Sterling stirling play

Abstract

In this paper we analyze a recently proposed impartial combinatorial ruleset that is played on a permutation of the set $$\left[ n\right] $$ . We call this ruleset Stirling Shave. A procedure utilizing the ordinal sum operation is given to determine the nim value of a given normal play position. Additionally, we enumerate the number of permutations of $$\left[ n\right] $$ which are $$\mathcal {P}$$ -positions. The formula given involves the Stirling numbers of the first-kind. We also give a complete analysis of the Misere version of Stirling Shave using Conway’s genus theory. An interesting by-product of this analysis is insight into how the ordinal sum operation behaves in Misere Play.