A new generation tree for permutations, preserving the number of fixed points

Philippe Duchon (1) , Romaric Duvignau (1)

1 LaBRI - Laboratoire Bordelais de Recherche en Informatique

We describe a new uniform generation tree for permutations with the specific property that, for most permutations, all of their descendants in the generation tree have the same number of fixed points. Our tree is optimal for the number of permutations having this property. We then use this tree to describe a new random generation algorithm for derangements, using an expected n+O(1) calls to a random number generator. Another application is a combinatorial algorithm for exact sampling from the Poisson distribution with parameter 1.

Mots clés: random generation; derangements; permutations; fixed points; generation tree

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Philippe Duchon, Romaric Duvignau. A new generation tree for permutations, preserving the number of fixed points. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) , Jun 2014, Chicago, United States. pp.679-690, ⟨10.46298/dmtcs.2433⟩. ⟨hal-01015604⟩