Restricted Tutte polynomials for some periodic oriented forests on infinite square lattice

Yvan Le Borgne (1) , Henri Derycke (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

For any finite graph, the Tutte polynomial is the generating function of spanning trees counted by their numbers of active external, respectively internal, edges. We consider two restrictions of this definition, either summing over a subset of spanning trees or counting only the activities in a subset of edges. Adding to the (infinite) square lattice one projective vertex in a (rational) direction θ, we define the restricted Tutte polynomial T θ,W×H (q, t) summing over some periodic spanning forests of period W × H and considering only activities on edges of the fundamental domain. Those polynomials are symmetric in q and t by self-duality of square lattice. Our main result is a family of bijections indexed by a finite number of θ proving that (T θ,W×H (q, 1)) θ does not depend on θ. Auto-duality preserving the number of trees per period and their common slope, we obtain refinements (T θ,W×H (w, z; q, t)) θ still symmetric in q and t. This work is motivated by results on the sandpile model presented in Section 3 at the end of this document. We focus first on the combinatorial result on an analogue of Tutte polynomial for the infinite square lattice. 1 Tutte polynomial For any finite connected graph G = (V, E), the Tutte polynomial [Tut54] is a classical graph invariant defined as follow: T G (q, t) := ∑ T q ext(T) t int(T) where T runs over the set T G of spanning trees of G and ext(T), respectively int(T), is the soon defined external, respectively internal, (Tutte) activity. Tutte activities depends on a arbitrary permutation of the edges of G also denoted as a total order < E. There are |E|! * [e-mail] and [e-mail] were supported by ENDEAR project.

Mots clés: Sandpile model; Square lattice; Tutte polynomial; Symmetric q; t-numbers

Domaines: Combinatoire [math.CO]

Yvan Le Borgne, Henri Derycke. Restricted Tutte polynomials for some periodic oriented forests on infinite square lattice. International conference series on Formal Power Series and Algebraic Combinatorics (FPSAC), Jul 2019, Ljubljana, Slovenia. ⟨hal-02199642⟩