# Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections

## TL;DR

This paper explores the relationship between permutation patterns and forbidden configurations in restricted elections, introducing enriched group-separable elections and establishing enumeration formulas, while also generalizing prior results to configurations with three voters.

## Contribution

It introduces enriched group-separable elections, provides their enumeration, and generalizes permutation pattern results to three-voter forbidden configurations.

## Key findings

- Defined enriched group-separable elections and derived their enumeration formulas.
- Established a connection between permutation patterns and forbidden configurations with three voters.
- Suggested a new approach for bounding restricted elections with complex forbidden configurations.

## Abstract

We investigate the connections between patterns in permutations and forbidden configurations in restricted elections, first discovered by Lackner and Lackner, in order to enhance the approach initiated by the two mentioned authors. More specifically, our achievements are essentially two. First, we define a new type of domain restriction, called enriched group-separable. Enriched group-separable elections are a subset of group-separable elections, which describe a special, still natural, situation that can arise in the context of group-separability. The exact enumeration of group-separable elections has been very recently determined by Karpov. Here we give a recursive characterization for enriched group-separable elections, from which we are able to find a recurrence relation and a closed formula expressing their number. Our second achievement is a generalization of a result of Lackner and Lackner, concerning the connection between permutation patterns and forbidden configurations with 3 voters. Our result relates forbidden configurations with the strong order on pairs of permutations, a notion which is still largely undeveloped, and suggests a potential approach for the determination of upper bounds for restricted elections whose forbidden configurations contains at least one configuration with 3 voters.