ALGEBRAIC COMBINATORICS

Universal Tutte characters via combinatorial coalgebras
Algebraic Combinatorics, Volume 1 (2018) no. 5, pp. 603-651.

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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/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
@article{ALCO_2018__1_5_603_0,
author = {Dupont, Cl\'ement and Fink, Alex and Moci, Luca},
title = {Universal {Tutte} characters via combinatorial coalgebras},
journal = {Algebraic Combinatorics},
pages = {603--651},
publisher = {MathOA foundation},
volume = {1},
number = {5},
year = {2018},
doi = {10.5802/alco.35},
zbl = {06987760},
mrnumber = {3887405},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.35/}
}
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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