Counting Specific Classes of Relations Regarding Fixed Points and Reflexive Points

TL;DR

This paper investigates the probability that a randomly chosen relation or function on an n-element set has a fixed point (for functions) or a reflexive point (for relations) across multiple relational classes, including functions, partial functions, total and arbitrary relations, permutations, involutions, and idempotent/transitive partial functions. It develops closed-form counting formulas for the number of objects with k fixed/reflexive points, derives associated probabilities, and analyzes their asymptotic behavior as n grows, with several results showing convergence toward 1 - 1/e in relevant cases. The authors also validate the theory with RelView experiments and provide detailed observations on the distribution of fixed points across classes, including derangements and kernel-related questions. The work advances the understanding of fixed-point phenomena in finite relational structures and identifies several open problems and directions for future research, such as kernels and injective mappings with fixed points. The findings have implications for random-relations theory and combinatorial enumeration in computer science and discrete mathematics.

Abstract

Given a finite and non-empty set $X$ and randomly selected specific functions and relations on $X$, we investigate the existence and non-existence of fixed points and reflexive points, respectively. First, we consider the class of functions, weaken it to the classes of partial functions, total relations and general relations and also strengthen it to the class of permutations. Then we investigate the class of involutions and the subclass of proper involutions. Finally, we treat idempotent functions, partial idempotent functions and related concepts. We count relations, calculate corresponding probabilities and also calculate the limiting values of the latter in case that the cardinality of $X$ tends to infinity. All these results have been motivated and also supported by numerous experiments performed with the RelView tool.

Figures (10)

• Figure 1: Number of functions and functions with fixed points
• Figure 2: Number of permutations and permutations with fixed points
• Figure 3: Number of partial functions and partial functions with fixed points
• Figure 4: Number of total relations on an $n$-set with $k$ reflexive points
• Figure 5: Subfactorials $!n$ for $0 \leq n \leq 10$

Theorems & Definitions (27)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Theorem 3.1
• Corollary 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4