# Avoiding patterns and making the best choice

## TL;DR

This paper explores a variation of the secretary problem where candidate rankings avoid certain permutation patterns, developing strategies for optimal candidate selection under these constraints, especially for size three permutations.

## Contribution

It introduces a new approach to the secretary problem with pattern-avoiding permutations and characterizes optimal strategies for these cases, including a novel threshold-based method.

## Key findings

- Optimal strategies are fully described for size three pattern-avoiding permutations.
- The 'disappointment-free' (321-avoiding) case involves a new threshold-based strategy.
- The study extends the secretary problem to non-uniform rank distributions with pattern restrictions.

## Abstract

We study a variation of the game of best choice (also known as the secretary problem or game of googol) under an additional assumption that the ranks of interview candidates are restricted using permutation pattern-avoidance. We develop some general machinery for investigating interview orderings with a non-uniform rank distribution, and give a complete description of the optimal strategies for the pattern-avoiding games under each of the size three permutations. The optimal strategy for the "disappointment-free" (i.e. 321-avoiding) interviews has a form that seems to be new, involving thresholds based on value-saturated left-to-right maxima in the permutation.