Alternating permutations containing the pattern 123 or 321 exactly once

TL;DR

This paper counts alternating permutations that contain the patterns 123 or 321 exactly once, providing explicit formulas for their enumeration based on permutation length.

Contribution

It offers the first explicit formulas for counting alternating permutations with exactly one occurrence of patterns 123 or 321.

Findings

Derived formulas for a_(2m)(123) and a_(2m)(321).

Derived formulas for a_(2m+1)(123) and a_(2m+1)(321).

Confirmed the counts for permutations containing these patterns exactly once.

Abstract

Inspired by a recent note of Zeilberger (arXiv:1110.4379), Alejandro Morales asked whether one can count alternating (i.e., up-down) permutations that contain the pattern 123 or 321 exactly once. In this note we answer the question in the affirmative; in particular, we show that for m > 1, a_(2m)(123) = 10 (2m)!/((m - 2)! (m + 3)!), a_(2m)(321) = 4(m - 2) (2m + 3)!/((m + 1)! (m + 4)!), and a_(2m + 1)(123) = a_(2m + 1)(321) = 3(3m + 4)(m - 1) (2m + 2)!/((m + 1)! (m + 4)!) where a_n(p) is the number of alternating permutations of length n containing the pattern p exactly once.