A Symmetric Function Resolution of the Number of Permutations With Respect to Block-Stable Elements

Abstract.

We consider a question raised by Suhov and Voice from quantum information theory and quantum computing. An element of a partition of {1, ..., n} is said to be block-stable for $\pi \in \mathfrak{S}_n$ if it is not moved to another block under the action of π. The problem concerns the determination of the generating series $S_{k_1 , \ldots k_r } (u)$ for elements of $\mathfrak{S}_n$ with respect to the number of block-stable elements of a canonical partition of a finite n-set, with block sizes k₁, ..., k _r , in terms of the moment (power) sums p _q (k₁, ..., k _r ). We also consider the limit $\lim _{n,r \to \infty } ( - 1)^n S_{k_1 , \ldots k_r } (1 - r)/r^n$ subject to the condition that $\lim _{n,r \to \infty } p_q (k_1 , \ldots k_r )/r$ exists for q = 1, 2,....