Multiple Hook Removing Game Whose Starting Position is a Rectangular Young Diagram with the Unimodal Numbering

Abstract

We introduce a new impartial game, named Multiple Hook Removing Game (MHRG for short). We also determine the $\mathcal{G}$-values of some game positions (including the starting positions) in MHRG$(m,n)$, the MHRG whose starting position is the rectangular Young diagram of size $m\times n$ with the unimodal numbering. In addition, we prove that MHRG$(m,n)$ is isomorphic, as games, to MHRG$(m,n+1)$ (if $m\le n$ and $m+n$ is even), and give a relationship between MHRG$(n,n+1)$ (and MHRG$(n,n)$) and HRG$(S_n)$, the Hook Removing Game in terms of shifted Young diagrams.