Séminaire Lotharingien de Combinatoire, 78B.4 (2017), 12 pp.

Spencer Backman and Matthias Lenz

A Convolution Formula for Tutte Polynomials of Arithmetic Matroids and Other Combinatorial Structures

Abstract. We generalize the convolution formula for the Tutte polynomial of Kook-Reiner-Stanton and Etienne-Las Vergnas to a more general setting that includes both arithmetic matroids and delta-matroids. As corollaries, we obtain new proofs of two positivity results for pseudo-arithmetic matroids and a combinatorial interpretation of the arithmetic Tutte polynomial at infinitely many points in terms of arithmetic flows and colorings. We also exhibit connections with a decomposition of Dahmen-Micchelli spaces and lattice point counting in zonotopes. Subsequently, we investigate the following problem: given a representable arithmetic matroid, when is the arithmetic matroid obtained by taking the kth power of its multiplicity function again representable? Bajo-Burdick-Chmutov have recently discovered that Arithmetic matroids of this type arise in the study of CW complexes. We also solve a related problem for the Grassmannian.

Received: November 14, 2016. Accepted: February 17, 2017. Final version: April 1, 2017.

The following versions are available: