Motzkin Numbers Research Articles

New sequence spaces derived from the Catalan–Motzkin matrix and related matrix transformations

UDC 512.6 We consider the domain of a conservative matrix involving Catalan and Motzkin numbers in the sequence spaces c and c 0 . Moreover, the α -, β -, and γ -duals are given and some matrix mappings are presented on the resulting spaces.

Supercongruences involving Motzkin numbers and central trinomial coefficients

Supercongruences involving Motzkin numbers and central trinomial coefficients

Certain topological properties for almost convergent Catalan-Motzkin sequence space

The purpose of this paper is to introduce a new almost convergent sequence space by means of a matrix involving Catalan and Motzkin numbers. It is demonstrated that this novel sequence space and the space of all almost convergent sequences are linearly isomorphic and the $\beta$-dual of this new space is deduced. In addition, by defining Catalan-Motzkin core of a complex-valued sequence, certain inclusion relations are derived for it.

Compact operators on the new Motzkin sequence spaces

<p>This study aims to construct the BK-spaces $ \ell_p(\mathcal{M}) $ and $ \ell_{\infty}(\mathcal{M}) $ as the domains of the conservative Motzkin matrix $ \mathcal{M} $ obtained by using Motzkin numbers. It investigates topological properties, obtains Schauder basis, and then gives inclusion relations. Additionally, it expresses $ \alpha $-, $ \beta $-, and $ \gamma $-duals of these spaces and submits the necessary and sufficient conditions of the matrix classes between the described spaces and the classical spaces. In the last part, the characterization of certain compact operators is given with the aid of the Hausdorff measure of non-compactness.</p>

Motzkin Combinatorics in Linear Degenerations of the Flag Variety

Abstract We establish an explicit combinatorial/homological characterization of supports for linear degenerations of flag varieties. For such purpose, we introduce the concept of an excessive multisegment. It provides a new class of combinatorial objects counted by Motzkin numbers.

A new statistic on Dyck paths for counting 3-dimensional Catalan words

A 3-dimensional Catalan word is a word on three letters so that the subword on any two letters is a Dyck path. For a given Dyck path D, a recently defined statistic counts the number of Catalan words with the property that any subword on two letters is exactly D. In this paper, we enumerate Dyck paths with this statistic equal to certain values, including all primes. The formulas obtained are in terms of Motzkin numbers and Motzkin ballot numbers.

Congruence Properties of Combinatorial Sequences via Walnut and the Rowland-Yassawi-Zeilberger Automaton

Certain famous combinatorial sequences, such as the Catalan numbers and the Motzkin numbers, when taken modulo a prime power, can be computed by finite automata. Many theorems about such sequences can therefore be proved using Walnut, which is an implementation of a decision procedure for proving various properties of automatic sequences. In this paper we explore some results (old and new) that can be proved using this method.

The pop-stack-sorting operator on Tamari lattices

Motivated by the pop-stack-sorting map on the symmetric groups, Defant defined an operator PopM:M→M for each complete meet-semilattice M byPopM(x)=⋀({y∈M:y⋖x}∪{x}). This paper concerns the dynamics of PopTamn, where Tamn is the n-th Tamari lattice.We say an element x∈Tamn is t-Pop-sortable if PopMt(x) is the minimal element and we let ht(n) denote the number of t-Pop-sortable elements in Tamn. We find an explicit formula for the generating function ∑n≥1ht(n)zn and verify Defant's conjecture that it is rational. We furthermore prove that the size of the image of PopTamn is the Motzkin number Mn, settling a conjecture of Defant and Williams.

Laguerre inequalities for discrete sequences

The Turán inequality, the Laguerre inequality and their m-rd generalizations have been proved to be closely relative with the Laguerre-Pólya class and Riemann hypothesis. Since these two inequalities are equivalent to log-concavity of the discrete sequences, we consider whether their generalizations hold for discrete sequences. Recently, Chen, Jia and Wang proved that the partition function satisfies the Turán inequality of order 2 and thus the 3-rd Jensen polynomials associated with the partition function have only real zeros. Griffin, Ono, Rolen and Zagier proved the n-th Jensen polynomials associated with the Maclaurin coefficients of the function in the Laguerre-Pólya class and the partition function have only real zeros except finite terms. In this paper, we show the Laguerre inequality of order 2 is true for the partition function, the overpartition function, the Bernoulli numbers, the derangement numbers, the Motzkin numbers, the Fine numbers, the Franel numbers and the Domb numbers.

Pattern-restricted permutations composed of 3-cycles

In this paper, we characterize and enumerate pattern-avoiding permutations composed of only 3-cycles. In particular, we answer the question for the six patterns of length 3. We find that the number of permutations composed of n 3-cycles that avoid the pattern 231 (equivalently 312) is given by 3n−1, while the generating function for the number of those that avoid the pattern 132 (equivalently 213) is given by a formula involving the generating functions for the well-known Motzkin numbers and Catalan numbers. The number of permutations composed of n 3-cycles that avoid the pattern 321 is characterized by a weighted sum involving statistics on Dyck paths of semilength n.

A Combinatorial Bijection on k-Noncrossing Partitions

For any integer k ≥ 2, we prove combinatorially the following Euler (binomial) transformation identity $${\rm{NC}}_{n + 1}^{(k)}(t) = t\sum\limits_{i = 0}^n {\left({\matrix{n \cr i \cr}} \right)} {\rm{NW}}_i^{(k)}(t),$$ where NC (k)m (t) (resp. NW (k)m (t)) is the sum of weights, tnumber of blocks, of partitions of {1,…,m} without k-crossings (resp. enhanced k-crossings). The special k = 2 and t = 1 case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for k = 3 and t = 1, arising naturally in a recent study of pattern avoidance in ascent sequences and inversion sequences, was proved only analytically.

On the log-convexity of the difference sequence of a log-convex sequence

Let $\{z_n\}_{n\ge 0}$ be a log-convex sequence, where $z_{n+1}-z_n>0$ for $n\ge 0$. In this paper, we give several sufficient conditions for the log-convexity of the sequence $\{z_{n+1}-z_n\}_{n\ge 0}$. As applications, we discuss log-convexity for a series of sequences involving derangement numbers, Motzkin numbers, Fine numbers, numbers of counting directed animals, and so on.

On Motzkin numbers and central trinomial coefficients

The Motzkin numbers Mn=∑k=0n(n2k)(2kk)/(k+1) (n=0,1,2,…) and the central trinomial coefficients Tn (n=0,1,2,…) given by the constant term of (1+x+x−1)n, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with n any positive integer:2n∑k=1n(2k+1)Mk2∈Z,n2(n2−1)6|∑k=0n−1k(k+1)(8k+9)TkTk+1, and also∑k=0n−1(k+1)(k+2)(2k+3)Mk23n−1−k=n(n+1)(n+2)MnMn−1.

Pattern avoiding alternating involutions

We enumerate and characterize some classes of alternating and reverse alternating involutions avoiding a single pattern of length three or four. If on one hand the case of patterns of length three is trivial, on the other hand, the length four case is more challenging and involves sequences of combinatorial interest, such as Motzkin and Fibonacci numbers.

Hankel determinants of linear combinations of moments of orthogonal polynomials, II

We present a formula that expresses the Hankel determinants of a linear combination of length d+1 of moments of orthogonal polynomials in terms of a dtimes d determinant of the orthogonal polynomials. This formula exists somehow hidden in the folklore of the theory of orthogonal polynomials but deserves to be better known, and be presented correctly and with full proof. We present four fundamentally different proofs, one that uses classical formulae from the theory of orthogonal polynomials, one that uses a vanishing argument and is due to Elouafi (J Math Anal Appl 431:1253–1274, 2015) (but given in an incomplete form there), one that is inspired by random matrix theory and is due to Brézin and Hikami (Commun Math Phys 214:111–135, 2000), and one that uses (Dodgson) condensation. We give two applications of the formula. In the first application, we explain how to compute such Hankel determinants in a singular case. The second application concerns the linear recurrence of such Hankel determinants for a certain class of moments that covers numerous classical combinatorial sequences, including Catalan numbers, Motzkin numbers, central binomial coefficients, central trinomial coefficients, central Delannoy numbers, Schröder numbers, Riordan numbers, and Fine numbers.

Pattern avoidance in biwords

In this paper, we study pattern avoidance on the set of biwords with no repetitions in each block and prove that the number of those biwords avoiding π‾ is independent of the choice of π∈S3, which extends Knuth's classic result on permutations avoiding π∈S3. We present the proof in both bijective and inductive methods. As applications, we will give new combinatorial interpretations on Catalan, Riordan, and Motzkin numbers via pattern avoidance of biwords. Using the elementary theory of symmetric functions, we investigate the generating functions of biwords avoiding several specific patterns.

Crossings and Nestings Over Some Motzkin Objects and $q$-Motzkin Numbers

We examine the enumeration of certain Motzkin objects according to the numbers of crossings and nestings. With respect to continued fractions, we compute and express the distributions of the statistics of the numbers of crossings and nestings over three sets, namely the set of $4321$-avoiding involutions, the set of $3412$-avoiding involutions, and the set of $(321,3\bar{1}42)$-avoiding permutations. To get our results, we exploit the bijection of Biane restricted to the sets of $4321$- and $3412$-avoiding involutions which was characterized by Barnabei et al. and the bijection between $(321,3\bar{1}42)$-avoiding permutations and Motzkin paths, presented by Chen et al.. Furthermore, we manipulate the obtained continued fractions to get the recursion formulas for the polynomial distributions of crossings and nestings, and it follows that the results involve two new $q$-Motzkin numbers.

Analytic properties of combinatorial triangles related to Motzkin numbers

The Motzkin numbers count the number of lattice paths which go from (0,0) to (n,0) using steps (1,1),(1,0) and (1,−1) and never go below the x-axis. Let Mn,k be the number of such paths with exactly k horizontal steps. We investigate the analytic properties of various combinatorial triangles related to the Motzkin triangle [Mn,k]n,k≥0, including their total positivity, the real-rootedness and interlacing property of the generating functions of their rows, and the asymptotic normality (by central and local limit theorems) of these triangles. We also prove several identities related to these triangles.

A combinatorial proof of the log-convexity of sequences in Riordan arrays

A Riordan array \(R=[r_{n,k}]_{n,k\ge 0}\) can be characterized by two sequences \(A=(a_n)_{n\ge 0}\) and \(Z=(z_n)_{n\ge 0}\) such that \(r_{0,0}=1, r_{0,k}=0~(k\ge 1)\) and $$\begin{aligned} r_{n+1,0}=\sum _{j\ge 0} z_j r_{n,j}, \quad r_{n+1,k+1}=\sum _{j\ge 0} a_j r_{n,k+j} \end{aligned}$$for \(n,k\ge 0\). Using an algebraic approach, Chen, Liang and Wang showed that the sequence \((r_{n,0})_{n\ge 0}\) is log-convex if the production matrix \([\zeta ,A]\) is TP\(_2\), where \(\zeta =[z_0,z_1,z_2,\ldots ]'\) and \(A=[a_{n-k}]_{n,k\ge 0}\) is the Toeplitz matrix of the A-sequence. In this paper, we present an injective proof of this result from the point of view of weighted Łukasiewicz paths, which gives a combinatorial proof of the log-convexity of many combinatorial sequences, including the Catalan numbers, the Motzkin numbers, the Schröder numbers, the central binomial coefficients, and the restricted hexagonal numbers, in a unified approach. This method also present an injective proof of the strong q-log-convexity of many well-known combinatorial polynomials.

Catalan-like number sequences and Hausdorff moment sequences

Many well-known Catalan-like sequences turn out to be Stieltjes moment sequences (Liang et al. (2016)). However, a Stieltjes moment sequence is in general not determinate; Liang et al. suggested a further analysis about whether these moment sequences are determinate and how to obtain the associated measures. In this paper we find necessary conditions for a Catalan-like sequence to be a Hausdorff moment sequence. As a consequence, we will see 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. We can also identify the smallest interval including the support of the unique representing measure. Since Hausdorff moment sequences are determinate and a representing measure for above mentioned sequences are already known, we could almost complete the analysis raised by Liang et al. In addition, subsequences of Catalan-like number sequences are also considered; we will see a necessary and sufficient condition for subsequences of Stieltjes Catalan-like number sequences to be Stieltjes Catalan-like number sequences. We will also study a representing measure for a linear combination of consecutive terms in Catalan-like number sequences.