In this paper, we propose a notion of colored Motzkin paths and establish a bijection between the $n$-cell standard Young tableaux (SYT) of bounded height and the colored Motzkin paths of length $n$. This result not only gives a lattice path interpretation of the standard Young tableaux but also reveals an unexpected intrinsic relation between the set of SYTs with at most $2d+1$ rows and the set of SYTs with at most 2d rows.
The enumeration of standard Young tableaux (SYTs) of shape λ can be easily computed by the hook-length formula. In 1981, Amitai Regev proved that the number of SYTs having at most three rows with n entries equals the nth Motzkin number M_n. In 2006, Regev conjectured that the total number of SYTs of skew shape λ/(2, 1) over all partitions λ having at most three parts with n entries is the difference of two Motzkin numbers, M_{n-1} - M_{n-3}. Ekhad and Zeilberger proved Regev's conjecture using a computer program. In 2009, S.-P. Eu found a bijection between Motzkin paths and SYTs of skew shape …
Motzkin paths of order-$\ell$ are a generalization of Motzkin paths that use steps $U=(1,1)$, $L=(1,0)$, and $D_i=(1,-i)$ for every positive integer $i \leq \ell$. We further generalize order-$\ell$ Motzkin paths by allowing for various coloring schemes on the edges of our paths. These $(\vecα,\vecβ)$-colored Motzkin paths may be enumerated via proper Riordan arrays, mimicking the techniques of Aigner in his treatment of Catalan-like numbers. After an investigation of their associated Riordan arrays, we develop bijections between $(\vecα,\vecβ)$-colored Motzkin paths and a variety of well-stud…
Let $\mathcal{T}_3$ 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" $\mathcal{T}_3 / (2,1,0)$ 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 …
We prove a quantitative version of the multi-colored Motzkin-Rabin theorem in the spirit of [BDWY12]: Let $V_1,\ldots,V_n \subset R^d$ be $n$ disjoint sets of points (of $n$ `colors'). Suppose that for every $V_i$ and every point $v \in V_i$ there are at least $δ|V_i|$ other points $u \in V_i$ so that the line connecting $v$ and $u$ contains a third point of another color. Then the union of the points in all $n$ sets is contained in a subspace of dimension bounded by a function of $n$ and $δ$ alone.
We calculate the total number of humps in Dyck and in Motzkin paths, and we give Standard-Young-Tableaux-interpretations of the numbers involved. One then observes the intriguing phenomena that the humps-calculations change the partitions in a strip to partitions in a hook.
The generating function and an explicit expression is derived for the (colored) Motzkin numbers of higher rank introduced recently. Considering the special case of rank one yields the corresponding results for the conventional colored Motzkin numbers for which in addition a recursion relation is given.
Noncrossing linked partitions arise in the study of certain transforms in free probability theory. We explore the connection between noncrossing linked partitions and colored Motzkin paths. A (3,2)-Motzkin path can be viewed as a colored Motzkin path in the sense that there are three types of level steps and two types of down steps. A large (3,2)-Motzkin path is defined to be a (3,2)-Motzkin path for which there are only two types of level steps on the x-axis. We establish a one-to-one correspondence between the set of noncrossing linked partitions of [n+1] and the set of large (3,2)-Motzkin p…
Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence $(λ^0,λ^1,...,λ^{2n})$ of Young diagrams starting with the empty shape and ending with the empty shape such that $λ^{i}$ is obtained from $λ^{i-1}$ by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook ta…
We extend the notion of $k$-ribbon tableaux to the Fibonacci lattice, a differential poset defined by R. Stanley in 1975. Using this notion, we describe an insertion algorithm that takes $k$-colored permutations to pairs of $k$-ribbon Fibonacci tableaux of the same shape, and we demonstrate a color-to-spin property, similar to that described by Shimozono and White for ribbon tableaux. We give an evacuation algorithm which relates the pair of $k$-ribbon Fibonacci tableaux obtained through the insertion algorithm to the pair of $k$-ribbon Fibonacci tableaux obtained using Fomin's growth diagrams…
We give a combinatorial interpretation of a matrix identity on Catalan numbers and the sequence $(1, 4, 4^2, 4^3, ...)$ which has been derived by Shapiro, Woan and Getu by using Riordan arrays. By giving a bijection between weighted partial Motzkin paths with an elevation line and weighted free Motzkin paths, we find a matrix identity on the number of weighted Motzkin paths and the sequence $(1, k, k^2, k^3, ...)$ for any $k \geq 2$. By extending this argument to partial Motzkin paths with multiple elevation lines, we give a combinatorial proof of an identity recently obtained by Cameron and N…
We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength $n$ that is shown to be in bijection with the set of all SYT with $n$ boxes. In addition, we present bijections from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of…
A notion of cyclic descents on standard Young tableaux (SYT) of rectangular shape was introduced by Rhoades, and extended to certain skew shapes by Adin, Elizalde and Roichman. The cyclic descent set restricts to the usual descent set when the largest value is ignored, and has the property that the number of SYT of a given shape with a given cyclic descent set D is invariant under cyclic shifts of the entries of D. Adin, Reiner and Roichman proved that a skew shape has a cyclic descent map if and only if it is not a connected ribbon. Unfortunately, their proof is nonconstructive. Recently Huan…
Flips of diagonals in colored triangle-free triangulations of a convex polygon are interpreted as moves between two adjacent chambers in a certain graphic hyperplane arrangement. Properties of geodesics in the associated flip graph are deduced. In particular, it is shown that: (1) every diagonal is flipped exactly once in a geodesic between distinguished pairs of antipodes; (2) the number of geodesics between these antipodes is equal to twice the number of Young tableaux of a truncated shifted staircase shape.
A new descent set statistic on involutions, defined geometrically via their interpretation as matchings, is introduced in this paper, and shown to be equi-distributed with the standard one. This concept is then applied to construct explicit cyclic descent extensions on involutions, standard Young tableaux and Motzkin paths. Schur-positivity of the associated quasisymmetric functions follows.
We define and study multi-colored dimer models on a segment and on a circle. The multivariate generating functions for the dimer models satisfy the recurrence relations similar to the one for Fibonacci numbers. We give closed formulae for the generating functions. We show that, in the large size limit with specializations of the formal variables, the generating functions exhibit the summations appearing in generalized Rogers--Ramanujan identities. Further, the generating functions of the dimer models have infinite product formulae for general values of formal variables in the large size limit.…
In this paper, we introduce a new family of generalized colored Motzkin paths, where horizontal steps are colored by means of $F_{k,l}$ colors, where $F_{k,l}$ is the $l$th $k$-Fibonacci number. We study the enumeration of this family according to the length. For this, we use infinite weighted automata.
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis ind…
This paper solves an open question of Mortimer and Prellberg asking for an explicit bijection between two families of walks. The first family is formed by what we name triangular walks, which are two-dimensional walks moving in six directions (0°, 60°, 120°, 180°, 240°, 300°) and confined within a triangle. The other family is comprised of two-colored Motzkin paths with bounded height, in which the horizontal steps may be forbidden at maximal height. We provide several new bijections. The first one is derived from a simple inductive proof, taking advantage of a $2^n$-to-one function from gener…
Recently Blasiak gave a combinatorial rule for the Kronecker coefficient $g_{λμν}$ when $μ$ is a hook shape by defining a set of colored Yamanouchi tableaux with cardinality $g_{λμν}$ in terms of a process called conversion. We give a characterization of colored Yamanouchi tableaux that does not rely on conversion, which leads to a simpler formulation and proof of the Kronecker rule for one hook shape.
The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights of affine Kac--Moody algebras are not known in most cases. In this paper, we study weight multiplicities for both finite and affine cases of classical types for certain infinite families of highest weights modules. We introduce new classes of Young tableaux, called the {\em $($spin$)$ rigid tableaux}, and prove that they are equinumerous to the w…
Chen, Deng, Du, Stanley, and Yan introduced the notion of $k$-crossings and $k$-nestings for set partitions, and proved that the sizes of the largest $k$-crossings and $k$-nestings in the partitions of an $n$-set possess a symmetric joint distribution. This work considers a generalization of these results to set partitions whose arcs are labeled by an $r$-element set (which we call \emph{$r$-colored set partitions}). In this context, a $k$-crossing or $k$-nesting is a sequence of arcs, all with the same color, which form a $k$-crossing or $k$-nesting in the usual sense. After showing that the …
For $\ell \geq 1$ and $k \geq 2$, we consider certain admissible sequences of $k-1$ lattice paths in a colored $\ell \times \ell$ square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of partitions of $\ell$ with height $\leq k$, which is also the number of $(k+1)k\cdots21$-avoiding permutations of $\{1, 2, \ldots, \ell\}$. Finally, we apply this result to the representation theory of the affine Lie algebra $\widehat{sl}(n)$ and show that this quantity gives the multiplicity of certain maximal domin…
In this paper we give a bijection between the class of permutations that can be drawn on an X-shape and a certain set of permutations that appears in [Knuth] in connection to sorting algorithms. A natural generalization of this set leads us to the definition of almost-increasing permutations, which is a one-parameter family of permutations that can be characterized in terms of forbidden patterns. We find generating functions for almost-increasing permutations by using their cycle structure to map them to colored Motzkin paths. We also give refined enumerations with respect to the number of cyc…
