Promotion and Evacuation on Standard Young Tableaux of Rectangle and Staircase Shape

Abstract

(Dual-)promotion and (dual-)evacuation are bijections on $SYT(\lambda)$ for any partition $\lambda$. Let $c^r$ denote the rectangular partition $(c,\ldots,c)$ of height $r$, and let $sc_k$ ($k>2$) denote the staircase partition $(k,k-1,\ldots,1)$. We demonstrate a promotion- and evacuation-preserving embedding of $SYT(sc_k)$ into $SYT(k^{k+1})$. We hope that this result, together with results by Rhoades on rectangular tableaux, can help to demonstrate the cyclic sieving phenomenon of promotion action on $SYT(sc_k)$.