$k$-Ribbon Fibonacci Tableaux

TL;DR

This paper generalizes ribbon tableaux to the Fibonacci lattice, introduces an insertion algorithm for $k$-colored permutations, and explores properties like color-to-spin, evacuation, and Knuth relations in this new setting.

Contribution

It extends ribbon tableau theory to Fibonacci lattices, providing new algorithms and relations for $k$-colored permutations and $k$-ribbon Fibonacci tableaux.

Findings

Introduces $k$-ribbon Fibonacci tableaux on the Fibonacci lattice.

Develops an insertion algorithm linking $k$-colored permutations to tableaux.

Establishes a color-to-spin property and analogues of Knuth relations.

Abstract

We extend the notion of $k$-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes $k$-colored permutations to pairs of $k$-ribbon Fibonacci tableaux of the same shape, and we demonstrate a color-to-spin property, similar to that described by Shimozono and White for ribbon tableaux. We give an evacuation algorithm which relates the pair of $k$-ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of $k$-ribbon Fibonacci tableaux obtained using Fomin's growth diagrams. In addition, we present an analogue of Knuth relations for $k$-colored permutations and $k$-ribbon Fibonacci tableaux.