On the Number of Subpermutations with Fixed Orbit Size

Abstract

Consider an n-set, say \(X_n = {1,2,…,n}\). An exponential generating function and recurrence relation for the number of subpermutations of \(X_n\), whose orbits are of size at most \(k \geq 0\) are obtained. Similar results for
the number of nilpotent subpermutations of nilpotency index at most \(k\), and exactly \k\) are also given, along with arithmetic and asypmtotic formulas for these numbers. \(1\) \(2\)