# A New Quantity Counted by OEIS Sequence A006012

## TL;DR

This paper proves a conjecture linking a specific recursive sequence to the count of permutations avoiding four complex patterns, enhancing understanding of pattern-avoiding permutations.

## Contribution

It establishes a proof that the recursive sequence from OEIS sequence A006012 counts permutations avoiding four particular generalized patterns.

## Key findings

- Confirmed the sequence counts pattern-avoiding permutations
- Connected recursive sequence to permutation pattern avoidance
- Enhanced understanding of permutation pattern enumeration

## Abstract

We prove an existing conjecture that the sequence defined recursively by $a_1=1, a_2=2, a_n=4a_{n-1}-2a_{n-2}$ counts the number of length-$n$ permutations avoiding the four generalized permutation patterns 1-32-4, 1-42-3, 2-31-4, and 2-41-3.