Two first-order logics of permutations

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 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 direction. First, we prove that, for all k ≥ 1, the set of k-stack sortable permutations in the sense of West 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.

Counting 3-stack-sortable permutations

“…For example, if π ∈ S n , then leg(π) = n+1 if and only if π avoids 231. The legal spaces of 145326 are (0, 1), (1,2), (4,5), (5,6), (6,7), so leg(145326) = 5. Imagine adding a new point somewhere to the left of all points in the plot of a permutation π.…”

“…The following illustration shows that s(4162) = 1426. 1 We say a permutation π is t-stack-sortable if s t (π) is an increasing permutation, where s t denotes the t-fold iterate of s. Let W t (n) be the set of t-stack-sortable permutations in S n , and let W t (n, k) = {π ∈ W t (n) : des(π) = k} and W t (n, k, p) = {π ∈ W t (n, k) : peak(π) = p}. Let W t (n) = |W t (n)|, W t (n, k) = |W t (n, k)|, and W t (n, k, p) = |W t (n, k, p)|.…”

“…Combinatorial proofs of Zeilberger's theorem emerged later in [19,32,33,36]. Some authors have investigated the enumeration of 2-stack-sortable permutations according to various statistics 1 The permutations that we call t-stack-sortable are often called "West t-stack-sortable," but we have dropped the word "West" for brevity. The term "t-stack-sortable" is sometimes used to refer to permutations that can be sorted using Knuth's (nondeterministic) algorithm with t stacks in series; that use of the term is different from ours.…”

“…There is very little known about t-stack-sortable permutations when t ≥ 3 is fixed.Úlfarsson [44] characterized 3-stack-sortable permutations in terms of new "decorated patterns," but the characterization is too unwieldy to yield any additional information. The recent paper [1] shows that for every t ≥ 1, the set of t-stack-sortable permutations can be described by a sentence in a first-order logical theory that the authors call TOTO. The paper [17] also investigates t-stacksortable permutations when t = n − r for some fixed r (focusing on the case in which r = 4).…”

Logical limit laws for layered permutations and related structures

“…We now discuss some connections to previous work. First-order properties of finite permutations (when viewed as pairs of linear orders) were studied in [1]. There the existence of a zero-one law was disproven, and it was asked whether or not permutations could admit a logical limit law; the answer to this turns out to be negative as well (as shown in [8]).…”