129 research outputs found
An edge-weighted hook formula for labelled trees
A number of hook formulas and hook summation formulas have previously
appeared, involving various classes of trees. One of these classes of trees is
rooted trees with labelled vertices, in which the labels increase along every
chain from the root vertex to a leaf. In this paper we give a new hook
summation formula for these (unordered increasing) trees, by introducing a new
set of indeterminates indexed by pairs of vertices, that we call edge weights.
This new result generalizes a previous result by F\'eray and Goulden, that
arose in the context of representations of the symmetric group via the study of
Kerov's character polynomials. Our proof is by means of a combinatorial
bijection that is a generalization of the Pr\"ufer code for labelled trees.Comment: 25 pages, 9 figures. Author-produced copy of the article to appear in
Journal of Combinatorics, including referee's suggestion
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
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