Curious cyclic sieving on increasing tableaux

We prove a cyclic sieving result for the set of 3 × 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, 1) with m+1 boxes, whereas K-promotion here only has order m.

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