The expected degree of noninvertibility of compositions of functions and a related combinatorial identity

TL;DR

This paper derives an exact formula for the expected degree of noninvertibility in compositions of functions and introduces a new combinatorial identity related to Stirling numbers.

Contribution

It provides a precise formula for the expected noninvertibility of function compositions and uncovers a novel combinatorial identity involving Stirling transforms.

Findings

Exact formula for expected noninvertibility of composed functions

New combinatorial identity involving Stirling numbers

Generalization of the degree of noninvertibility concept

Abstract

Recently, Defant and Propp [2020] defined the degree of noninvertibility of a function $f\colon X\to Y$ between two finite nonempty sets by $\text{deg}(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. We obtain an exact formula for the expected degree of noninvertibility of the composition of $t$ functions for every $t\in \mathbb{N}$. An equivalent formulation for the definition of the degree of noninvertibility is then the starting point for a generalization yielding a seemingly new combinatorial identity involving the Stirling transform of the signed Stirling numbers of the first kind.