Generating Permutations with Restricted Containers

TL;DR

This paper introduces a generalized framework called $\\mathcal{C}$-machines for generating permutations, providing methods to derive and solve functional equations, and analyzing properties like rationality and algebraicity of their generating functions.

Contribution

It develops a unified approach to study permutation classes generated by $\\mathcal{C}$-machines, including solving their functional equations and exploring their enumerative properties.

Findings

Functional equations can be solved using kernel method or guessing.

Some classes have rational or algebraic generating functions.

Certain classes lack D-finite generating functions despite extensive enumeration.

Abstract

We investigate a generalization of stacks that we call $\mathcal{C}$-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that $\mathcal{C}$-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by $\mathcal{C}$-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions.