Related papers: Injectively $k$-colored rooted forests

We present formulas for the Clebsch-Gordan coefficients and the Racah coefficients for the root of unity representations ($N$-dimensional representations with $q^{2N}=1$) of $U_q(sl(2))$. We discuss colored vertex models and colored IRF…

Given a set of $k$-colored points in the plane, we consider the problem of finding $k$ trees such that each tree connects all points of one color class, no two trees cross, and the total edge length of the trees is minimized. For $k=1$,…

An injective $k$-edge-coloring of a graph $G$ is an assignment of colors, i.e. integers in $\{1, \ldots , k\}$, to the edges of $G$ such that any two edges each incident with one distinct endpoint of a third edge, receive distinct colors.…

The Catalan triangle, as well as a Fuss-Catalan triangle, enter a problem of counting particular tied arc diagrams. This setting allows us to prove some combinatorial properties of these triangles.

A nice factorization is given for the characteristic polynomials of intervals in some posets of leaf-labeled forests of rooted binary trees.

We study three different kinds of embeddings of tree patterns: weakly-injective, ancestor-preserving, and lca-preserving. While each of them is often referred to as injective embedding, they form a proper hierarchy and their computational…

We count unlabeled k-trees by properly coloring them in k+1 colors and then counting orbits of these colorings under the action of the symmetric group on the colors.

We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear…

Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the second-order super-Catalan numbers (u_{n})_{n>=0}=(3,2,3,6,14,36,...): they count "aligned cubic trees" on n internal vertices. Here we give…

For a graph G, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of…

It is well-known that every planar or projective planar graph can be 3-colored so that each color class induces a forest. This bound is sharp. In this paper, we show that there are in fact exponentially many 3-colorings of this kind for any…

In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled…

In this paper, we give a sampling algorithm for the Potts model using Markov chains. Based on the sampling algorithm, we give \emph{FPRAS}es for the Potts model and the number of $k$-colorings of the graph.

Let $K$ be a field, complete with respect to a discrete non-archimedian valuation and let $k$ be the residue field. Consider a system $F$ of $n$ polynomial equations in $K\vars$. Our first result is a reformulation of the classical Hensel's…

We develop direct bijections between the set $F_n^k$ of minimal factorizations of the long cycle $(0\,1\,\cdots\, kn)$ into $(k+1)$-cycle factors and the set $R_n^k$ of rooted labelled forests on vertices $\{1,\ldots,n\}$ with edges…

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…

A rooted tree $\vec{R}$ is a rooted subtree of a tree $T$ if the tree obtained by replacing the directed edges of $\vec{R}$ by undirected edges is a subtree of $T$. We study the problem of assigning minimum number of colors to a given set…

We establish combinatorial interpretations of several identities for the Catalan and Fine numbers and, along the way, we present some new bijections of independent interest. Briefly, we show that C_{n} = 1/(n+1) Sum_{k} (n+1)choose(2k+1)…

We count the number of closed walks on a vertex in a regular tree using the Catalan's triangle and also the Borel's triangle, showing another combinatorial structure counted by these two array of numbers.

We give a new proof of the $k$-fold convolution of the Catalan numbers. This is done by enumerating a certain class of polygonal dissections called $k$-in-$n$ dissections. Furthermore, we give a formula for the average number of cycles in a…