Abstract
Given a finite set $X$ and a function $f:X\to X$, we define the \emph{degree of noninvertibility} of $f$ to be $\displaystyle\deg(f)=\frac{1}{|X|}\sum_{x\in X}|f^{-1}(f(x))|$. This is a natural measure of how far the function $f$ is from being bijective. We compute the degrees of noninvertibility of some specific discrete dynamical systems, including the Carolina solitaire map, iterates of the bubble sort map acting on permutations, bubble sort acting on multiset permutations, and a map that we call "nibble sort." We also obtain estimates for the degrees of noninvertibility of West's stack-sorting map and the Bulgarian solitaire map. We then turn our attention to arbitrary functions and their iterates. In order to compare the degree of noninvertibility of an arbitrary function $f:X\to X$ with that of its iterate $f^k$, we prove that \[\max_{\substack{f:X\to X\\ |X|=n}}\frac{\deg(f^k)}{\deg(f)^\gamma}=\Theta(n^{1-1/2^{k-1}})\] for every real number $\gamma\geq 2-1/2^{k-1}$. We end with several conjectures and open problems.
Highlights
Functions between finite sets play a fundamental role in classical enumerative and algebraic combinatorics, as they are often used to transfer combinatorial information from one set of objects to another; such functions are typically bijective
A different branch of combinatorics studies the iteration of maps f : X → X, where X is some finite set; this is the field of combinatorial dynamics
In the spirit of combinatorial dynamics, but with relevance to enumerative combinatorics, we introduce and explore a natural way of measuring how far a map is from being injective
Summary
Functions between finite sets play a fundamental role in classical enumerative and algebraic combinatorics, as they are often used to transfer combinatorial information from one set of objects to another; such functions are typically bijective. We state the definition for arbitrary maps between finite sets, but we will focus our attention in most of the article on discrete dynamical systems In this case, we are measuring how far the function is from being bijective. Given sets X, Y, Y , Z of the same cardinality, we say that two functions f : X → Y and g : Y → Z are pseudoconjugate if there exist bijections h : Y → Z and h : X → Y such that h ◦ f = g ◦ h It is immediate from the previous proposition that two pseudoconjugate maps must have the same degree
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