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An edge-weighted hook formula for labelled trees


Féray, Valentin; Goulden, I P (2014). An edge-weighted hook formula for labelled trees. Journal of Combinatorics, 5(2):245-269.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2014
Deposited On:27 Jan 2015 16:26
Last Modified:05 Apr 2016 18:47
Publisher:International Press, Boston
ISSN:2156-3527
Publisher DOI:https://doi.org/10.4310/JOC.2014.v5.n2.a6

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