Motzkin Numbers Research Articles

Automated discovery and proof of congruence theorems for partial sums of combinatorial sequences

Many combinatorial sequences (e.g. the Catalan and the Motzkin numbers) may be expressed as the constant term of , for some Laurent polynomials P(x) and Q(x) in the variable x with integer coefficients. Denoting such a sequence by , we obtain a general formula that determines the congruence class, modulo p, of the indefinite sum , for any prime p, and any positive integer r, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime p.

Avoiding patterns in irreducible permutations

We explore the classical pattern avoidance question in the case of irreducible permutations, <i>i.e.</i>, those in which there is no index $i$ such that $\sigma (i+1) - \sigma (i)=1$. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length $n-1$ and the sets of irreducible permutations of length $n$ (respectively fixed point free irreducible involutions of length $2n$) avoiding a pattern $\alpha$ for $\alpha \in \{132,213,321\}$. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations.

Solvable structures of a simple model of earthquakes

This paper propose an extension of Random Domino Automaton by introducing extra parameter related to a very short-distance interactions between clusters which allows to isolate influence two mechanisms of clusters grow, namely enlarging and coalescence. In this setting statistically stationary state under assumption of independence of clusters is investigated and respective set of equations is derived.Moreover, a special solvable case is studied in detail and it is shown how to derive Motzkin numbers out of the automaton for a family of parameters without considering a limit.

Orthogonal Polynomials Approach to the Hankel Transform of Sequences Based on Motzkin Numbers

In this paper, we use a method based on orthogonal polynomials to give closed-form evaluations of the Hankel transform of sequences based on the Motzkin numbers. It includes linear combinations of consecutive two, three, and four Motzkin numbers. In some cases, we were able to derive the closed-form evaluation of the Hankel transform, while in the others we showed that the Hankel transform satisfies a particular difference equation. As a corollary, we reobtain known results and show some new results regarding the Hankel transform of Motzkin and shifted Motzkin numbers. Those evaluations also give an idea on how to apply the method based on orthogonal polynomials on the sequences having zero entries in their Hankel transform.

Motzkin Paths, Motzkin Polynomials and Recurrence Relations

We consider the Motzkin paths which are simple combinatorial objects appearing in many contexts. They are counted by the Motzkin numbers, related to the well known Catalan numbers. Associated with the Motzkin paths, we introduce the Motzkin polynomial, which is a multi-variable polynomial "counting" all Motzkin paths of a certain type. Motzkin polynomials (also called Jacobi-Rogers polynomials) have been studied before, but here we deduce some properties based on recurrence relations. The recurrence relations proved here also allow an efficient computation of the Motzkin polynomials. Finally, we show that the matrix entries of powers of an arbitrary tridiagonal matrix are essentially given by Motzkin polynomials, a property commonly known but usually stated without proof.

Kronecker coefficients for some near-rectangular partitions

We give formulae for computing Kronecker coefficients occurring in the expansion of sμ⁎sν, where both μ and ν are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study s(n,n−1,1)⁎s(n,n), s(n−1,n−1,1)⁎s(n,n−1), s(n−1,n−1,2)⁎s(n,n), s(n−1,n−1,1,1)⁎s(n,n) and s(n,n,1)⁎s(n,n,1). Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers.

Effective algorithms for computing triangular operator in Schubert calculus

We develop two parallel algorithms progressively based on C++ to compute a triangle operator problem, which plays an important role in the study of Schubert calculus. We also analyse the computational complexity of each algorithm by using combinatorial quantities, such as the Catalan number, the Motzkin number, and the central binomial coefficients. The accuracy and efficiency of our algorithms have been justified by numerical experiments.

Log-convexity of Aigner–Catalan–Riordan numbers

Let T=[tn,k]n,k≥0 be an infinite lower triangular matrix defined byt0,0=1,tn+1,0=∑j=0nzjtn,j,tn+1,k+1=∑j=knaj,ktn,j for n,k≥0, where all zj,aj,k are nonnegative and aj,k=0 unless j≥k≥0. We show that the sequence (tn,0)n≥0 is log-convex if the coefficient matrix [ζ,A] is TP2, where ζ=[z0,z1,z2,…]′ and A=[ai,j]i,j≥0. This gives a unified proof of the log-convexity of many well-known combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the central binomial coefficients, the Schröder numbers, the Bell numbers, and so on.

EXACT p-ADIC ORDERS FOR DIFFERENCES OF MOTZKIN NUMBERS

For any prime p, we establish congruences modulo pn+1 for the difference of the pn+1 th and pn th Motzkin numbers and determine the p-adic order of the difference. The results confirm recent conjectures on the order. The applied techniques involve the use of congruences for the differences of certain Catalan numbers and binomial coefficients, congruential identities for sums of Catalan numbers, central binomial and trinomial coefficients, infinite incongruent disjoint covering systems and the solution of congruential recurrences.

On Block Design and the Special Class of (123)-avoiding Aunu Permutation Pattern: A Slanted Flag Model

This paper concerns an important application of the (123)-avoiding class of Aunu Permutation studied by earlier by Usman, Magami and Garba under various topics such as enumeration, thin cyclic desighn and association schemes. .In this report, some method of construction are used using accents and descents as enumerated by positive and negative units in a manner that resembles, in some sense, the Motzkin numbers. This method produced some graph models which when superimposed produced the required design of a slanted flag. This paper therefore uncovers yet another rich property of this class of permutation pattern: applications in design theory which interested researchers can further explore.

Infinitely log-monotonic combinatorial sequences

We introduce the notion of infinitely log-monotonic sequences. By establishing a connection between completely monotonic functions and infinitely log-monotonic sequences, we show that the sequences of the Bernoulli numbers, the Catalan numbers and the central binomial coefficients are infinitely log-monotonic. In particular, if a sequence {an}n⩾0 is log-monotonic of order two, then it is ratio log-concave in the sense that the sequence {an+1/an}n⩾0 is log-concave. Furthermore, we prove that if a sequence {an}n⩾k is ratio log-concave, then the sequence {ann}n⩾k is strictly log-concave subject to an initial condition. As consequences, we show that the sequences of the derangement numbers, the Motzkin numbers, the Fine numbers, the central Delannoy numbers, the numbers of tree-like polyhexes and the Domb numbers are ratio log-concave. For the case of the Domb numbers Dn, we confirm a conjecture of Sun on the log-concavity of the sequence {Dnn}n⩾1.

Motzkin algebras

We introduce an associative algebra Mk(x) whose dimension is the 2k-th Motzkin number. The algebra Mk(x) has a basis of “Motzkin diagrams”, which are analogous to Brauer and Temperley–Lieb diagrams. We show for a particular value of x that the algebra Mk(x) is the centralizer algebra of the quantum enveloping algebra Uq(gl2) acting on the k-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible Uq(gl2)-modules. We prove that Mk(x) is cellular in the sense of Graham and Lehrer and construct indecomposable Mk(x)-modules which are the left cell modules. When Mk(x) is a semisimple algebra, these modules provide a complete set of representatives of isomorphism classes of irreducible Mk(x)-modules. We compute the determinant of the Gram matrix of a bilinear form on the cell modules and use these determinants to show that Mk(x) is semisimple exactly when x is not the root of certain Chebyshev polynomials.

A direct method for evaluating some nice Hankel determinants and proofs of several conjectures

In this paper, we propose a direct method to evaluate Hankel determinants for some generating functions satisfying a certain type of quadratic equations, which cover generating functions of Catalan numbers, Motzkin numbers and Schröder numbers. Additionally, four recent conjectures proposed by Cigler (2011) [3] are proved.

Parametric Catalan numbers and Catalan triangles

Here presented a generalization of Catalan numbers and Catalan triangles associated with two parameters based on the sequence characterization of Bell-type Riordan arrays. Among the generalized Catalan numbers, a class of large generalized Catalan numbers and a class of small generalized Catalan numbers are defined, which can be considered as an extension of large Schröder numbers and small Schröder numbers, respectively. Using the characterization sequences of Bell-type Riordan arrays, some properties and expressions including the Taylor expansions of generalized Catalan numbers are given. A few combinatorial interpretations of the generalized Catalan numbers are also provided. Finally, a generalized Motzkin numbers and Motzkin triangles are defined similarly. An interrelationship among parametrical Catalan triangle, Pascal triangle, and Motzkin triangle is presented based on the sequence characterization of Bell-type Riordan arrays.

Motzkin numbers out of Random Domino Automaton

Motzkin numbers are derived from a special case of Random Domino Automaton – recently proposed a slowly driven system being a stochastic toy model of earthquakes. It is also a generalisation of 1D Drossel–Schwabl forest-fire model. A solution of the set of equations describing stationary state of Random Domino Automaton in inverse-power case is presented. A link with Motzkin numbers allows to present explicit form of asymptotic behaviour of the automaton.

The Double Riordan Group

The Riordan group is a group of infinite lower triangular matrices that are defined by two generating functions, $g$ and $f$. The kth column of the matrix has the generating function $gf^k$. In the Double Riordan group there are two generating function $f_1$ and $f_2$ such that the columns, starting at the left, have generating functions using $f_1$ and $f_2$ alternately. Examples include Dyck paths with level steps of length 2 allowed at even height and also ordered trees with differing degree possibilities at even and odd height(perhaps representing summer and winter). The Double Riordan group is a generalization not of the Riordan group itself but of the checkerboard subgroup. In this context both familiar and far less familiar sequences occur such as the Motzkin numbers and the number of spoiled child trees. The latter is a slightly enhanced cousin of ordered trees which are counted by the Catalan numbers.

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

Let g ≥ 2 be an integer and let (un)n≥1 be a sequence of integers which satisfies a relation un+1 = h(n)un 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 un 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 un+2 = h1(n)un+1 + h2(n)un with two nonconstant rational functions \({h_1(X), h_2(X) \in {\mathbb Q} [X]}\). This class includes the Apery, Delannoy, Motzkin, and Schroder numbers.

Some determinants of path generating functions

We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers—most notably of Catalan, ballot, and of Motzkin numbers—that appeared previously in the literature.

Hankel determinants of sums of consecutive Motzkin numbers

We use combinatorial methods to evaluate Hankel determinants for the sequence of sums of consecutive t-Motzkin numbers. More specifically, we consider the following determinant: det m i + j + r t + m i + j + r + 1 t 0 ⩽ i , j ⩽ n - 1 , where t is a real number and m k t is the total weight of all paths from ( 0 , 0 ) to ( k , 0 ) that stay above the x-axis and use up and down steps of weight one and level steps of weight t.

The 2-log-convexity of the Apéry Numbers

We present an approach to proving the 2-log-convexity of sequences satisfying three-term recurrence relations. We show that the Apery numbers, the Cohen-Rhin numbers, the Motzkin numbers, the Fine numbers, the Franel numbers of orders and and the large Schroder numbers are all 2-log-convex. Numerical evidence suggests that all these sequences are -log-convex for any possibly except for a constant number of terms at the beginning.