Alexander Burstein's Lovely Combinatorial Proof of John Noonan's Beautiful Formula that the number of n-permutations that contain the Pattern 321 Exactly Once Equals (3/n)(2n)!/((n-3)!(n+3)!)

TL;DR

This paper presents a combinatorial proof of John Noonan's formula counting permutations with exactly one occurrence of the pattern 321, providing a new elegant proof of a known enumeration result.

Contribution

Alexander Burstein offers a novel combinatorial proof of Noonan's formula, which was previously proven analytically, thus enriching the combinatorial understanding of pattern-avoiding permutations.

Findings

Confirmed the formula for permutations with exactly one 321 pattern

Provided a constructive combinatorial proof

Enhanced understanding of permutation pattern enumeration

Abstract

In 1996, my brilliant student John Noonan, discovered, and proved that there are 3(2n)!/(n(n+3)!(n-3)!) ways to line-up n people of different heights in such a way that out of the n(n-1)(n-2)/6 possible triples of people exactly one is such that the tallest stands (not necessarily immediately) in front of the second-tallest, who in turn, stands (not necessarily immediately) in front of the shortest. In that article, I promised a prize of 25 dollars for a nice combinatorial proof. Alex Burstein gave such a proof. On Oct. 14, 2011, I talked about Alex's lovely proof at the Howard U. math colloquium, and publicly presented the 25-dollar prize. The present note is the outcome. Congratulations Alex!