Asymptotics of the extremal excedance set statistic. (May 2015)

• Record Type: Journal Article
• Authors: de Andrade, Rodrigo Ferraz; Lundberg, Erik; Nagle, Brendan
• Is Part Of: European journal of combinatorics. Volume 46(2015:May)
• Page Start: 75
• Page End: 88
• Publication Date: 2015-05
• Subjects: Combinatorial analysis -- Periodicals; Analyse combinatoire -- Périodiques; Combinatorial analysis; Periodicals; Electronic journals; 511.6
• DOI: 10.1016/j.ejc.2014.11.008
• Languages: English
• ISSNs: 0195-6698

Abstract

Abstract: For non-negative integers r and s with r + s = n − 1, let [ b r a s ] denote the number of permutations π ∈ S n which have the property that π ( i ) > i if, and only if, i ∈ [ r ] = { 1, …, r } . Answering a question of Clark and Ehrenborg (2010), we determine asymptotics on [ b r a s ] when r = ⌊ ( n − 1 ) / 2 ⌋ : [ b ⌊ ( n − 1 ) / 2 ⌋ a ⌈ ( n − 1 ) / 2 ⌉ ] = ( 1 2 log 2 ( 1 − log 2 ) + o ( 1 ) ) ( 1 2 log 2 ) n n ! . We also determine asymptotics on [ b r a s ] for a suitably related r = Θ ( s ) → ∞ . Our proof depends on multivariate asymptotic methods of R. Pemantle and M.C. Wilson. We also consider two applications of our main result. One, we determine asymptotics on the number of permutations π ∈ S n which simultaneously avoid the vincular patterns 21-34 and 34-21, i.e., for which π is order-isomorphic to neither (2, 1, 3, 4) nor (3, 4, 2, 1) on any coordinates 1 ≤ i < i + 1 < j < j + 1 ≤ n . We also determine asymptotics on the number of n -cycles π ∈ C n which avoid stretching pairs, i.e., those for which 1 ≤ π ( i ) < i < j < π ( j ) ≤ n .