36 research outputs found
Catalan-like numbers and Stieltjes moment sequences
We provide sufficient conditions under which the Catalan-like numbers are
Stieltjes moment sequences. As applications, we show that many well-known
counting coefficients, including the Bell numbers, the Catalan numbers, the
central binomial coefficients, the central Delannoy numbers, the factorial
numbers, the large and little Schr\"oder numbers, are Stieltjes moment
sequences in a unified approach.Comment: 7 page
A Simple Proof of the Aztec Diamond Theorem
Based on a bijection between domino tilings of an Aztec diamond and
non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is
given in terms of Hankel determinants of the large and small Schr\"oder
numbers.Comment: 8 pages, 4 figure
Total positivity of recursive matrices
Let be an infinite lower triangular matrix defined
by the recurrence where
unless and are all nonnegative. Many well-known
combinatorial triangles are such matrices, including the Pascal triangle, the
Stirling triangle (of the second kind), the Bell triangle, the Catalan
triangles of Aigner and Shapiro. We present some sufficient conditions such
that the recursive matrix is totally positive. As applications we give the
total positivity of the above mentioned combinatorial triangles in a unified
approach
Rotundus: triangulations, Chebyshev polynomials, and Pfaffians
We introduce and study a cyclically invariant polynomial which is an analog
of the classical tridiagonal determinant usually called the continuant. We
prove that this polynomial can be calculated as the Pfaffian of a
skew-symmetric matrix. We consider the corresponding Diophantine equation and
prove an analog of a famous result due to Conway and Coxeter. We also observe
that Chebyshev polynomials of the first kind arise as Pfaffians.Comment: 8 page
Total positivity of Riordan arrays
We present sufficient conditions for total positivity of Riordan arrays. As
applications we show that many well-known combinatorial triangles are totally
positive and many famous combinatorial numbers are log-convex in a unified
approach
Four transformations on the Catalan triangle
In this paper, we define four transformations on the classical Catalan
triangle with
. The first three ones are based on the
determinant and the forth is utilizing the permanent of a square matrix. It not
only produces many known and new identities involving Catalan numbers, but also
provides a new viewpoint on combinatorial triangles.Comment: 13page
Single Polygon Counting for Fixed Nodes in Cayley Tree: Two Extremal Cases
We denote a polygon as a connected component in Cayley tree of order 2
containing certain number of fix vertices. We found an exact formula for a
polygon counting problem for two cases, in which, for the first case the
polygon contain a full connected component of a Cayley tree and for the second
case the polygon contain two fixed vertices. From these formulas, which is in
the form of finite linear combination of Catalan numbers, one can find the
asymptotic estimation for a counting problem.Comment: 18 pages, 5 figure
Some exact solutions of the Volterra lattice
We study solutions of the Volterra lattice satisfying the stationary equation
for its non-autonomous symmetry. It is shown that the dynamics in and
are governed by the continuous and discrete Painlev\'e equations, respectively.
The class of initial data leading to regular solutions is described. For the
lattice on the half-line, these solutions are expressed in terms of the
confluent hypergeometric function. The Hankel transform of the coefficients of
the corresponding Taylor series is computed on the basis of the Wronskian
representation of the solution.Comment: 17 pages, 4 figure
Matrix Identities on Weighted Partial Motzkin Paths
We give a combinatorial interpretation of a matrix identity on Catalan
numbers and the sequence which has been derived by
Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between
weighted partial Motzkin paths with an elevation line and weighted free Motzkin
paths, we find a matrix identity on the number of weighted Motzkin paths and
the sequence for any . By extending this
argument to partial Motzkin paths with multiple elevation lines, we give a
combinatorial proof of an identity recently obtained by Cameron and Nkwanta. A
matrix identity on colored Dyck paths is also given, leading to a matrix
identity for the sequence .Comment: 15 pages, 3figure
Catalan-like numbers and Hausdorff moment sequences
In this paper we show that many well-known counting coefficients, including
the Catalan numbers, the Motzkin numbers, the central binomial coefficients,
the central Delannoy numbers are Hausdorff moment sequences in a unified
approach. In particular we answer a conjecture of Liang at al. which such
numbers have unique representing measures. The smallest interval including the
support of representing measure is explicitly found. Subsequences of
Catalan-like numbers are also considered. We provide a necessary and sufficient
condition for a pattern of subsequences that if sequences are the Stieltjes
Catalan-like numbers, then their subsequences are Stieltjes Catalan-like
numbers. Moreover, a representing measure of a linear combination of
consecutive Catalan-like numbers is studied