The number of bar{3}bar{1}542-avoiding permutations

TL;DR

This paper proves a conjecture linking barred pattern-avoiding permutations to a known sequence, providing a bijective proof and explicit enumeration based on left-to-right maxima and Stirling partition numbers.

Contribution

It confirms a conjecture by Lara Pudwell and establishes a bijective enumeration formula for barred pattern-avoiding permutations.

Findings

Permutations avoiding barred pattern 3̅1̅542 are counted by OEIS sequence A047970.

Derived an explicit formula involving Stirling partition numbers and left-to-right maxima.

Provided a bijective proof connecting pattern avoidance with combinatorial parameters.

Abstract

We confirm a conjecture of Lara Pudwell and show that permutations of [n] that avoid the barred pattern bar{3}bar{1}542 are counted by OEIS sequence A047970. In fact, we show bijectively that the number of bar{3}bar{1}542 avoiders of length n with j+k left-to-right maxima, of which j initiate a descent in the permutation and k do not, is {n}-choose-{k} j! StirlingPartition{n-j-k}{j}, where StirlingPartition{n}{j} is the Stirling partition number.