Counting the Number of Multi-player Partizan Cold Games Born by Day d

Abstract

In combinatorial games, few results are known about the overall structure of multi-player games. Recently, it has been proved that multi-player games lasting at most $d$ moves, also known as the games born by day $d$, forms a completely distributive lattice with respect to every partial order relation $\le _C$, where $C$ is an arbitrary coalition of players. In this paper, we continue our investigation concerning multi-player games and, using the strings of multi-player Hackenbush in order to construct multi-player cold games, we calculate lower and upper bounds on $S_n[d]$ equal to the number of $n$-player partizan cold games born by day $d$. In particular, we prove that if $n$ is fixed, then $S_n[d] \in \Theta (2{nd})$, but in order to establish the exact value of $S_n[d]$ further efforts are necessary.