The permutation classes Av(1234,2341) and Av(1243,2314)

TL;DR

This paper analyzes the structure and enumerates two specific permutation classes defined by forbidden patterns, using graph decomposition and the kernel method to derive their algebraic generating functions.

Contribution

It introduces a novel application of the 'adding a slice' approach and kernel method to enumerate these permutation classes, providing explicit algebraic generating functions.

Findings

Derived algebraic generating functions for both classes

Established structural decompositions using Hasse graphs

Extended enumeration techniques to permutation classes

Abstract

We investigate the structure of the two permutation classes defined by the sets of forbidden patterns {1234,2341} and {1243,2314}. By considering how the Hasse graphs of permutations in these classes can be built from a sequence of rooted source graphs, we determine their algebraic generating functions. Our approach is similar to that of "adding a slice", used previously to enumerate various classes of polyominoes and other combinatorial structures. To solve the relevant functional equations, we make extensive use of the kernel method.