Curious cyclic sieving on increasing tableaux

TL;DR

This paper establishes a cyclic sieving phenomenon for a specific set of increasing tableaux under K-promotion, revealing a surprising connection to the q-hook formula for certain standard tableaux shapes.

Contribution

It introduces a new cyclic sieving result for increasing tableaux with a novel link to the q-hook formula for toothbrush-shaped tableaux.

Findings

Cyclic sieving holds for 3 x k increasing tableaux under K-promotion.

The sieving polynomial is derived from the q-hook formula for toothbrush shape tableaux.

K-promotion has order m, matching the maximum entry in the tableaux.

Abstract

We prove a cyclic sieving result for the set of $3 \times k$ packed increasing tableaux with maximum entry $m :=3+k$ under K-promotion. The "curiosity" is that the sieving polynomial arises from the $q$-hook formula for standard tableaux of "toothbrush shape" $(2^3, 1^{k-2})$ with $m+1$ boxes, whereas K-promotion here only has order $m$.