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Number of cycles in the graph of 312-avoiding permutations

Ehrenborg, Richard and Kitaev, Sergey and Steingrimsson, Einar (2014) Number of cycles in the graph of 312-avoiding permutations. In: Conference on Formal Power Series & Algebraic Combinatorics, 2014-06-29 - 2014-07-03.

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The graph of overlapping permutations is defined in a way analogous to the De Bruijn graph on strings of symbols. However, instead of requiring the tail of one permutation to equal the head of another for them to be connected by an edge, we require that the head and tail in question have their letters appear in the same order of size. We give a formula for the number of cycles of length d in the subgraph of overlapping 312-avoiding permutations. Using this we also give a refinement of the enumeration of 312-avoiding affine permutations. Le graphique de permutations qui se chevauchent est d&xE;9finie d'une mani&xE;8re analogue &xE;0 celle du graphe de De Bruijn sur des cha&xEEnes;de symboles. Cependant, au lieu d'exiger la queue d'une permutation d'&xE;9galer la t&xEAte;d'un autre pour qu'ils soient reli&xE;9s par un bord, nous avons besoin que la t&xEAte;et la queue en question ont leurs lettres apparaissent dans le m&xEAme;ordre de grandeur. Nous donnons une formule pour le nombre de cycles de longueur d dans le sous-graphe de chevauchement 312-&xE;9vitant permutations. L'utilisation de ce nous donnent &xE;9galement un raffinement de l'&xE;9num&xE;9ration de 312-&xE;9vitant permutations affines.