Geodesic
Cumulants of the q-semicircular law, Tutte polynomials, and heaps
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012).

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The q-semicircular law as introduced by Bożejko and Speicher interpolates between the Gaussian law and the semicircular law, and its moments have a combinatorial interpretation in terms of matchings and crossings. We prove that the cumulants of this law are, up to some factor, polynomials in q with nonnegative coefficients. This is done by showing that they are obtained by an enumeration of connected matchings, weighted by the evaluation at (1,q) of a Tutte polynomial. The two particular cases q=0 and q=2 have also alternative proofs, related with the fact that these particular evaluation of the Tutte polynomials count some orientations on graphs. Our methods also give a combinatorial model for the cumulants of the free Poisson law. 
@article{DMTCS_2012_special_263_a60,
     author = {Josuat-Verg\`es, Matthieu},
     title = {Cumulants of the q-semicircular law, {Tutte} polynomials, and heaps},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012)},
     year = {2012},
     doi = {10.46298/dmtcs.3074},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3074/}
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Josuat-Vergès, Matthieu. Cumulants of the q-semicircular law, Tutte polynomials, and heaps. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012), DMTCS Proceedings vol. AR, 24th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2012) (2012). doi : 10.46298/dmtcs.3074. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3074/

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