# Combinatorial specifications for juxtapositions of permutation classes

## TL;DR

This paper presents a method to derive combinatorial specifications for juxtaposed permutation classes, enabling enumeration of complex grid classes with predictable generating function types based on the original class.

## Contribution

It introduces a systematic way to obtain specifications for juxtaposed classes and shows that rational or algebraic enumeration properties are preserved, allowing effective enumeration of certain grid classes.

## Key findings

- Specification for juxtaposed classes can be derived from original class specifications.
- Enumeration properties like rationality or algebraicity are preserved under the process.
- Method can be iterated to enumerate complex grid classes with at most one non-monotone cell.

## Abstract

We show that, given a suitable combinatorial specification for a permutation class $\mathcal{C}$, one can obtain a specification for the juxtaposition (on either side) of $\mathcal{C}$ with Av(21) or Av(12), and that if the enumeration for $\mathcal{C}$ 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' $k\times 1$ 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.