Rook placements and Jordan forms of upper-triangular nilpotent matrices

The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. We present a combinatorial formula for computing the number F_\lambda(q) of matrices of Jordan type lambda as a weighted sum over standard Young tableaux. We also study a connection between these matrices and non-attacking rook placements, which leads to a refinement of the formula for F_\lambda(q).Comment: 25 pages, 6 figure

A categorification of the chromatic symmetric polynomial

International audienceThe Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$*($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology.Le polynôme chromatique symétrique d’un graphe $G$ est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie $H$*($G$) de modules gradués $S_n$, dont la série bigraduée de Frobeniusse $Frob_G(q,t)$ réduit au polynôme chromatique symétrique à $q=t=1$. Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie

q-Rook placements and Jordan forms of upper-triangular nilpotent matrices

The set of $n$ by $n$ upper-triangular nilpotent matrices with entries in a finite field $F_q$ has Jordan canonical forms indexed by partitions $λ \vdash n$. We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number$F_λ (q)$ of matrices of fixed Jordan type as a weighted sum over rook placements

A categorification of the chromatic symmetric polynomial

The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$*($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology