9 research outputs found
Two first-order logics of permutations
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 (Theory Of Two Orders) and
(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 , the set of -stack sortable permutations in the sense of West
is expressible in , and that a logical sentence describing this
set can be obtained automatically. Previously, descriptions of this set were
only known for . Next, we characterize permutation classes inside
which it is possible to express in that some given points form
a cycle. Lastly, we show that sets of permutations that can be described both
in and are in some sense trivial. This gives a
mathematical evidence that permutations-as-bijections and permutations-as-words
are somewhat different objects.Comment: v2: minor changes, following a referee repor
Twin-width and permutations
Inspired by a width invariant defined on permutations by Guillemot and Marx,
the twin-width invariant has been recently introduced by Bonnet, Kim,
Thomass\'e, and Watrigant. We prove that a class of binary relational
structures (that is: edge-colored partially directed graphs) has bounded
twin-width if and only if it is a first-order transduction of a~proper
permutation class. As a by-product, it shows that every class with bounded
twin-width contains at most pairwise non-isomorphic -vertex
graphs
Prolific structures in combinatorial classes
Under what circumstances might every extension of a combinatorial structure contain more copies of another one than the original did? This property, which we call prolificity, holds universally in some cases (e.g., finite linear orders) and only trivially in others (e.g., permutations). Integer compositions, or equivalently layered permutations, provide a middle ground. In that setting, there are prolific compositions for a given pattern if and only if that pattern begins and ends with 1. For each pattern, there are methods that identify conditions that allow classification of the texts that are prolific for the pattern. This notion is also extendable to other combinatorial classes. In the context of permutations that are sums of cycles we can also establish minimal elements for the set of prolific permutations based on the bijective correspondence between these permutations and compositions, with a slightly different containment order. We also take a brief step into the more general world of permutations that avoid the pattern 321 and attempt to establish some preliminary results
Two first-order logics of permutations
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 TOTO (Theory Of Two Orders) and TOOB (Theory Of One Bijection) respectively. We consider various expressibility questions in these theories.
Our main results go in three different directions. First, we prove that, for all k ≥ 1, the set of k-stack sortable permutations in the sense of We s t is expressible in TOTO, and that a logical sentence describing this set can be obtained automatically. Previously, descriptions of this set were only known for k ≤ 3. Next, we characterize permutation classes inside which it is possible to express in TOTO that some given points form a cycle. Lastly, we show that sets of permutations that can be described both in TOOB and TOTO are in some sense trivial. This gives a mathematical evidence that permutations-as-bijections and permutations-as-words are somewhat different objects