Universal Tutte characters via combinatorial coalgebras

Dupont, Clément; Fink, Alex; Moci, Luca

doi:10.5802/alco.35

Dupont, Clément ¹; Fink, Alex ²; Moci, Luca ³

¹ IMAG Université de Montpellier CNRS Montpellier France
² School of Mathematical Sciences Queen Mary University of London UK
³ IMJ-PRG Université Paris-Diderot Paris 7 Paris France

Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 603-651.

Abstract

This work discusses the extraction of meaningful invariants of combinatorial objects from coalgebra or bialgebra structures. The Tutte polynomial is an invariant of graphs well known for the formula which computes it recursively by deleting and contracting edges, and for its universality with respect to similar recurrence. We generalize this to all classes of combinatorial objects with deletion and contraction operations, associating to each such class a universal Tutte character by a functorial procedure. We show that these invariants satisfy a universal property and convolution formulae similar to the Tutte polynomial. With this machinery we recover classical invariants for delta-matroids, matroid perspectives, relative and colored matroids, generalized permutohedra, and arithmetic matroids. We also produce some new invariants along with new convolution formulae.

Received: 2017-11-24
Revised: 2018-05-17
Accepted: 2018-07-11
Published online: 2018-11-30

MR Zbl

DOI: 10.5802/alco.35

Classification: 16T10, 16T15, 05B35, 05C31
Keywords: coalgebra, bialgebra, Tutte polynomial, dichromatic polynomial, Las Vergnas polynomial, Bollobás–Riordan polynomial, arithmetic Tutte polynomial, minors system, convolution formula

Author's affiliations:

Dupont, Clément ¹; Fink, Alex ²; Moci, Luca ³

¹ IMAG Université de Montpellier CNRS Montpellier France
² School of Mathematical Sciences Queen Mary University of London UK
³ IMJ-PRG Université Paris-Diderot Paris 7 Paris France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Dupont, Clément; Fink, Alex; Moci, Luca. Universal Tutte characters via combinatorial coalgebras. Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 603-651. doi : 10.5802/alco.35. https://alco.centre-mersenne.org/articles/10.5802/alco.35/

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