Motzkin Path Research Articles

Erratum to “Lefschetz Theory for Exterior Algebras and Fermionic Diagonal Coinvariants”

This erratum corrects the proof of the main result [1, Thm. 5.2] of [1, Sec. 5]. While this result is correct as stated, its proof is flawed. We adopt the notation of [1, Sec. 5]. The total order |$\prec $| is not a term order, so that [1, Lem. 5.3] loses meaning. In particular, [1, Lem. 5.1] is false because the depth |$d(\sigma )$| is not multiplicative. For |$n = 6$|⁠, the elements |$u, v \in \wedge \{ \Theta _{6}, \Xi _{6}\}$| given by |$u=\xi _{1} \theta _{3}\xi _{3} \theta _{4} \theta _{5} \xi _{5}$| and |$v=\theta _{1} \theta _{2}\xi _{2} \theta _{4} \theta _{6} \xi _{6}$| have the following lattice path representations as in [1, Sec. 5]: We correct the proof of [1, Thm. 5.2] as follows. We shall calculate a Gröbner basis for the ideal |$I_{n} = \langle \delta _{n}\rangle \subset \wedge \{\Theta _{n}, \Xi _{n} \}$| where |$\delta _{n}= \sum _{i=1}^{n} \theta _{i} \xi _{i}$| with respect to the lexicographical term order |$<_{\textrm{lex}}$|⁠. For each Motzkin path |$\sigma $| as in [1, Sec. 5], we define |$j(\sigma )$| to be the |$x$|-coordinate where the depth |$d(\sigma )$| is achieved the 1st time. We have |$d(\sigma )=0$| if and only if |$j(\sigma )=0$|⁠. If |$u, v$| are the Motzkin paths (or monomials) in Example 1, then |$j(u)=5$| and |$j(v)=2$|⁠. Given a Motzkin path |$\sigma =(s_{1}, s_{2},\dots , s_{n})$| and |$i \leq n$|⁠, we define |$k_{i}$| to be the difference between the |$y$|-coordinate of the starting point of |$s_{i}$| and |$d(\sigma )$|⁠. For example, |$k_{1}=-d(\sigma )$| and |$k_{j(\sigma )+1}=0$|⁠. For |$i\leq j(\sigma )$|⁠, we introduce the exterior algebra elements ... ... The initial ideal |$\operatorname{in}_{lex}(\delta _{n}^{k})$| with respect the lexicographical term order contains all monomials |$\sigma $| with depth |$d(\sigma )\leq -k$|⁠. We claim that the leading monomial of |$p(\sigma )\delta _{n}^{-d(\sigma )}$| divides the monomial |$\textrm{wt}(\sigma )$|⁠, and we prove this statement for all |$n$| by induction on |$j(\sigma )$|⁠. The base case |$j(\sigma )=0$| is trivial because all monomials belong to the ideal generated by |$\delta _{n}^{0}=1$|⁠. For the inductive step, we remove the 1st step |$s_{1}$| from |$\sigma $| to get a new path |$\tau =(s_{2}, \dots , s_{n})$| involving only the variables |$\theta _{2}, \dots , \theta _{n}, \xi _{2}, \dots , \xi _{n}$|⁠. Notice that |$p(\sigma )=p_{1}(\sigma )p(\tau )$|⁠. We divide proof in three cases according to the 1st step |$s_{1}$|⁠. Case 1:|$s_{1} = (1,1)$|is an up step. We have |$d(\tau )=d(\sigma )-1$|⁠, |$\textrm{wt}(\sigma )=\textrm{wt}(\tau )$|⁠, and ... Case 2:|$s_{1} = (1,0)$|is a horizontal step. We assume that the horizontal step |$s_{1}$| is labelled with |$\theta $|⁠; the other case is identical. We have |$d(\tau )=d(\sigma )$|⁠, |$\textrm{wt}(\sigma )=\theta _{1} \textrm{wt}(\tau )$|⁠, and ... ... Case 3:|$s_{1} = (1,-1)$|is a down step. We have |$d(\tau )=d(\sigma )+1$|⁠, |$\textrm{wt}(\sigma )=\theta _{1}\xi _{1} \textrm{wt}(\tau )$|⁠, |$p(\sigma )=p(\tau )$|⁠, and ... We conclude that |$\textrm{wt}(\sigma ) \in \textrm{in}_{\textrm{lex}}(\delta _{n}^{-d(\sigma )}) \subseteq \textrm{in}_{\textrm{lex}}(\delta _{n}^{k})$| for all |$k \leq -d(\sigma )$| and the proof is complete. The above theorem substitutes [1, Lem. 5.3]. The 2nd part of the proof of [1, Thm. 5.2] is correct and can be left unchanged. The set |$\{p(\sigma )\delta _{n} \mid \sigma \textnormal{ s.t.} d(\sigma )=-1\}$| is a Gröbner basis for the ideal |$I_{n}= \langle \delta _{n} \rangle $| with respect to the lexicographical term order.

Motzkin path decompositions of functionals in noncommutative probability

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional [Formula: see text] are naturally described in terms of Motzkin paths identified with reduced Motzkin words [Formula: see text]. Using this fact, we demonstrate that the mixed moments of order [Formula: see text] of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of the orthogonal replicas of these variables with respect to [Formula: see text] with summation extending over [Formula: see text], the set of reduced Motzkin paths of length [Formula: see text]. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice [Formula: see text] into sublattices [Formula: see text] of partitions which are monotonically adapted to colors in [Formula: see text]. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.

Computation of Al-Salam Carlitz and Askey-Wilson Moments using Motzkin Paths

In this paper we study the moments of polynomials from the Askey scheme, and we focus on Askey-Wilson polynomials. More precisely, we give a combinatorial proof for the case where $d=0$. Their values have already been computed by Kim and Stanton in 2015, however, the proof is not completely combinatorial, which means that an explicit bijection has not been exhibited yet. In this work, we use a new combinatorial approach for the simpler case of Al-Salam-Carlitz, using a sign reversing involution that directly operates on Motzkin path. We then generalize this method to Askey-Wilson polynomials with $d=0$ only, providing the first fully combinatorial proof for that case.

Peakless Motzkin paths with marked level steps at fixed height

A Motzkin path is a path starting and ending on the x-axis which uses up, down and level steps and never goes below the x-axis. We consider Motzkin paths where any number of level steps on the x-axis are allowed to be marked. It is known that the number of such paths corresponds to a matrix which turns out to be a pseudo-involution in the Riordan group. We provide a combinatorial proof of this result by means of a sign-reversing involution on pairs of signed marked Motzkin paths. We also extend this result to the class of matrices that count Motzkin paths where level steps at a fixed height are allowed to be marked.

Counting self-conjugate $$(s,s+1,s+2)$$-core partitions

We are concerned with counting self-conjugate $(s,s+1,s+2)$-core partitions. A Motzkin path of length $n$ is a path from $(0,0)$ to $(n,0)$ which stays above the $x$-axis and consists of the up $U=(1,1)$, down $D=(1,-1)$, and flat $F=(1,0)$ steps. We say that a Motzkin path of length $n$ is symmetric if its reflection about the line $x=n/2$ is itself. In this paper, we show that the number of self-conjugate $(s,s+1,s+2)$-cores is equal to the number of symmetric Motzkin paths of length $s$, and give a closed formula for this number.

Refined enumerations of alternating sign triangles

This article introduces and investigates a refinement of alternating sign trapezoids by means of Catalan objects and Motzkin paths. Alternating sign trapezoids are a generalisation of alternating sign triangles, which were recently introduced by Ayyer, Behrend and Fischer. We show that the number of alternating sign trapezoids associated with a Catalan object (resp. a Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of these polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a constant term identity for the refined counting of alternating sign trapezoids.

Fluctuations of random Motzkin paths

It is known that after scaling a random Motzkin path converges in distribution to a Brownian excursion. We prove that the fluctuations of the counting processes of the ascent steps, the descent steps and the level steps converge jointly to linear combinations of two independent processes: a Brownian motion and a Brownian excursion. The proofs rely on the Laplace transforms and an integral representation based on an identity connecting non-crossing pair partitions and joint moments of an explicit non-homogeneous Markov process.

Minors of a class of Riordan arrays related to weighted partial Motzkin paths

A partial Motzkin path is a path from (0,0) to (n,k) in the XOY-plane that does not go below the X-axis and consists of up steps U=(1,1), down steps D=(1,−1) and horizontal steps H=(1,0). A weighted partial Motzkin path is a partial Motzkin path with the weight assignment that all up steps and down steps are weighted by 1, the horizontal steps are endowed with a weight x if they are lying on X-axis, and endowed with a weight y if they are not lying on X-axis. Denote by Mn,k(x,y) the weight function of all weighted partial Motzkin paths from (0,0) to (n,k), and M=(Mn,k(x,y))n≥k≥0 the infinite lower triangular matrices. In this paper, we consider the sums of minors of second order of the matrix M, and obtain a lot of interesting determinant identities related to M, which are proved by bijections using weighted partial Motzkin paths. When the weight parameters (x,y) are specialized, several new identities are obtained related to some classical sequences involving Catalan numbers. Besides, in the alternating cases we also give some new explicit formulas for Catalan numbers.

GENERATING FUNCTIONS FOR PLATEAUS IN MOTZKIN PATHS

A plateau in a Motzkin path is a sequence of three steps: an up step, a horizontal step, then a down step. We find three different forms for the bivariate generating function for plateaus in Motzkin paths, then generalize to longer plateaus. We conclude by describing a further generalization: a continued fraction form from which one can easily derive new multivariate generating functions for various kinds of path statistics. Several examples of generating functions are given using this technique.

Noncrossing linked partitions and large [formula omitted]-Motzkin paths

Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and (3,2)-Motzkin paths, where a (3,2)-Motzkin path can be viewed as a Motzkin path for which there are three kinds of horizontal steps and two kinds of down steps. A large (3,2)-Motzkin path is a (3,2)-Motzkin path for which there are only two kinds of horizontal steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of {1,…,n+1} and the set of large (3,2)-Motzkin paths of length n, which leads to a simple explanation of the well-known relation between the large and the little Schröder numbers.

Adsorbing Motzkin paths

The adsorption of a Motzkin path with lifted endpoints onto an adsorbing line is examined as a model of polymer adsorption. The partition function of this model is determined exactly, and the free energy is derived by examining the partition function. The location of the critical point in the model is given exactly in one case, and as the solution of a nonlinear equation in another case. The phase diagram of the model is examined, and I show that the transition is in general a first-order adsorption transition, except in cases where one or both endpoints of the path are restricted to be close to the adsorbing line.

A note on 2-distant noncrossing partitions and weighted Motzkin paths

We prove a conjecture of Drake and Kim: the number of 2 -distant noncrossing partitions of { 1 , 2 , … , n } is equal to the sum of weights of Motzkin paths of length n , where the weight of a Motzkin path is a product of certain fractions involving Fibonacci numbers. We provide two proofs of their conjecture: one uses continued fractions and the other is combinatorial.

Pulled Motzkin paths

In this paper the models of pulled Dyck paths in Janse van Rensburg (2010 J. Phys. A: Math. Theor. 43 215001) are generalized to pulled Motzkin path models. The generating functions of pulled Motzkin paths are determined in terms of series over trinomial coefficients and the elastic response of a Motzkin path pulled at its endpoint (see Orlandini and Whittington (2004 J. Phys. A: Math. Gen. 37 5305–14)) is shown to be R(f) = 0 for forces pushing the endpoint toward the adsorbing line and R(f) = f(1 + 2cosh f))/(2sinh f) → f as f → ∞, for forces pulling the path away from the X-axis. In addition, the elastic response of a Motzkin path pulled at its midpoint is shown to be R(f) = 0 for forces pushing the midpoint toward the adsorbing line and R(f) = f(1 + 2cosh (f/2))/sinh (f/2) → 2f as f → ∞, for forces pulling the path away from the X-axis. Formal combinatorial identities arising from pulled Motzkin path models are also presented. These identities are the generalization of combinatorial identities obtained in directed paths models to their natural trinomial counterparts.

Peaks and Valleys in Motzkin Paths

A Motzkin path is a non-negative lattice path in N2 starting at the origin, where only three types of steps are allowed: the diagonal up step (1, 1) called u, the diagonal down step (1, −1) called d and the horizontal step (1, 0) called h. We consider paths of size n, ending at the point (n, 0). A peak is defined to be a node between the following steps: uh, hh, hd and ud, and a valley is a node between dh, hh, hu and du. A sharp peak or a sharp valley is the node between the steps ud and du respectively. We also define a low peak to be a peak on level one and a low valley to be a valley on level zero. In this paper, we find the asymptotic expressions for the average number of peaks, sharp peaks, low peaks, low sharp peaks and similarly for valleys on Motzkin paths of size n as n →∞.

Potential polynomials and Motzkin paths

A Motzkin path of length n is a lattice path from ( 0 , 0 ) to ( n , 0 ) in the plane integer lattice Z × Z consisting of horizontal-steps ( 1 , 0 ) , up-steps ( 1 , 1 ) , and down-steps ( 1 , − 1 ) , which never passes below the x -axis. A u - segment (resp. h - segment) of a Motzkin path is a maximal sequence of consecutive up-steps (resp. horizontal-steps). The present paper studies two kinds of statistics on Motzkin paths: “number of u -segments” and “number of h -segments”. The Lagrange inversion formula is utilized to represent the weighted generating function for the number of Motzkin paths according to the two statistics as a sum of the partial Bell polynomials or the potential polynomials. As an application, a general framework for studying compositions are also provided.

Motzkin path models of long chain polymers in slits

We consider Motzkin path models for polymers confined to a slit. The path interacts with each of the two confining lines, and we define parameters a and b to characterize the strengths of the interactions with the two lines. We consider the cases where (i) each vertex in a confining line contributes to the energy and (ii) where each edge in a confining line contributes to the energy. For the vertex and edge versions of Motzkin paths, we can find the generating functions and give rigorous explicit expressions for the free energy at some special points in the (a, b)-plane, and asymptotically (i.e. for large slit widths) elsewhere. We find regions where the force between the lines is long range and repulsive, short range and repulsive and short range and attractive. Our results indicate that the general form of the phase diagram is model independent, and similar to previous results for a Dyck path model, although the details do depend on the underlying configurational model. We also contrast the method used here to find the generating function with the transfer matrix and heap methods.

Minimal and maximal plateau lengths in Motzkin paths

The minimal length of a plateau (a sequence of horizontal steps, preceded by an up- and followed by a down-step) in a Motzkin path is known to be of interest in the study of secondary structures which in turn appear in mathematical biology. We will treat this and the related parameters &lt;i&gt; maximal plateau length, horizontal segment &lt;/i&gt;and &lt;i&gt;maximal horizontal segment &lt;/i&gt;as well as some similar parameters in unary-binary trees by a pure generating functions approach―-Motzkin paths are derived from Dyck paths by a substitution process. Furthermore, we provide a pretty general analytic method to obtain means and limiting distributions for these parameters. It turns out that the maximal plateau and the maximal horizontal segment follow a Gumbel distribution.

Some Motzkin path models of random and periodic copolymers

Motzkin paths are one of several directed walk models of polymers which have been used to investigate configurational properties. They have the advantage, like Dyck paths, that the models can often be solved exactly, and they have a further advantage over Dyck paths when considering physical systems involving an interface or surface. We illustrate their utility by considering several physical situations where Motzkin path models can be used. We consider the localization of periodic and random copolymers at an interface, under the influence of an applied force, and the unzipping (by an applied force) of a duplex polymer with a random monomer sequence. For the random cases the quenched system cannot be solved exactly and we use the Morita approximation but show that this gives a bound on the force required in the quenched system.

Forces in Motzkin paths in a wedge

Entropic forces in models of Motzkin paths in a wedge geometry are considered as models of forces in polymers in confined geometries. A Motzkin path in the square lattice is a path from the origin to a point in the line Y = X while it never visits sites below this line, and it is constrained to give unit length steps only in the north and east directions and steps of length in the north-east direction. Motzkin path models may be generalized to ensembles of NE-oriented paths above the line Y = rX, where r > 0 is an irrational number. These are paths giving east, north and north-east steps from the origin in the square lattice, and confined to the r-wedge formed by the Y-axis and the line Y = rX. The generating function gr of these paths is not known, but if r > 1, then I determine its radius of convergence to be and if r (0, 1), then tr = 1/3. The entropic force the path exerts on the line Y = rX may be computed from this. An asymptotic expression for the force for large values of r is given by In terms of the vertex angle α of the r-wedge, the moment of the force about the origin has leading terms as α → 0+ and F(α) = 0 if α [π/4, π/2]. Moreover, numerical integration of the force shows that the total work done by closing the wedge is 1.085 07... lattice units. An alternative ensemble of NE-oriented paths may be defined by slightly changing the generating function gr. In this model, it is possible to determine closed-form expressions for the limiting free energy and the force. The leading term in an asymptotic expansions for this force agrees with the leading term in the asymptotic expansion of the above model, and the subleading term only differs by a factor of 2.

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.