Cyclic sieving for a class of rectangular domino tableaux

TL;DR

This paper explores the cyclic sieving phenomenon in domino tableaux, providing enumeration results, proving a new CSP for 2-by-n shapes, and conjecturing a broader CSP for rectangular shapes, linking to Fibonacci and Catalan numbers.

Contribution

It introduces a new CSP for 2-by-n domino tableaux, extends enumeration to rectangular shapes, and proposes a conjecture for a general CSP involving these objects.

Findings

Enumeration of 2-by-n domino tableaux

Proof of a new cyclic sieving phenomenon for these tableaux

Connections to Fibonacci and Catalan number identities

Abstract

The cyclic sieving phenomenon (CSP) provides valuable data about symmetry classes of cyclic actions, and has applications to representation theory. In this paper, we enumerate domino tableaux of shape 2-by-n, and use this result to prove a new CSP on these objects. We then enumerate the rectangular domino tableaux of any dimensions, and conjecture a more general CSP on rectangular domino tableaux. As a consequence of the enumerative results, we obtain several identities involving Fibonacci and Catalan numbers.