Combinatorial Specifications for Juxtapositions of Permutation Classes

Abstract

We show that, given a suitable combinatorial specification for a permutation class C, one can obtain a specification for the juxtaposition (on either side) of C with Av(21) or Av (12), and that if the enumeration for C is given by a rational or algebraic generating function, so is the enumeration for the juxtaposition. Furthermore this process can be iterated, thereby providing an effective method to enumerate any 'skinny' k × 1 grid class in which at most one cell is non-monotone, with a guarantee on the nature of the enumeration given the nature of the enumeration of the non-monotone cell.In this paper, we consider the effect on combinatorial specifications of juxtapositions. The juxtaposition of two permutations can be thought of as a special kind of merge, in which only the values of the two permutations can be interleaved arbitrarily, and not the positions. However, our primary motivation in studying juxtapositions is towards a generalisation in another direction, namely the broader study of grid classes.Our main result is as follows. Definitions are given in Section 2, but we note here that context-free specifications give rise to algebraic generating functions, while regular ones give rational generating functions.Theorem 1.1. Let C be a permutation class that admits a bottom-to-top combinatorial specification S , and let M ∈ {Av(21), Av(12)}.(i) If S tracks the rightmost entry, then there exists a combinatorial specification for C | M that tracks the rightmost entry.(ii) Similarly, if S tracks the leftmost entry, then there exists one for M | C that tracks the leftmost entry.(iii) If S tracks both the leftmost and the rightmost entries, then there exists specifications for C | M and M | C that do.(iv) If S is context-free (resp. regular), then the specifications for C | M and M | C are context-free (regular).Since the resulting specifications satisfy the same conditions as the theorem requires of C, the process can be iterated, thereby allowing us to generate combinatorial specifications for classes of the form given in Figure 1.Figure 1: The k × 1 grid classes in Corollary 1.2, in which C possesses a rightmost-and leftmost-entry tracking specification, and M i ∈ {Av(21), Av(12)} for all i = j.Corollary 1.2. If a permutation class C possesses a specification S that tracks the rightmost and leftmost entries, then so does any k × 1 grid class of the form given in Figure 1. In particular, if C possesses an algebraic or rational generating function, then so too does the k × 1 grid class.Permutation classes to which our method applies include any class C that contains only finitely many simple permutations, but it is not limited to this (for example, in Section 4 we present a suitable specification for Av(321)). By its nature, there is a parallel between our bottom-to-top specifications and the insertion encoding of Albert, Linton and Ruškuc [4], and we explore this further in Section 4.Prior to this paper, the most general result for k × 1 grids concerns the case where the class C above is itself also monoton...

Independence of permutation limits at infinitely many scales

“…Given a tiered permuton Γ v , the skinny monotone grid class Grid(v) contains every permutation that can be sampled from Γ v as described in Section 2.1 (that is, it consists of every possible Γ v -random permutation). For more on skinny grid classes, see [4,Chapter 3] and [7]. We can assemble a sequence of permutations (η j ) j∈N so that, for every skinny monotone grid class Grid(v), there is a scale f v such that (η j ) converges at scale f v to Γ v .…”

Independence of permutation limits at infinitely many scales

“…Given a tiered permuton Γ v , the skinny monotone grid class Grid(v) contains every permutation that can be sampled from Γ v as described in Section 2.1 (that is, it consists of every possible Γ vrandom permutation). For more on skinny grid classes, see [2, Chapter 3] and [5].…”

Labelled Well-Quasi-Order in Juxtapositions of Permutation Classes

“…The juxtaposition of permutations was first introduced in Atkinson's foundational work [2], and has since been studied in terms of enumeration (see, for example, [8]) since it represents a natural yet non-trivial way to combine two permutation classes. Indeed, juxtapositions are a special case of grid classes, which we define in the next section.…”