Lattice paths and branched continued fractions. II: Multivariate Lah polynomials and Lah symmetric functions. (English) Zbl 1458.05017
Summary: We introduce the generic Lah polynomials \(L_{n,k}(\phi)\), which enumerate unordered forests of increasing ordered trees with a weight \(\phi_i\) for each vertex with \(i\) children. We show that, if the weight sequence \(\phi\) is Toeplitz-totally positive, then the triangular array of generic Lah polynomials is totally positive and the sequence of row-generating polynomials \(L_n(\phi,y)\) is coefficientwise Hankel-totally positive. Upon specialization we obtain results for the Lah symmetric functions and multivariate Lah polynomials of positive and negative type. The multivariate Lah polynomials of positive type are also given by a branched continued fraction. Our proofs use mainly the method of production matrices; the production matrix is obtained by a bijection from ordered forests of increasing ordered trees to labeled partial Łukasiewicz paths. We also give a second proof of the continued fraction using the Euler-Gauss recurrence method.
For Part I see [M. Pétréolle et al. “Lattice paths and branched continued fractions: an infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity”, Preprint, arXiv:1807.03271].
For Part I see [M. Pétréolle et al. “Lattice paths and branched continued fractions: an infinite sequence of generalizations of the Stieltjes-Rogers and Thron-Rogers polynomials, with coefficientwise Hankel-total positivity”, Preprint, arXiv:1807.03271].
MSC:
| 05A15 | Exact enumeration problems, generating functions |
| 05E05 | Symmetric functions and generalizations |
| 11A55 | Continued fractions |
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 11B73 | Bell and Stirling numbers |
Online Encyclopedia of Integer Sequences:
Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.Triangle of coefficients of Bessel polynomials (exponents in decreasing order).
Triangle read by rows: the Bell transform of the triple factorial numbers A008544 without column 0.
Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
Triangle of Lah numbers.
The convolution matrix of the double factorial of odd numbers (A001147).
Triangle read by rows, the Bell transform of the triple factorial numbers A007559(n+1) without column 0.
Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.
Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 0.
Coefficient triangle of generalized Laguerre polynomials (a=1).
Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.
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