Asymptotics of the extremal excedance set statistic
نویسندگان
چکیده
For non-negative integers r and s with r + s = n − 1, let [bras] denote the number of permutations π ∈ Sn 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 [bras] when r = b(n− 1)/2c: [ bb(n−1)/2cad(n−1)/2e ] = ( 1 2 log 2 √ (1− log 2) + o(1) )( 1 2 log 2 )n n!. We also determine asymptotics on [bras] 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 ∈ Sn 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 ∈ Cn which avoid stretching pairs, i.e., those for which 1 ≤ π(i) < i < j < π(j) ≤ n.
منابع مشابه
Proof of a Conjecture of Ehrenborg and Steingrímsson on Excedance Statistic
Very recently, Ehrenborg and Steingrı́msson [7] studied enumerative properties of the excedance statistic. Let Sn denote the permutation group on the set {1, 2, . . . , n} and π = π1 π2 · · ·πn ∈ Sn . An excedance in π is an index i such that πi > i . Following [7], we encode the excedance set of a permutation as a word in the letters a and b. The excedance word w(π) of π is the ab-word w1w2 · ·...
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عنوان ژورنال:
- Eur. J. Comb.
دوره 46 شماره
صفحات -
تاریخ انتشار 2015