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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 *k*th 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.

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