Some Games related to Permutation Group Statistics and their type \(B\) Analogues

A coin tossing game — with a biased coin with probability \(q\) for the tail — for \(n\) persons was discussed by Moritz and Williams in \(1987\), in which the probability for players to go out in a prescribed order is described by what is commonly called the “major index” (due to Major MacMahon), which is an important statistic for the permutation group \(\mathcal{S}_n\). We first describe a variation on this game, for which the same question is answered in terms of the better known statistic “length function” in the sense of Coxeter group theory (also called “inversion number” in combinatorial literature). This entails a new bijection implying the old equality (due to MacMahon) of the generating functions for these two statistics.

Next we describe a game for \(2n\) persons where the ‘same’ question is answered in terms of the Coxeter length function for the reflection group of type \(B_n\). We conclude with some miscellaneous results and questions.