A generalized Eulerian triangle from staircase tableaux and tree-like tableaux. (English) Zbl 1433.05318
Summary: Motivated by the classical Eulerian triangle and triangular arrays from staircase tableaux and tree-like tableaux, we study a generalized Eulerian array \([ T_{n , k} ]_{n , k \geq 0}\), which satisfies the recurrence relation:
\[ T_{n , k} = \lambda( a_1 k + a_2) T_{n - 1 , k} + [( b_1 - d a_1) n -( b_1 - 2 d a_1) k + b_2 - d( a_1 - a_2)] T_{n - 1 , k - 1} + \frac{ d ( b_1 - d a_1 )}{ \lambda}(n - k + 1) T_{n - 1 , k - 2},\]
where \(T_{0 , 0} = 1\) and \(T_{n , k} = 0\) unless \(0 \leq k \leq n\). We derive some properties of \([ T_{n , k} ]_{n , k \geq 0} \), including the explicit formulae of \(T_{n , k}\) and the exponential generating function of the generalized Eulerian polynomial \(T_n(q)\), and the ordinary generating function of \(T_n(q)\) in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of \(T_n(q)\), stability of the iterated Turán-type polynomial \(T_{n + 1}(q) T_{n - 1}(q) - T_n^2(q)\). Furthermore, we also prove the \(q\)-Stieltjes moment property and \(3-q\)-log-convexity of \(T_n(q)\) and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for \(\gamma \)-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get \(\gamma \)-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types \(A\) and \(B\) in a unified manner. We also derive \(\gamma \)-positivity for a symmetric sub-array of \([ T_{n , k} ]_{n , k \geq 0}\), which in particular gives a unified proof of \(\gamma \)-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 A. Nunge [Sémin. Lothar. Comb. 80B, 80B.57, 12 p. (2018; Zbl 1492.11048)] about the unimodality from segmented permutations.
\[ T_{n , k} = \lambda( a_1 k + a_2) T_{n - 1 , k} + [( b_1 - d a_1) n -( b_1 - 2 d a_1) k + b_2 - d( a_1 - a_2)] T_{n - 1 , k - 1} + \frac{ d ( b_1 - d a_1 )}{ \lambda}(n - k + 1) T_{n - 1 , k - 2},\]
where \(T_{0 , 0} = 1\) and \(T_{n , k} = 0\) unless \(0 \leq k \leq n\). We derive some properties of \([ T_{n , k} ]_{n , k \geq 0} \), including the explicit formulae of \(T_{n , k}\) and the exponential generating function of the generalized Eulerian polynomial \(T_n(q)\), and the ordinary generating function of \(T_n(q)\) in terms of the Jacobi continued fraction expansion, and real rootedness and log-concavity of \(T_n(q)\), stability of the iterated Turán-type polynomial \(T_{n + 1}(q) T_{n - 1}(q) - T_n^2(q)\). Furthermore, we also prove the \(q\)-Stieltjes moment property and \(3-q\)-log-convexity of \(T_n(q)\) and that the triangular convolution preserves Stieltjes moment property of sequences. In addition, we also give a criterion for \(\gamma \)-positivity in terms of the Jacobi continued fraction expansion. In consequence, we get \(\gamma \)-positivity of a generalized Narayana polynomial, which implies that of Narayana polynomials of types \(A\) and \(B\) in a unified manner. We also derive \(\gamma \)-positivity for a symmetric sub-array of \([ T_{n , k} ]_{n , k \geq 0}\), which in particular gives a unified proof of \(\gamma \)-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 A. Nunge [Sémin. Lothar. Comb. 80B, 80B.57, 12 p. (2018; Zbl 1492.11048)] about the unimodality from segmented permutations.
MSC:
| 05E05 | Symmetric functions and generalizations |
| 05A15 | Exact enumeration problems, generating functions |
| 05B30 | Other designs, configurations |
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11A55 | Continued fractions |
Keywords:
continued fractions; real zeros; log-concavity; Hankel-total positivity; Stieltjes moment property; \(q\)-log-convexity; \(\gamma\)-positivity; Eulerian numbers; Eulerian polynomials; Narayana polynomialsCitations:
Zbl 1492.11048Software:
OEISReferences:
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