Cyclic sieving for a class of rectangular domino tableaux

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.

💡 Research Summary

The paper investigates domino tableaux—fillings of Young diagrams with domino-shaped tiles labeled by the integers 1,…,n such that rows and columns increase strictly—and establishes a new instance of the cyclic sieving phenomenon (CSP) for these objects. The authors first focus on the rectangular shape 2 × n. They observe that any domino tiling of a 2 × n rectangle corresponds to a Fibonacci number Fₙ, but a single tiling may admit several labelings when it contains a “k‑stack” (k consecutive horizontal domino pairs) with k ≥ 2. Lemma 3.3 shows that the number of labelings of a k‑stack equals the Catalan number Cₖ. Crucially, Lemma 3.4 proves that a domino tableau of shape 2 × n is uniquely determined by the set of labels on its bottom horizontal dominoes. Using this fact, the authors construct a bijection Φ between DT(2 × n) and the family of ⌊n/2⌋‑element subsets of