1,395 research outputs found
Enumerative properties of Ferrers graphs
We define a class of bipartite graphs that correspond naturally with Ferrers
diagrams. We give expressions for the number of spanning trees, the number of
Hamiltonian paths when applicable, the chromatic polynomial, and the chromatic
symmetric function. We show that the linear coefficient of the chromatic
polynomial is given by the excedance set statistic.Comment: 12 page
Power sum expansion of chromatic quasisymmetric functions
The chromatic quasisymmetric function of a graph was introduced by Shareshian
and Wachs as a refinement of Stanley's chromatic symmetric function. An
explicit combinatorial formula, conjectured by Shareshian and Wachs, expressing
the chromatic quasisymmetric function of the incomparability graph of a natural
unit interval order in terms of power sum symmetric functions, is proven. The
proof uses a formula of Roichman for the irreducible characters of the
symmetric group.Comment: Final version, 9 pages; comments by a referee incorporate
Bruhat order, smooth Schubert varieties, and hyperplane arrangements
The aim of this article is to link Schubert varieties in the flag manifold
with hyperplane arrangements. For a permutation, we construct a certain
graphical hyperplane arrangement. We show that the generating function for
regions of this arrangement coincides with the Poincare polynomial of the
corresponding Schubert variety if and only if the Schubert variety is smooth.
We give an explicit combinatorial formula for the Poincare polynomial. Our main
technical tools are chordal graphs and perfect elimination orderings.Comment: 14 pages, 2 figure
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