttoni, Heiko Pälike, Maria Letizia Pampaloni, Cesare A. Papazzoni, Paul N. Pearson, Johannes Pignatti, Isabella Premoli Silva, Raffi, Domenico Rio, Lorenzo Rook, Diana Sahy, David J.A. Spofforth, Cristin...

Extending the notion of pattern avoidance in permutations, we study matchings and set partitions whose arc diagram representation avoids a given configuration of three arcs. These configurations, which generalize 3-crossings and 3-nestings, have an interpretation, in the case of matchings, in terms of patterns in full rook placements on Ferrers boards. We enumerate 312-avoiding matchings and pa...

We give two combinatorial interpretations of the Matrix Ansatz of the PASEP in terms of lattice paths and rook placements. This gives two (mostly) combinatorial proofs of a new enumeration formula for the partition function of the PASEP. Besides other interpretations, this formula gives the generating function for permutations of a given size with respect to the number of ascents and occurrence...

The simplicial rook graph SR(d, n) is the graph whose vertices are the lattice points in the nth dilate of the standard simplex in R, with two vertices adjacent if they differ in exactly two coordinates. We prove that the adjacency and Laplacian matrices of SR(3, n) have integral spectra for every n. We conjecture that SR(d, n) is integral for all d and n, and give a geometric construction of a...

Given an aperiodic random walk on a finite graph, an expression will be derived for the hitting times in terms of the eigenvalues of the transition matrix. The process of diagonalizing the transition matrix and its associated fundamental matrix will be discussed. These results will then be applied to a random walk on a rook graph. Lastly, a cover time bound depending on the hitting times will b...