2022/08/02 by Egge, Eric S.
#05A05 #05A15 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2208.01484
A Fishburn permutation is a permutation which avoids the bivincular pattern (231, \1\, \1\), while an ascent sequence is a sequence of nonnegative integers in which each entry is less than or equal to one more than the number of ascents to its left. Fishburn permutations and ascent sequences are linked by a bijection g of Bousquet-Mélou, Claesson, Dukes, and Kitaev. We write Fn(σ1,…,σk) to denote the set of Fishburn permutations of length n which avoid each of σ1,…,σk and we write An(α1,…,αk) to denote the set of ascent sequences which avoid each of α1,…,αk. We settle a conjecture of Gil and Weiner by showing that g restricts to a bijection between Fn(3412) and An(201). Building on work of Gil and Weiner, we use elementary techniques to enumerate Fn(123) with respect to inversion number and number of left-to-right maxima, obtaining expressions in terms of q-binomial coefficients, and to enumerate Fn(123,σ) for all σ. We use generating tree techniques to study the generating functions for Fn(321, 1423), Fn(321,3124), and Fn(321,2143) with respect to inversion number and number of left-to-right maxima. We use these results to show |Fn(321,1423)| = |Fn(321,3124)| = Fn+2 - n - 1, where Fn is a Fibonacci number, and |Fn(321,2143)| = 2n-1. We conclude with a variety of conjectures and open problems.