Tetromino tilings and the Tutte polynomial

Jesper Lykke Jacobsen (1, 2)

1 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
2 SPhT - Service de Physique Théorique

We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each tile is assigned a weight that depends on its orientation and position on the lattice. For a particular choice of the weights, the generating function of tilings is shown to be the evaluation of the multivariate Tutte polynomial Z_G(Q,v) (known also to physicists as the partition function of the Q-state Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the edge weights v can take arbitrary values depending on the tile weights.

Mots clés: Tetromino tilings; Tutte polynomial; Potts model

Domaines: Combinatoire [math.CO]; Physique mathématique [math-ph]; Mécanique statistique [cond-mat.stat-mech]

Jesper Lykke Jacobsen. Tetromino tilings and the Tutte polynomial. Journal of Physics A: Mathematical and Theoretical, 2007, 40, pp.1439-1446. ⟨10.1088/1751-8113/40/7/002⟩. ⟨hal-00095986v2⟩