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

Authors:	D M Jackson M Yip

Institution:	(1) Department of Combinatorics and Optimization, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada

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,.... Received January 31, 2006

Keywords:	05E05 81V99
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