Motzkin Numbers Research Articles

Restricted involutions and Motzkin paths

We show how a bijection due to Biane between involutions and labelled Motzkin paths yields bijections between Motzkin paths and two families of restricted involutions that are counted by Motzkin numbers, namely, involutions avoiding 4321 and 3412. As a consequence, we derive characterizations of Motzkin paths corresponding to involutions avoiding either 4321 or 3412 together with any pattern of length 3. Furthermore, we exploit the described bijection to study some notable subsets of the set of restricted involutions, namely, fixed point free and centrosymmetric restricted involutions.

Skew-standard tableaux with three rows

Let T 3 be the three-rowed strip. Recently Regev conjectured that the number of standard Young tableaux with n − 3 entries in the “skew three-rowed strip” T 3 / ( 2 , 1 , 0 ) is m n − 1 − m n − 3 , a difference of two Motzkin numbers. This conjecture, together with hundreds of similar identities, were derived automatically and proved rigorously by Zeilberger via his powerful program and WZ method. It appears that each one is a linear combination of Motzkin numbers with constant coefficients. In this paper we will introduce a simple bijection between Motzkin paths and standard Young tableaux with at most three rows. With this bijection we answer Zeilberger's question affirmatively that there is a uniform way to construct bijective proofs for all of those identities.

Log-convexity of combinatorial sequences from their convexity

A sequence (xn)n0 of positive real numbers is log-convex if the inequality x 2 xn−1xn+1 is valid for all n 1. We show here how the problem of establishing the log-convexity of a given combinatorial sequence can be reduced to examining the ordinary convexity of related sequences. The new method is then used to prove that the sequence of Motzkin numbers is log-convex.

On the order of the recursion relation of Motzkin numbers of higher rank

For Motzkin paths with up- and down-steps of heights 1 and 2, the minimal recursion is of order 6, not of order 4, as conjectured by Schork.

Some identities on the Catalan, Motzkin and Schröder numbers

In this paper, some identities between the Catalan, Motzkin and Schröder numbers are obtained by using the Riordan group. We also present two combinatorial proofs for an identity related to the Catalan numbers with the Motzkin numbers and an identity related to the Schröder numbers with the Motzkin numbers, respectively.

Asymptotics of RNA Shapes

RNA shapes, introduced by Giegerich et al. (2004), provide a useful classification of the branching complexity for RNA secondary structures. In this paper, we derive an exact value for the asymptotic number of RNA shapes, by relying on an elegant relation between non-ambiguous, context-free grammars, and generating functions. Our results provide a theoretical upper bound on the length of RNA sequences amenable to probabilistic shape analysis (Steffen et al., 2006; Voss et al., 2006), under the assumption that any base can basepair with any other base. Since the relation between context-free grammars and asymptotic enumeration is simple, yet not well-known in bioinformatics, we give a self-contained presentation with illustrative examples. Additionally, we prove a surprising 1-to-1 correspondence between pi-shapes and Motzkin numbers.

Catalan and Motzkin numbers modulo 4 and 8

In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8.

Power-law persistence characterizes traveling waves in coupled circle maps with repulsive coupling

We study persistence in coupled circle maps with repulsive (inhibitory) coupling, and find that it offers an effective way to characterize the synchronous, traveling wave and spatiotemporally chaotic states of the system. In the traveling wave state, persistence decays as a power law and, in contrast to earlier observations in dynamical systems, this power-law scaling does not occur at the transition point alone, but over the entire dynamical phase (with the same exponent). We give a cellular automata model displaying the qualitative features of the traveling wave regime and provide an argument based on the theory of Motzkin numbers in combinatorics to explain the observed scaling.

A Bijection on Dyck Paths and its Cycle Structure

The known bijections on Dyck paths are either involutions or have notoriously intractable cycle structure. Here we present a size-preserving bijection on Dyck paths whose cycle structure is amenable to complete analysis. In particular, each cycle has length a power of 2. A new manifestation of the Catalan numbers as labeled forests crops up en route as does the Pascal matrix mod 2. We use the bijection to show the equivalence of two known manifestations of the Motzkin numbers. Finally, we consider some statistics on the new Catalan manifestation and the identities they interpret.

On the complexity of Jensen's algorithm for counting fixed polyominoes

Recently I. Jensen published a novel transfer-matrix algorithm for computing the number of polyominoes in a rectangular lattice. However, his estimation of the computational complexity of the algorithm ( O ( ( 2 ) n ) , where n is the size of the polyominoes), was based only on empirical evidence. In contrast, our research provides some solid proof. Our result is based primarily on an analysis of the number of some class of strings that plays a significant role in the algorithm. It turns out that this number is closely related to Motzkin numbers. We provide a rigorous computation that roughly confirms Jensen's estimation. We obtain the bound O ( n 5 / 2 ( 3 ) n ) on the running time of the algorithm, while the actual number of polyominoes is about C 4.06 n / n , for some constant C > 0 .

Congruences for Catalan and Motzkin numbers and related sequences

We prove various congruences for Catalan and Motzkin numbers as well as related sequences. The common thread is that all these sequences can be expressed in terms of binomial coefficients. Our techniques are combinatorial and algebraic: group actions, induction, and Lucas’ congruence for binomial coefficients come into play. A number of our results settle conjectures of Cloitre and Zumkeller. The Thue–Morse sequence appears in several contexts.

On integrality and periodicity of the Motzkin numbers

In this note we investigate arithmetic properties of the Motzkin numbers. We prove that among all fractional sequences of Motzkin type, the only integral ones are integral multiples of the sequence of Motzkin numbers; in fact, we prove a stronger result. We also show that the sequence of Motzkin numbers is nonperiodic modulo any positive integer larger than one.

Human and constructive proof of combinatorial identities: an example from Romik

It has become customary to prove binomial identities by means of the method for automated proofs as developed by Petkovšek, Wilf and Zeilberger. In this paper, we wish to emphasize the role of "human'' and constructive proofs in contrast with the somewhat lazy attitude of relaying on "automated'' proofs. As a meaningful example, we consider the four formulas by Romik, related to Motzkin and central trinomial numbers. We show that a proof of these identities can be obtained by using the method of coefficients, a human method only requiring hand computations.

Normalizers of ad-nilpotent ideals

Let b be a Borel subalgebra of a complex simple Lie algebra g . An ideal c ⊂ b is called ad -nilpotent, if it is contained in [ b , b ] . The normalizer of c in g is a standard parabolic subalgebra of g . We give several descriptions of the normalizer: (1) using the weight of an ideal, or (2) the affine Weyl group and integer points in a certain simplex, or (3) a relationship with dominant regions of the Shi arrangement. We also characterise the ad -nilpotent ideals whose normalizer is equal to b . For s l ( n ) and s p ( 2 n ) , explicit enumerative results are obtained, which demonstrate a connection with the Motzkin and Riordan numbers, number of directed animals and trinomial coefficients.

The statistic “number of udu's” in Dyck paths

In the present paper, we consider the statistic “number of udu's” in Dyck paths. The enumeration of Dyck paths according to semilength and various other parameters has been studied in several papers. However, the statistic “number of udu's” has been considered only recently. Let D n denote the set of Dyck paths of semilength n and let T n , k , L n , k , H n , k and W n , k ( r ) denote the number of Dyck paths in D n with k udu's, with k udu's at low level, at high level, and at level r ⩾ 2 , respectively. We derive their generating functions, their recurrence relations and their explicit formulas. A new setting counted by Motzkin numbers is also obtained. Several combinatorial identities are given and other identities are conjectured.

Enumerative aspects of secondary structures

A secondary structure is a planar, labeled graph on the vertex set {1,…, n} having two kind of edges: the segments [ i, i+1], for 1⩽ i⩽ n−1 and arcs in the upper half-plane connecting some vertices i, j, i⩽ j, where j− i> l, for some fixed integer l. Any two arcs must be totally disjoint. We enumerate secondary structures with respect to their size n, rank l and order k (number of arcs), obtaining recursions and, in some cases, explicit formulae in terms of Motzkin, Catalan, and Narayana numbers. We give the asymptotics for the enumerating sequences and prove their log-convexity, log-concavity and unimodality. It is shown how these structures are connected with hypergeometric functions and orthogonal polynomials.

Four Classes of Pattern-Avoiding Permutations Under One Roof: Generating Trees with Two Labels

Many families of pattern-avoiding permutations can be described by a generating tree in which each node carries one integer label, computed recursively via a rewriting rule. A typical example is that of $123$-avoiding permutations. The rewriting rule automatically gives a functional equation satisfied by the bivariate generating function that counts the permutations by their length and the label of the corresponding node of the tree. These equations are now well understood, and their solutions are always algebraic series. Several other families of permutations can be described by a generating tree in which each node carries two integer labels. To these trees correspond other functional equations, defining 3-variate generating functions. We propose an approach to solving such equations. We thus recover and refine, in a unified way, some results on Baxter permutations, $1234$-avoiding permutations, $2143$-avoiding (or: vexillary) involutions and $54321$-avoiding involutions. All the generating functions we obtain are D-finite, and, more precisely, are diagonals of algebraic series. Vexillary involutions are exceptionally simple: they are counted by Motzkin numbers, and thus have an algebraic generating function. In passing, we exhibit an interesting link between Baxter permutations and the Tutte polynomial of planar maps.

Dyck Paths with Peaks Avoiding or Restricted to a Given Set

In this paper we focus on Dyck paths with peaks avoiding or restricted to an arbitrary set of heights. The generating functions of such types of Dyck paths can be represented by continued fractions. We also discuss a special case that requires all peak heights to either lie on or avoid a congruence class (or classes) modulo k. The case when k= 2 is especially interesting. The two sequences for this case are proved, combinatorially as well as algebraically, to be the Motzkin numbers and the Riordan numbers. We introduce the concept of shift equivalence on sequences, which in turn induces an equivalence relation on avoiding and restricted sets. Several interesting equivalence classes whose representatives are well‐known sequences are given as examples.

Motzkin Paths and Reduced Decompositions for Permutations with Forbidden Patterns

We obtain a characterization of $(321, 3\bar{1}42)$-avoiding permutations in terms of their canonical reduced decompositions. This characterization is used to construct a bijection for a recent result that the number of $(321,3\bar{1}42)$-avoiding permutations of length $n$ equals the $n$-th Motzkin number, due to Gire, and further studied by Barcucci, Del Lungo, Pergola, Pinzani and Guibert. Similarly, we obtain a characterization of $(231,4\bar{1}32)$-avoiding permutations. For these two classes, we show that the number of descents of a permutation equals the number of up steps on the corresponding Motzkin path. Moreover, we find a relationship between the inversion number of a permutation and the area of the corresponding Motzkin path.

Calculus proofs of some combinatorial inequalities

Using calculus we show how to prove some combinatorial inequalities of the type log-concavity or log-convexity. It is shown by this method that binomial coefficients and Stirling numbers of the first and second kinds are log-concave, and that Motzkin numbers and secondary structure numbers of rank 1 are log-convex. In fact, we prove via calculus a much stronger result that a natural continuous ``patchwork'' (i.e. corresponding dynamical systems) of Motzkin numbers and secondary structures recursions are increasing functions. We indicate how to prove asymptotically the log-convexity for general secondary structures. Our method also applies to show that sequences of values of some orthogonal polynomials, and in particular the sequence of central Delannoy numbers, are log-convex.