17 research outputs found
Legendre–Stirling permutations
AbstractWe first give a combinatorial interpretation of Everitt, Littlejohn, and Wellman’s Legendre–Stirling numbers of the first kind. We then give a combinatorial interpretation of the coefficients of the polynomial (1−x)3k+1∑n=0∞{{n+kn}}xn analogous to that of the Eulerian numbers, where {{nk}}are Everitt, Littlejohn, and Wellman’s Legendre–Stirling numbers of the second kind. Finally we use a result of Bender to show that the limiting distribution of these coefficients as n approaches infinity is the normal distribution
On the asymptotic normality of the Legendre-Stirling numbers of the second kind
For the Legendre-Stirling numbers of the second kind asymptotic formulae are
derived in terms of a local central limit theorem. Thereby, supplements of the
recently published asymptotic analysis of the Chebyshev-Stirling numbers are
established. Moreover, we provide results on the asymptotic normality and
unimodality for modified Legendre-Stirling numbers
Positivity properties of Jacobi-Stirling numbers and generalized Ramanujan polynomials
Generalizing recent results of Egge and Mongelli, we show that each diagonal
sequence of the Jacobi-Stirling numbers \js(n,k;z) and \JS(n,k;z) is a
P\'olya frequency sequence if and only if and study the
-total positivity properties of these numbers. Moreover, the polynomial
sequences \biggl\{\sum_{k=0}^n\JS(n,k;z)y^k\biggr\}_{n\geq 0}\quad \text{and}
\quad \biggl\{\sum_{k=0}^n\js(n,k;z)y^k\biggr\}_{n\geq 0} are proved to be
strongly -log-convex. In the same vein, we extend a recent result of
Chen et al. about the Ramanujan polynomials to Chapoton's generalized Ramanujan
polynomials. Finally, bridging the Ramanujan polynomials and a sequence arising
from the Lambert function, we obtain a neat proof of the unimodality of the
latter sequence, which was proved previously by Kalugin and Jeffrey.Comment: 17 pages, 2 tables, the proof of Lemma 3.3 is corrected, final
version to appear in Advances in Applied Mathematic
Differential operators, grammars and Young tableaux
In algebraic combinatorics and formal calculation, context-free grammar is
defined by a formal derivative based on a set of substitution rules. In this
paper, we investigate this issue from three related viewpoints. Firstly, we
introduce a differential operator method. As one of the applications, we deduce
a new grammar for the Narayana polynomials. Secondly, we investigate the normal
ordered grammars associated with the Eulerian polynomials. Thirdly, motivated
by the theory of differential posets, we introduce a box sorting algorithm
which leads to a bijection between the terms in the expansion of and
a kind of ordered weak set partitions, where is a smooth function in the
indeterminate and is the derivative with respect to . Using a map
from ordered weak set partitions to standard Young tableaux, we find an
expansion of in terms of standard Young tableaux. Combining this with
the theory of context-free grammars, we provide a unified interpretations for
the Ramanujan polynomials, Andr\'e polynomials, left peak polynomials, interior
peak polynomials, Eulerian polynomials of types and , -Eulerian
polynomials, second-order Eulerian polynomials, and Narayana polynomials of
types and in terms of standard Young tableaux. Along the same lines, we
present an expansion of the powers of in terms of standard Young
tableaux, where is a positive integer. In particular, we provide four
interpretations for the second-order Eulerian polynomials. All of the above
apply to the theory of formal differential operator rings.Comment: 38 page