On formal fractions associated with the symmetric groups

AbstractWith any permutation o in Sn, the symmetric group on {1, 2,…, n}, a fraction in the variables x0 , x1 ,…, xn is associated in such a way that the length of the bar between xk−1 and x1, … xn is σ(k). This fraction is evaluated in the free commutative group generated by x0 , x1, …, xn. The evaluated value of this fraction is denoted by Vn(σ). Two permutations σ, τ ϵ Sn are defined to be equivalent if Vn(σ) = Vn(τ). The cardinality of the largest class under this equivalence relation is denoted by Mn. The values of Mn are investigated in this paper. The result is M2n = E2n and M2n-1 = [22n(22n − 1)/2n]B2n, where E2n and B2n denote Euler and Bernoulli numbers. The asymptotic behavior of Mn is also studied. The result is limn→∞(Mn/nMn−1) = 2/π