119 research outputs found
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 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 *() of graded -modules, whose graded Frobenius series reduces to the chromatic symmetric function at . We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology.Le polynôme chromatique symétrique d’un graphe 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 *() de modules gradués , dont la série bigraduée de Frobeniusse réduit au polynôme chromatique symétrique à . 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 by upper-triangular nilpotent matrices with entries in a finite field has Jordan canonical forms indexed by partitions . We study a connection between these matrices and non-attacking q-rook placements, which leads to a combinatorial formula for the number 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 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 *() of graded -modules, whose graded Frobenius series reduces to the chromatic symmetric function at . We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology
- …