Young–Fibonacci insertion, tableauhedron and Kostka numbers

This work is first concerned with some properties of the Young–Fibonacci insertion algorithm and its relation with Fominʹs growth diagrams. It also investigates a relation between the combinatorics of Young–Fibonacci tableaux and the study of Okadaʹs algebras associated to the Young–Fibonacci lattice. The original algorithm was introduced by Roby and we redefine it in such a way that both the insertion and recording tableaux of any permutation are conveniently interpreted as saturated chains in the Young–Fibonacci lattice. Using our conventions, we give a simpler proof of a property of Killpatrickʹs evacuation algorithm for Fibonacci tableaux. It also appears that this evacuation is no longer needed in making Robyʹs and Fominʹs constructions coincide. We provide the set of Young–Fibonacci tableaux of size n with a structure of graded poset called tableauhedron, induced by the weak order of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients of the transition matrix from the analogue of complete symmetric functions to analogue of the Schur functions in Okadaʹs algebra associated to the Young–Fibonacci lattice. We prove a similar result relating usual Kostka numbers with four partial orders on Young tableaux, studied by Melnikov and Taskin.