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 · · ·wn−1 of length n − 1, where wi = b if i is an excedance in π and wi = a otherwise. Denote the number of permutations in Sn with excedance word w by the bracket [w]. Ehrenborg and Steingrı́msson have shown, among other things, that the sequence {[ba]}k=0 is unimodal and that for any ab-word u the sequence {[ua n ]}n≥0 is log-concave. Furthermore, they conjectured the following. CONJECTURE 1.1 ([7]). For any three ab-words u, v and w the following four inequalities hold: [uvw][uavaw] ≤ [uavw][uvaw] (1) [uvw][uavbw] ≥ [uavw][uvbw] (2) [uvw][ubvaw] ≥ [ubvw][uvaw] (3) [uvw][ubvbw] ≤ [ubvw][uvbw]. (4) It is easy to see that inequality (1) implies the log-concavity of the sequence {[uaw]}n≥0. Moreover, Ehrenborg and Steingrı́msson have observed that Conjecture 1.1 implies the logconcavity of the sequence {[ubvaw]}k=0. So Conjecture 1.1 can be viewed as a general log-concavity property of the excedance statistic. The main object of this paper is to verify Conjecture 1.1.