Séminaire Lotharingien de Combinatoire, B61Ae (2009), 33 pp.

Alexander D. Scott and Alan D. Sokal

Some Variants of the Exponential Formula, with Application to the Multivariate Tutte Polynomial (alias Potts Model)

Abstract. We prove some variants of the exponential formula and apply them to the multivariate Tutte polynomials (also known as Potts-model partition functions) of graphs. We also prove some further identities for the multivariate Tutte polynomial, which generalize an identity for counting connected graphs found by Riordan, Nijenhuis, Wilf and Kreweras and in more general form by Leroux and Gessel, and an identity for the inversion enumerator of trees found by Mallows, Riordan and Kreweras. Finally, we prove a generalization of Möbius inversion on the partition lattice.

Received: February 17, 2009. Accepted: July 14, 2009. Final Version: July 15, 2009.

The following versions are available: