the k-Dyck paths and ordinary Dyck paths as special cases; ii) giving a geometric interpretation of the dinv statistic of a~k-Dyck path. Our bounce construction is inspired by Loehr’s construction and Xin-Zhang’s linear algorithm for inverting the sweep map on ~k-Dyck paths. Our dinv interpretation is inspired by Garsia-Xin’s visual proof of2.From Dyck paths with 2-colored hills to Dyck paths We de ne a mapping ˚: D(2)!D+ that has a simple non-recursive description; for every 2D(2), the path ˚( ) is constructed in two steps as follows: (˚1)Transform each H2 (hill with color 2) of into a du(a valley at height 1).A Dyck path is a lattice path in the first quadrant of the x y -plane that starts at the origin and ends on the x -axis and has even length. This is composed of the same number of North-East ( X) and South-East ( Y) steps. A peak and a valley of a Dyck path are the subpaths X Y and Y X, respectively. A peak is symmetric if the valleys ...The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: \\emph{area} (the area under the path) and \\emph{rank} (the rank in the lattice). While area for Dyck paths has been …set of m-Dyck paths and the set of m-ary planar rooted trees, we may deﬁne a Dyckm algebra structure on the vector space spanned by the second set. But the description of this Dyckm algebra is much more complicated than the one deﬁned on m-Dyck paths. Our motivation to work on this type of algebraic operads is two fold.k-Dyck paths correspond to (k+ 1)-ary trees, and thus k-Dyck paths of length (k+ 1)nare enumerated by Fuss–Catalan numbers (see [FS09, Example I.14]) which are given by …Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo. k. Clemens Heuberger, Sarah J. Selkirk, Stephan Wagner. For fixed non-negative integers k, t, and n, with t < k, a k_t -Dyck path of length (k+1)n is a lattice path that starts at (0, 0), ends at ( (k+1)n, 0), stays weakly above the line y = -t, and consists of ...\(\square \) As we make use of Dyck paths in the sequel, we now set up relevant notations. A Dyck path of semilength n is a lattice path that starts at the origin, ends at (2n, 0), has steps \(U = (1, 1)\) and \(D = (1, -1),\) and never falls below the x-axis.A peak in a Dyck path is an up-step immediately followed by a down-step. The height of a …Decompose this Dyck word into a sequence of ascents and prime Dyck paths. A Dyck word is prime if it is complete and has precisely one return - the final step. In particular, the empty Dyck path is not prime. Thus, the factorization is unique. This decomposition yields a sequence of odd length: the words with even indices consist of up steps ... We relate the combinatorics of periodic generalized Dyck and Motzkin paths to the cluster coefficients of particles obeying generalized exclusion statistics, and obtain explicit expressions for the counting of paths with a fixed number of steps of each kind at each vertical coordinate. A class of generalized compositions of the integer path length …can be understood for Dyck paths by decomposing a Dyck path p according to its point of last return, i.e., the last time the path touches the line y = x before reaching (n, n). If the path never touches the line y = x except at the endpoints we consider (0, 0) to be the point of last return. See Figure 6.5.That article finds general relationships between a certain class of orthogonal polynomials and weighted Motzkin paths, which are a generalization of Dyck paths that allow for diagonal jumps. In particular, Viennot shows that the elements of the inverse coefficient matrix of the polynomials are related to the sum of the weights of all Motzkin ... Note that setting \(q=0\) in Theorem 3.3 yields the classical bijection between 2-Motzkin paths of length n and Dyck paths of semilength \(n+1\) (see Deutsch ). Corollary 3.4 There is a bijection between the set of (3, 2)-Motzkin paths of length n and the set of small Schröder paths of semilength \(n+1\). Corollary 3.51 Dyck Paths 1.1 Decomposing Dyck paths Deﬁnition. ADyck pathis a path on Z2 from (0;0) to (n;0) that never steps below the line y= 0 with steps from the set f(1;1);(1; 1)g. …$\begingroup$ This is related to a more general question already mentioned here : Lattice paths and Catalan Numbers, or slightly differently here How can I find the number of the shortest paths between two points on a 2D lattice grid?. This is called a Dyck path. It's a very nice combinatorics subject. $\endgroup$ –A Dyck path of semilength n is a lattice path in the Euclidean plane from (0,0) to (2n,0) whose steps are either (1,1) or (1,−1) and the path never goes below the x-axis. The height H of a Dyck path is the maximal y-coordinate among all points on the path. The above graph (c) shows a Dyck path with semilength 5 and height 2.Dyck paths and standard Young tableaux (SYT) are two of the most central sets in combinatorics. Dyck paths of semilength n are perhaps the best-known family counted by the Catalan number \ (C_n\), while SYT, beyond their beautiful definition, are one of the building blocks for the rich combinatorial landscape of symmetric functions.A Dyck Path is a series of up and down steps. The path will begin and end on the same level; and as the path moves from left to right it will rise and fall, never dipping below the …The middle path of length \( 4 \) in paths 1 and 2, and the top half of the left peak of path 3, are the Dyck paths on stilts referred to in the proof above. This recurrence is useful because it can be used to prove that a sequence of numbers is the Catalan numbers. Famous watercolor artists include Albrecht Durer, Peter Paul Rubens, Van Dyck, Thomas Gainsborough and Eugene Delacroix. The earliest known use of watercolor exists in cave paintings.tice. The m-Tamari lattice is a lattice structure on the set of Fuss-Catalan Dyck paths introduced by F. Bergeron and Pr eville-Ratelle in their combinatorial study of higher diagonal coinvariant spaces [6]. It recovers the classical Tamari lattice for m= 1, and has attracted considerable attention in other areas such as repre-An interesting case are e.g. Dyck paths below the slope $2/3$ (this corresponds to the so called Duchon's club model), for which we solve a conjecture related to the asymptotics of the area below ...The simplest lattice path problem is the problem of counting paths in the plane, with unit east and north steps, from the origin to the point (m, n). (When not otherwise specified, our paths will have these steps.) The number of such paths is the binomial co- efficient m+n . We can find more interesting problems by counting these paths accordingInspired by Thomas-Williams work on the modular sweep map, Garsia and Xin gave a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a coprime pairs $(m,n)$ of positive integers. We find their idea naturally extends for general Dyck paths. Indeed, we define a class of Order sweep maps on general Dyck paths, …We exhibit a bijection between 132-avoiding permutations and Dyck paths. Using this bijection, it is shown that all the recently discovered results on generating functions for 132-avoiding permutations with a given number of occurrences of the pattern $12... k$ follow directly from old results on the enumeration of Motzkin paths, among …k-Dyck paths correspond to (k+ 1)-ary trees, and thus k-Dyck paths of length (k+ 1)nare enumerated by Fuss–Catalan numbers (see [FS09, Example I.14]) which are given by …Dyck path is a staircase walk from bottom left, i.e., (n-1, 0) to top right, i.e., (0, n-1) that lies above the diagonal cells (or cells on line from bottom left to top right). The task is to count the number of Dyck Paths from (n-1, 0) to (0, n-1). Examples :A Dyck path is a staircase walk from (0,0) to (n,n) which never crosses (but may touch) the diagonal y=x. The number of staircase walks on a grid with m horizontal lines and n vertical lines is given by (m+n; m)=((m+n)!)/(m!n!) (Vilenkin 1971, Mohanty 1979, Narayana 1979, Finch 2003).tice. The m-Tamari lattice is a lattice structure on the set of Fuss-Catalan Dyck paths introduced by F. Bergeron and Pr eville-Ratelle in their combinatorial study of higher diagonal coinvariant spaces [6]. It recovers the classical Tamari lattice for m= 1, and has attracted considerable attention in other areas such as repre-Here is a solution using Dyck paths. Bijections for the identity The title identity counts 2n-step lattice paths of upsteps and downsteps (a) by number 2k of steps before the path's last return to ground level, and (b) by number 2k of steps lying above ground level.The set of Dyck paths of length $2n$ inherits a lattice structure from a bijection with the set of noncrossing partitions with the usual partial order. In this paper, we study the joint distribution of two statistics for Dyck paths: \\emph{area} (the area under the path) and \\emph{rank} (the rank in the lattice). While area for Dyck paths has been …Counting Dyck Paths A Dyck path of length 2n is a diagonal lattice path from (0;0) to (2n;0), consisting of n up-steps (along the vector (1;1)) and n down-steps (along the vector (1; 1)), such that the path never goes below the x-axis. We can denote a Dyck path by a word w 1:::w 2n consisting of n each of the letters D and U. The conditionDyck Paths# This is an implementation of the abstract base class sage.combinat.path_tableaux.path_tableau.PathTableau. This is the simplest implementation of a path tableau and is included to provide a convenient test case and for pedagogical purposes. In this implementation we have sequences of nonnegative integers.Dyck path which starts at (0,0) and goes up as much as possible by staying under the original Dyck path, then goes straight to the y= x line and “bounces back” again as much as possible as drawn on Fig. 3. The area sequence of the bounce path is the bounce sequence which can be computed directly from the area sequence of the Dyck path.Dyck paths count paths from (0, 0) ( 0, 0) to (n, n) ( n, n) in steps going east (1, 0) ( 1, 0) or north (0, 1) ( 0, 1) and that remain below the diagonal. How many of these pass through a given point (x, y) ( x, y) with x ≤ y x ≤ y? combinatorics Share Cite Follow edited Sep 15, 2011 at 2:59 Mike Spivey 54.8k 17 178 279 asked Sep 15, 2011 at 2:35A Dyck path of semilength n is a diagonal lattice path in the first quadrant with up steps u = 1, 1 , rises, and down steps = 1, −1 , falls, that starts at the origin (0, 0), ends at (2n, 0), and never passes below the x-axis. The Dyck path of semilength n we will call an n-Dyck path.CORE – Aggregating the world’s open access research papersA Dyck path consists of up-steps and down-steps, one unit each, starts at the origin and returns to the origin after 2n steps, and never goes below the x-axis. The enumeration …2.With our chosen conventions, a lattice path taht corresponds to a sequence with no IOUs is one that never goes above the diagonal y = x. De nition 4.5. A Dyck path is a lattice path from (0;0) to (n;n) that does not go above the diagonal y = x. Figure 1: all Dyck paths up to n = 4 Proposition 4.6 ([KT17], Example 2.23).Dyck paths that have exactly one return step are said to be primitive. A peak (valley)in a (partial) Dyck path is an occurrence of ud(du). By the levelof apeak (valley)we mean the level of the intersection point of its two steps. A pyramidin a (partial) Dyck path is a section of the form uhdh, a succession of h up steps followed immediately byA Dyck path with air pockets is called prime whenever it ends with D k, k¥2, and returns to the x-axis only once. The set of all prime Dyck paths with air pockets of length nis denoted P n. Notice that UDis not prime so we set P ﬂ n¥3 P n. If U UD kPP n, then 2 ⁄k€n, is a (possibly empty) pre x of a path in A, and we de ne the Dyck path ...a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424.The cyclic descent set on Dyck path of length 2n restricts to the usual descent set when the largest value 2n is omitted, and has the property that the number of Dyck paths with a given cyclic descent set D\subset [2n] is invariant under cyclic shifts of the entries of D. In this paper, we explicitly describe cyclic descent sets for Motzkin paths.A Dyck path is a staircase walk from (0,0) to (n,n) that lies strictly below (but may touch) the diagonal y=x. The number of Dyck paths of order n is given by the Catalan number C_n=1/ (n+1) (2n; n), i.e., 1, 2, 5, 14, 42, 132, ... (OEIS A000108).the Dyck paths of arbitrary length are located in the Catalan lattice. In Figure 1, we show the diagonal paths in the i × j grid and the monotone paths in the l × r grid. There are other versions. For example, the reader can obtain diago-nal-monotonic paths in the l × j grid (diagonal upsteps and vertical downsteps).Two other Strahler distributions have been discovered with the logarithmic height of Dyck paths and the pruning number of forests of planar trees in relation with molecular biology. Each of these three classes are enumerated by the Catalan numbers, but only two bijections preserving the Strahler parameters have been explicited: by Françon ...A Dyck path with air pockets is called prime whenever it ends with D k, k¥2, and returns to the x-axis only once. The set of all prime Dyck paths with air pockets of length nis denoted P n. Notice that UDis not prime so we set P ﬂ n¥3 P n. If U UD kPP n, then 2 ⁄k€n, is a (possibly empty) pre x of a path in A, and we de ne the Dyck path ...(n;n)-Labeled Dyck paths We can get an n n labeled Dyck pathby labeling the cells east of and adjacent to a north step of a Dyck path with numbers in (P). The set of n n labeled Dyck paths is denoted LD n. Weight of P 2LD n is tarea(P)qdinv(P)XP. + 2 3 3 5 4) 2 3 3 5 4 The construction of a labeled Dyck path with weight t5q3x 2x 2 3 x 4x 5. Dun ...Dyck paths and Motzkin paths. For instance, Dyck paths avoiding a triple rise are enumerated by the Motzkin numbers [7]. In this paper, we focus on the distribution and the popularity of patterns of length at most three in constrained Dyck paths deﬁned in [4]. Our method consists in showing how patterns are getting transferred from ...Are you tired of the same old tourist destinations? Do you crave a deeper, more authentic travel experience? Look no further than Tauck Land Tours. With their off-the-beaten-path adventures, Tauck takes you on a journey to uncover hidden ge...a(n) = the number of Dyck paths of semilength n+1 avoiding UUDU. a(n) = the number of Dyck paths of semilength n+1 avoiding UDUU = the number of binary trees without zigzag (i.e., with no node with a father, with a right son and with no left son). This sequence is the first column of the triangle A116424.Dyck Paths, Binary Words, and Grassmannian Permutations Avoiding an Increasing Pattern. October 2023 · Annals of Combinatorics. Krishna Menon ...When it comes to pursuing an MBA in Finance, choosing the right college is crucial. The quality of education, faculty expertise, networking opportunities, and overall reputation of the institution can greatly impact your career prospects in...A Dyck Path is a series of up and down steps. The path will begin and end on the same level; and as the path moves from left to right it will rise and fall, never dipping below the height it began on. You can see, in Figure 1, that paths with these limitations can begin to look like mountain ranges.We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words. A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021.Consider a Dyck path of length 2n: It may dip back down to ground-level somwhere between the beginning and ending of the path, but this must happen after an even number of steps (after an odd number of steps, our elevation will be odd and thus non-zero). So let us count the Dyck paths that rst touch down after 2m A Dyck path is a path in the first quadrant, which begins at the origin, ends at (2n,0) and consists of steps (1, 1) (North-East, called rises) and (1,-1) (South-East, called falls). We will refer to n as the semilength of the path. We denote by Dn the set of all Dyck paths of semilength n. By Do we denote the set consisting only of the empty path.Area, dinv, and bounce for k → -Dyck paths. Throughout this section, k → = ( k 1, k 2, …, k n) is a fix vector of n positive integers, unless specified otherwise. We define the three statistics for k → -Dyck paths. The area and bounce are defined using model 1, and the area and dinv are defined using model 3.The number of Dyck paths of length 2n 2 n and height exactly k k Ask Question Asked 4 years, 9 months ago Modified 4 years, 9 months ago Viewed 2k times 8 In A080936 gives the number of Dyck …A valley in a Dyck path is a local minimum, and a peak is a local maximum. A Dyck path is non-decreasing if the y-coordinates of the valleys of the path valley form anon-decreasing sequence.In this paper we provide some statistics about peaks and valleys in non-decreasing Dyck paths, such as their total number, the number of low and high …The length of a Dyck path is the length of the associated Dyck word (which is necessarily an even number). Consider the set \(\mathbf {D}_n\) of all Dyck paths of length 2 n ; it can be endowed with a very natural poset structure, by declaring \(P\le Q\) whenever P lies weakly below Q in the usual two-dimensional drawing of Dyck paths …Every nonempty Dyck path α can be uniquely decomposed in the form α = u β d γ, where β, γ ∈ D. This is the so called first return decomposition. If γ = ε, then α is a prime Dyck path. Every Dyck path can be uniquely decomposed into prime paths, called prime components. For example, the prime components of the Dyck path in Fig. 1 are ...Counting Dyck paths Catalan numbers The Catalan number is the number of Dyck paths, that is, lattice paths in n n square that never cross the diagonal: Named after Belgian mathematician Eug ene Charles Catalan (1814{1894), probably discovered by Euler. c n = 1 n + 1 2n n = (2n)! n!(n + 1)!: First values: 1;2;5;14;42;132:::\(\square \) As we make use of Dyck paths in the sequel, we now set up relevant notations. A Dyck path of semilength n is a lattice path that starts at the origin, ends at (2n, 0), has steps \(U = (1, 1)\) and \(D = (1, -1),\) and never falls below the x-axis.A peak in a Dyck path is an up-step immediately followed by a down-step. The height of a …A Dyck path is a lattice path in the first quadrant of the x y-plane that starts at the origin and ends on the x-axis and has even length.This is composed of the same number of North-East (X) and South-East (Y) steps.A peak and a valley of a Dyck path are the subpaths X Y and Y X, respectively.A peak is symmetric if the valleys determining the …use modiﬁed versions of the classical bijection from Dyck paths to SYT of shape (n,n). (4) We give a new bijective proof (Prop. 3.1) that the number of Dyck paths of semilength n that avoid three consecutive up-steps equals the number of SYT with n boxes and at most 3 rows. In addition, this bijection maps Dyck paths with s singletons to SYT Oct 1, 2016 · How would one show, without appealing to a bijection with a well known problem, that Dyck Paths satisfy the Catalan recurrence? Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. CORE – Aggregating the world’s open access research papersDyck paths and vacillating tableaux such that there is at most one row in each shape. These vacillating tableaux allow us to construct the noncrossing partitions. In Section 3, we give a characterization of Dyck paths obtained from pairs of noncrossing free Dyck paths by applying the Labelle merging algorithm. 2 Pairs of Noncrossing Free Dyck Paths set of m-Dyck paths and the set of m-ary planar rooted trees, we may deﬁne a Dyckm algebra structure on the vector space spanned by the second set. But the description of this Dyckm algebra is much more complicated than the one deﬁned on m-Dyck paths. Our motivation to work on this type of algebraic operads is two fold.Then we move to skew Dyck paths [2]. They are like Dyck paths, but allow for an extra step (−1,−1), provided that the path does not intersect itself. An equivalent model, deﬁned and described using a bijection, is from [2]: Marked ordered trees. They are like ordered trees, with an additional feature, namely each rightmost edge (exceptThus, every Dyck path can be encoded by a corresponding Dyck word of u’s and d’s. We will freely pass from paths to words and vice versa. Much is known about Dyck paths and their connection to other combinatorial structures like rooted trees, noncrossing partitions, polygon dissections, Young tableaux, and other lattice paths.Every nonempty Dyck path α can be uniquely decomposed in the form α = u β d γ, where β, γ ∈ D. This is the so called first return decomposition. If γ = ε, then α is a prime Dyck path. Every Dyck path can be uniquely decomposed into prime paths, called prime components. For example, the prime components of the Dyck path in Fig. 1 are ...Add style to your yard, and create a do-it-yourself sidewalk, a pretty patio or a brick path to surround your garden. Use this simple guide to find out how much brick pavers cost and where to find the colors and styles you love.Area, dinv, and bounce for k → -Dyck paths. Throughout this section, k → = ( k 1, k 2, …, k n) is a fix vector of n positive integers, unless specified otherwise. We define the three statistics for k → -Dyck paths. The area and bounce are defined using model 1, and the area and dinv are defined using model 3.Now, by dropping the first and last moves from a Dyck path joining $(0, 0)$ to $(2n, 0)$, grouping the rest into pairs of adjacent moves, we see that the truncated path becomes a modified Dyck path: Conversely, starting from any modified Dyck paths (using four types of moves in $\text{(*)}$ ) we can recover the Dyck path by reversing the …For the superstitious, an owl crossing one’s path means that someone is going to die. However, more generally, this occurrence is a signal to trust one’s intuition and be on the lookout for deception or changing circumstances.Bijections between bitstrings and lattice paths (left), and between Dyck paths and rooted trees (right) Full size image Rooted trees An (ordered) rooted tree is a tree with a specified root vertex, and the children of each …In this paper this will be done only for the enumeration of Dyck paths according to length and various other parameters but the same systematic approach can be applied to Motzkin paths, Schr6der paths, lattice paths in the upper half-plane, various classes of polyominoes, ordered trees, non-crossing par- titions, (the last two types of combinato...The simplest lattice path problem is the problem of counting paths in the plane, with unit east and north steps, from the origin to the point (m, n). (When not otherwise specified, our paths will have these steps.) The number of such paths is the binomial co- efficient m+n . We can find more interesting problems by counting these paths accordingA Dyck path is called restricted [Formula: see text]-Dyck if the difference between any two consecutive valleys is at least [Formula: see text] (right-hand side minus left-hand side) or if it has ...Algebraic structures defined on. -Dyck paths. We introduce natural binary set-theoretical products on the set of all -Dyck paths, which led us to define a non-symmetric algebraic operad $\Dy^m$, described on the vector space spanned by -Dyck paths. Our construction is closely related to the -Tamari lattice, so the products defining $\Dy^m$ are ...Have you started to learn more about nutrition recently? If so, you’ve likely heard some buzzwords about superfoods. Once you start down the superfood path, you’re almost certain to come across a beverage called kombucha.n Dyck Paths De nition (Dyck path) An n n Dyck path is a lattice path from (0; 0) to (n; n) consisting of east and north steps which stays above the diagonal y = x. The set of n n Dyck paths is denoted 1 2n Dn, and jDnj = Cn = . n+1 n (7; 7)-Dyck path Area of a Dyck Path De nition (area)k-Dyck paths correspond to (k+ 1)-ary trees, and thus k-Dyck paths of length (k+ 1)nare enumerated by Fuss–Catalan numbers (see [FS09, Example I.14]) which are given by …Irving and Rattan gave a formula for counting lattice paths dominated by a cyclically shifting piecewise linear boundary of varying slope. Their main result may be considered as a deep extension of well-known enumerative formulas concerning lattice paths from (0, 0) to (kn, n) lying under the line \(x=ky\) (e.g., the Dyck paths when \(k=1\)).A {\em k-generalized Dyck 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 (k, 0) for a given integer k ≥ 0, up-steps (1, 1) , and down-steps (1, −1), which never passes below the x-axis. The present paper studies three kinds of statistics on k -generalized Dyck .... 2.1. Combinatorics. A Dyck path is a lattice pCounting Dyck Paths A Dyck path of length 2n is a diagonal lat tice. The m-Tamari lattice is a lattice structure on the set of Fuss-Catalan Dyck paths introduced by F. Bergeron and Pr eville-Ratelle in their combinatorial study of higher diagonal coinvariant spaces [6]. It recovers the classical Tamari lattice for m= 1, and has attracted considerable attention in other areas such as repre- A Dyck path is a lattice path from (0, 0) to Touchard’s and Koshy’s identities are beautiful identities about Catalan numbers. It is worth noting that combinatorial interpretations for extended Touchard’s identity and extended Koshy’s identity can intuitively reflect the equations. In this paper, we give a new combinatorial proof for the extended Touchard’s identity by means of Dyck Paths. …Dyck paths that have exactly one return step are said to be primitive. A peak (valley)in a (partial) Dyck path is an occurrence of ud(du). By the levelof apeak (valley)we mean the level of the intersection point of its two steps. A pyramidin a (partial) Dyck path is a section of the form uhdh, a succession of h up steps followed immediately by ing Dyck paths. A Dyck path of length 2n...

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