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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.

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