A generalized Eulerian triangle from staircase tableaux and tree-like tableaux

Abstract

Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array [Tn,k]n,k≥0, which satisfies the recurrence relation:Tn,k=λ(a1k+a2)Tn−1,k+[(b1−da1)n−(b1−2da1)k+b2−d(a1−a2)]Tn−1,k−1+d(b1−da1)λ(n−k+1)Tn−1,k−2, where T0,0=1 and Tn,k=0 unless 0≤k≤n. We derive some properties of [Tn,k]n,k≥0, including the explicit formulae of Tn,k and the exponential generating function of the generalized Eulerian polynomial Tn(q), and the ordinary generating function of Tn(q) in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of Tn(q), stability of the iterated Turán-type polynomial Tn+1(q)Tn−1(q)−Tn2(q). Furthermore, we also prove the q-Stieltjes moment property and 3-q-log-convexity of Tn(q) and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for γ-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get γ-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types A and B in a unified manner. We also derive γ-positivity for a symmetric sub-array of [Tn,k]n,k≥0, which in particular gives a unified proof of γ-positivity of Eulerian polynomials of types A and B.Our results not only can immediately apply to Eulerian triangles of two kinds and arrays from staircase tableaux and tree-like tableaux, but also to segmented permutations and flag excedance numbers in colored permutations groups in a unified approach. In particular, we also confirm a conjecture of Nunge about the unimodality from segmented permutations.