Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].     

Standard Young tableaux and colored Motzkin paths

In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the n-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length n. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most $2d+1$ rows and the set of SYTs with at most 2d rows.     

Intersective polynomials and the primes

Intersective polynomials are polynomials in Z[x] having roots every modulus. For example, P₁(n)=n² and P₂(n)=n²−1 are intersective polynomials, but P₃(n)=n²+1 is not. The purpose of this note is to deduce, using results of Green and Tao (2006) [8] and Lucier (2006) [16], that for any intersective polynomial h, inside any subset of positive relative density of the primes, we can find distinct primes p₁,p₂ such that p₁−p₂=h(n) for some integer n. Such a conclusion also holds in the Chen primes (where by a Chen prime we mean a prime number p such that p+2 is the product of at most 2 primes).     

Counting strings in Dyck paths

This paper deals with the enumeration of Dyck paths according to the statistic “number of occurrences of τ”, for an arbitrary string τ. In this direction, the statistic “number of occurrences of τ at height j” is considered. It is shown that the corresponding generating function can be evaluated with the aid of Chebyshev polynomials of the second kind. This is applied to every string of length 4. Further results are obtained for the statistic “number of occurrences of τ at even (or odd) height”.     

Bivariate Gonarov polynomials and integer sequences

Univariate Gonarov polynomials arose from the Gonarov interpolation problem in numerical analysis.They provide a natural basis of polynomials for working with u-parking functions,which are integer sequences whose order statistics are bounded by a given sequence u.In this paper,we study multivariate Gonarov polynomials,which form a basis of solutions for multivariate Gonarov interpolation problem.We present algebraic and analytic properties of multivariate Gonarov polynomials and establish a combinatorial relation with integer sequences.Explicitly,we prove that multivariate Gonarov polynomials enumerate k-tuples of integers sequences whose order statistics are bounded by certain weights along lattice paths in Nk.It leads to a higher-dimensional generalization of parking functions,for which many enumerative results can be derived from the theory of multivariate Gonarov polynomials.     

Some matrix identities on colored Motzkin paths

Merlini and Sprugnoli (2017) give both an algebraic and a combinatorial proof for an identity proposed by Louis Shapiro by using Riordan arrays and a particular model of lattice paths. In this paper, we revisit the identity and emphasize the use of colored partial Motzkin paths as appropriate tool. By using colored Motzkin paths with weight defined according to the height of its last point, we can generalize the identity in several ways. These identities allow us to move from Fibonacci polynomials, Lucas polynomials, and Chebyshev polynomials, to the polynomials of the form ${(z+b)}^n$.     

Hankel transforms of generalized Motzkin numbers

We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.     

On the g-Ary Expansions of Apéry, Motzkin, Schröder and Other Combinatorial Numbers

Let g ≥ 2 be an integer and let (u _n ) _n≥1  be a sequence of integers which satisfies a relation u _n+1 = h(n)u _n for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of u _n in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u _n+2 = h ₁(n)u _n+1 + h ₂(n)u _n with two nonconstant rational functions ${h_1(X), h_2(X) \in {\mathbb Q} [X]}Let g ≥ 2 be an integer and let (u _n ) _n≥1  be a sequence of integers which satisfies a relation u _n+1 = h(n)u _n for a rational function h(X). For example, various combinatorial numbers as well as their products satisfy relations of this type. Here, we show that under some mild technical assumptions the number of nonzero digits of u _n in base g is large on a set of n of asymptotic density 1. We also extend this result to a class of sequences satisfying relations of second order u _n+2 = h ₁(n)u _n+1 + h ₂(n)u _n with two nonconstant rational functions h₁(X), h₂(X) ? \mathbb Q [X]{h_1(X), h_2(X) \in {\mathbb Q} [X]}. This class includes the Apéry, Delannoy, Motzkin, and Schr?der numbers.     

2-Binary trees: Bijections and related issues

A 2-binary tree is a binary rooted tree whose root is colored black and the other vertices are either black or white. We present several bijections concerning different types of 2-binary trees as well as other combinatorial structures such as ternary trees, non-crossing trees, Schröder paths, Motzkin paths and Dyck paths. We also obtain a number of enumeration results with respect to certain statistics.