Tableaux and Rank-selection in Fibonacci Lattices

Box-shaped tableaux and Jordan-Hölder sequences are used to solve a problem of R. Stanley on the enumeration of maximal chains in rank-selected subposets of the two Fibonacci lattices Fib(r) and Z(r),r≥1. Several tools for the analysis of box-tableaux are developed and bijections between tableaux and Jordan-Hölder sequences are described.