Evacuation and a geometric construction for Fibonacci tableaux

Abstract

Tableaux have long been used to study combinatorial properties of permutations and multiset permutations. Discovered independently by Robinson and Schensted and generalized by Knuth, the Robinson–Schensted correspondence has provided a fundamental tool for relating permutations to tableaux. In 1963, Schützenberger defined a process called evacuation on standard tableaux which gives a relationship between the pairs of tableaux ( P , Q ) resulting from the Schensted correspondence for a permutation and both the reverse and the complement of that permutation. Viennot gave a geometric construction for the Schensted correspondence and Fomin described a generalization of the correspondence which provides a bijection between permutations and pairs of chains in Young's lattice. In 1975, Stanley defined a Fibonacci lattice and in 1988 he introduced the idea of a differential poset. Roby gave an insertion algorithm, analogous to the Schensted correspondence, for mapping a permutation to a pair of Fibonacci tableaux. The main results of this paper are to give an evacuation algorithm for the Fibonacci tableaux that is analogous to the evacuation algorithm on Young tableaux and to describe a geometric construction for the Fibonacci tableaux that is similar to Viennot's geometric construction for Young tableaux.