Increasing Tableaux,Narayana Numbers and an Instance of the Cyclic Sieving Phenomenon

We give a counting formula for the set of rectangular increasing tableaux in terms of generalized Narayana numbers. We define small m–Schröder paths and give a bijection between the set of increasing rectangular tableaux and small m–Schröder paths, generalizing a result of Pechenik 4]. Using K–jeu de taquin promotion, we give a cyclic sieving phenomenon for the set of increasing hook tableaux.