The Number of Classes of Choice Functions under Permutation Equivalence

A choice function f of an n\Gammaset X is a function whose domain is the power set P(X), and whose range is X, such that f(A) 2 A for each A ` X. If k is a fixed positive integer, k n, by a k-restricted choice function we mean the restriction of some choice function to the collection of k\Gammasubsets of X. The symmetric group SX acts, by natural extension on the respective collections C(X) of choice functions, and C k (X) of k\Gammarestricted choice functions of X. In this paper we address the problem of finding the number of orbits in the two actions, and give closed form formulas for the respective numbers. This work was supported in part by NSA grant MSFPF-95-G-091, and by CCIS -- Univ. of Nebraska - Lincoln y This work was supported in part by CCIS -- Univ. of Nebraska - Lincoln 1 Introduction A choice function f on an n\Gammaset X is defined to be a function whose domain is the collection of all subsets of X, and whose range is X, such that for each subset A of X, f(A)..