Journal of Combinatorics

Volume 5 (2014)

Number 2

An edge-weighted hook formula for labelled trees

Pages: 245 – 269



Valentin Féray (Institüt für Mathematik, Universität Zürich, Switzerland)

I. P. Goulden (Department of Combinatorics & Optimization, University of Waterloo, Ontario, Canada)

Alain Lascoux


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 Féray 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 Prüfer code for labelled trees.


hook formula, tree enumeration, combinatorial bijection, generating function

Full Text (PDF format)