The Expected Degree of Noninvertibility of Compositions of Functions

Recently, Defant and Propp defined the degree of noninvertibility of a function f: X → Y between two finite nonempty sets by deg(f) =¹ ∑ |X| x∈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 ∈ N. Subsequently, we use the expected value to quantify a strengthening of a sort of a submultiplicativity property of the degree of noninvertibil-ity. Finally, we generalize an equivalent formulation of the degree of noninvertibility and obtain a combinatorial identity involving the Stirling numbers of the first and second kind.