1,063 research outputs found
Counting invertible Schr\"odinger Operators over Finite Fields for Trees, Cycles and Complete Graphs
We count invertible Schr\"odinger operators (perturbations by diagonal
matrices of the adjacency matrix) over finite fieldsfor trees, cycles and
complete graphs.This is achieved for trees through the definition and use of
local invariants (algebraic constructions of perhapsindependent
interest).Cycles and complete graphs are treated by ad hoc methods.Comment: Final version to appear in Electronic Journal of Combinatoric
Combinatorial families of multilabelled increasing trees and hook-length formulas
In this work we introduce and study various generalizations of the notion of
increasingly labelled trees, where the label of a child node is always larger
than the label of its parent node, to multilabelled tree families, where the
nodes in the tree can get multiple labels. For all tree classes we show
characterizations of suitable generating functions for the tree enumeration
sequence via differential equations. Furthermore, for several combinatorial
classes of multilabelled increasing tree families we present explicit
enumeration results. We also present multilabelled increasing tree families of
an elliptic nature, where the exponential generating function can be expressed
in terms of the Weierstrass-p function or the lemniscate sine function.
Furthermore, we show how to translate enumeration formulas for multilabelled
increasing trees into hook-length formulas for trees and present a general
"reverse engineering" method to discover hook-length formulas associated to
such tree families.Comment: 37 page
A simple model of trees for unicellular maps
We consider unicellular maps, or polygon gluings, of fixed genus. A few years
ago the first author gave a recursive bijection transforming unicellular maps
into trees, explaining the presence of Catalan numbers in counting formulas for
these objects. In this paper, we give another bijection that explicitly
describes the "recursive part" of the first bijection. As a result we obtain a
very simple description of unicellular maps as pairs made by a plane tree and a
permutation-like structure. All the previously known formulas follow as an
immediate corollary or easy exercise, thus giving a bijective proof for each of
them, in a unified way. For some of these formulas, this is the first bijective
proof, e.g. the Harer-Zagier recurrence formula, the Lehman-Walsh formula and
the Goupil-Schaeffer formula. We also discuss several applications of our
construction: we obtain a new proof of an identity related to covered maps due
to Bernardi and the first author, and thanks to previous work of the second
author, we give a new expression for Stanley character polynomials, which
evaluate irreducible characters of the symmetric group. Finally, we show that
our techniques apply partially to unicellular 3-constellations and to related
objects that we call quasi-constellations.Comment: v5: minor revision after reviewers comments, 33 pages, added a
refinement by degree of the Harer-Zagier formula and more details in some
proof
Decorated hypertrees
C. Jensen, J. McCammond and J. Meier have used weighted hypertrees to compute
the Euler characteristic of a subgroup of the automorphism group of a free
product. Weighted hypertrees also appear in the study of the homology of the
hypertree poset. We link them to decorated hypertrees after a general study on
decorated hypertrees, which we enumerate using box trees.---C. Jensen, J.
McCammond et J. Meier ont utilis\'e des hyperarbres pond\'er\'es pour calculer
la caract\'eristique d'Euler d'un sous-groupe du groupe des automorphismes d'un
produit libre. Un autre type d'hyperarbres pond\'er\'es appara\^it aussi dans
l'\'etude de l'homologie du poset des hyperarbres. Nous \'etudions les
hyperarbres d\'ecor\'es puis les comptons \`a l'aide de la notion d'arbre en
bo\^ite avant de les relier aux hyperarbres pond\'er\'es.Comment: nombre de pages : 3
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