The Distribution of the Combined Length of Spanned Cycles in a Random Permutation

For a random permutation π on {1, 2, . . . , n} for fixed n, and forM ⊆ {1, 2, . . . , n}, we analyse the distribution of the combined length L = L(π,M) of all cycles of π that contain at least one element of M . We give a simple, explicit formula for the probability of every possible value for L (backed by three proofs of distinct flavours), as well as closed-form formulae for its expectation and variance, showing that less than 1 |M |+1 of the elements 1, . . . , n are expected to be contained in cycles of π that are disjoint from M , with low probability for a large deviation from this fraction. We furthermore give a simple explicit formula for all rising-factorial moments of L. These results are applicable to the study of manipulation in matching markets. Given a random permutation on a fixed finite set of objects, we are interested in the combined length of all cycles of the permutation that intersect a given subset of these objects (or alternatively, of all cycles of the permutation that are disjoint from this given subset); when the subset consists of a single point, this quantity is simply the wellstudied length of the cycle that contains that point (for an analysis of this special case, see e.g. [1, p. 24]). The question of the distribution of this quantity arises during analysis of the limits of manipulation in matching markets; for more information, the interested reader is referred to [2]. We commence by precisely defining the problem at hand. Definition 1. Throughout this paper, we use the following standard notation. • P , {1, 2, 3, . . .} [5]; throughout this paper, k, `, ̃̀,m, m̃, n, ñ denote elements of P. • [n] , {1, 2, . . . , n} [5]. • [m,n] , {m,m+ 1, . . . , n} [5] . ([m,n] = ∅ if m > n.) • SN , {π : N 7→ N | π is a bijection} — the set of permutations of a set N . • Sn , S[n]. • Furthermore, we denote n , n · (n+ 1) · . . . · (n+ k − 1) = (n+k−1)! (n−1)! . ∗Einstein Institute of Mathematics and Center for the Study of Rationality, Hebrew University of Jerusalem, Israel. Email : [email protected].