Two first-order logics of permutations

TL;DR

This paper introduces two logical frameworks for analyzing finite permutations from different perspectives, demonstrating their expressibility limits and applications to permutation classes, revealing fundamental differences between the two views.

Contribution

It defines two first-order theories for permutations, proves the expressibility of k-stack sortable permutations, and characterizes classes where cycle points are definable, highlighting differences between permutation representations.

Findings

k-stack sortable permutations are expressible in TOTO for all k

Permutation classes with cycle-definability in TOTO are characterized

Permutations as bijections and as words have fundamentally different definability properties

Abstract

We consider two orthogonal points of view on finite permutations, seen as pairs of linear orders (corresponding to the usual one line representation of permutations as words) or seen as bijections (corresponding to the algebraic point of view). For each of them, we define a corresponding first-order logical theory, that we call $\mathsf{TOTO}$ (Theory Of Two Orders) and $\mathsf{TOOB}$ (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories. Our main results go in three different direction. First, we prove that, for all $k \ge 1$, the set of $k$-stack sortable permutations in the sense of West is expressible in $\mathsf{TOTO}$, and that a logical sentence describing this set can be obtained automatically. Previously, descriptions of this set were only known for $k \le 3$. Next, we characterize permutation classes inside which it is possible to express in $\mathsf{TOTO}$ that some given points form a cycle. Lastly, we show that sets of permutations that can be described both in $\mathsf{TOOB}$ and $\mathsf{TOTO}$ are in some sense trivial. This gives a mathematical evidence that permutations-as-bijections and permutations-as-words are somewhat different objects.