We use Riordan array theory to give characterizations of the Borel triangle and its associated polynomial sequence. We show that the Borel polynomials are the moment sequence for a family of orthogonal polynomials whose coefficient array is a Riordan array. The role of the Catalan matrix in defining the Borel triangle is examined. We generalize the Borel triangle to a family of two parameter triangles. Generating functions are expressed as Jacobi continued fractions, as well as the zeros of appropriate quadratic expressions. The Borel triangle is exhibited as a Hadamard product of matrices. We investigate the reversions of the triangles studied. We introduce the notion of Fuss-Borel triangles and Fuss-Catalan triangles. We end with some remarks on the Catalan triangle.
Borel's triangle is an array of integers closely related to the classical Catalan numbers. In this paper we study combinatorial statistics counted by Borel's triangle. We present various combinatorial interpretations of Borel's triangle in terms of lattice paths, binary trees, and pattern avoiding permutations and matchings, and derive a functional equation that is useful in analyzing the involved structures.
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 consider Tuenter polynomials as linear combinations of descending factorials and show that coefficients of these linear combinations are expressed via a Catalan triangle of numbers. We also describe a triangle of coefficients in terms of some polynomials.
In this paper we first generalize the numerical recurrence relation given by Hosoya to polynomials. Using this generalization we construct a Hosoya-like triangle for polynomials, where its entries are products of generalized Fibonacci polynomials (GFP). Examples of GFP are: Fibonacci polynomials, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials, Fermat polynomials, Jacobsthal polynomials, Vieta polynomials and other familiar sequences of polynomials. For every choice of a GFP we obtain a triangular array of polynomials. In this paper we extend the star of Da…
Using Chebyshev polynomialsof both kinds, we construct rational fractions which are convergents of the smallest root of $x^2-αx+1$ for $α=3,4,5,\dots$.Some of the underlying identities suggest an identity involving binomialcoefficients which leads to a triangular array sharing many propertieswith Pascal's triangle.
In this expository paper we collect some simple facts about analogues of Pascals triangle where the entries count subsets of the integers with an even or odd sum and show that they are related to Rogers-Szego polynomials. In particular we consider an interesting triangle due to Losanitsch from this point of view. We also sketch some extensions of these results to sets whose sums have fixed residues modulo a prime $p$.
It is well-known that affine (respectively projective) simple arrangements of n pseudo-lines may have at most n(n-2)/3 (respectively n(n-1)/3) triangles. However, these bounds are reached for only some values of n (mod 6). We provide the best polynomial bound for the affine and the projective case, and for each value of n (mod 6).
In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_λ(x;q)$ in some special cases. We associate a weighted graph $Π$ to $λ$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_λ(x;q)$ whenever $Π$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.
We show how some special production matrices may be used to define families of generalized Eulerian triangles. We furthermore show that these generalized Eulerian triangles are the coefficient arrays of polynomials which are the moments of families of orthogonal polynomials. Using the previously defined $\mathcal{T}$ transform, we associate these generalized Eulerian triangles to triangles defined by Catalan generating functions. Again, these new triangles are the coefficient arrays of polynomial moment sequences. The main tools used are the Sumudu transform, Jacobi continued fractions and Rio…
The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian polynomials. In this paper, we prove a $q$-analogue of this expansion for Carlitz's $q$-Eulerian polynomials as well as a similar formula for Chow-Gessel's $q$-Eulerian polynomials of type $B$. We shall give some applications of these two formulae, which involve two new sequences of polynomials in the variable $q$ with positive integral coefficients. An open p…
The present paper investigates properties of quasi-stable ideals and of Borel-fixed ideals in a polynomial ring $k[x_0,\dots,x_n]$, in order to design two algorithms: the first one takes as input $n$ and an admissible Hilbert polynomial $P(z)$, and outputs the complete list of saturated quasi-stable ideals in the chosen polynomial ring with the given Hilbert polynomial. The second algorithm has an extra input, the characteristic of the field $k$, and outputs the complete list of saturated Borel-fixed ideals in $k[x_0,\dots,x_n]$ with Hilbert polynomial $P(z)$. The key tool for the proof of bot…
In a recent work, the combinatorial interpretation of the polynomial alpha(n;k1,k2,...,kn) counting the number of Monotone Triangles with bottom row k1 < k2 < ... < kn was extended to weakly decreasing sequences k1 >= k2 >= ... >= kn. In this case the evaluation of the polynomial is equal to a signed enumeration of objects called Decreasing Monotone Triangles. In this paper we define Generalized Monotone Triangles - a joint generalization of both ordinary Monotone Triangles and Decreasing Monotone Triangles. As main result of the paper we prove that the evaluation of alpha(n;k1,k2,...,kn) at a…
We generalize a parity result of Fleishner and Stiebitz that being combined with Alon--Tarsi polynomial method allowed them to prove that a 4-regular graph formed by a Hamiltonian cycle and several disjoint triangles is always 3-choosable. Also we present a modification of polynomial method and show how it gives slightly more combinatorial information about colourings than direct application of Alon's Combinatorial Nullstellensatz.
For any northeast path $ν$, we define two bivariate polynomials associated with the $ν$-associahedron: the $F$- and the $H$-triangle. We prove combinatorially that we can obtain one from the other by an invertible transformation of variables. These polynomials generalize the classical $F$- and $H$-triangles of F.~Chapoton in type $A$. Our proof is completely new and has the advantage of providing a combinatorial explanation of the relation between the $F$- and $H$-triangle.
We are interested in the zero locus of a Chapoton's $F$-triangle as a polynomial in two real variables $x$ and $y$. An expectation is that (1) the $F$-triangle of rank $l$ as a polynomial in $x$ for each fixed $y\in[0,1]$, has exactly $l$ distinct real roots in $[0,1]$, and (2) $i$-th root $x_i(y)$ ($1 \le i \le l$) as a function on $y \in [0,1]$ is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of $F$-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above e…
We give three interpretations of the number $b$ of orbits of the Borel subgroup of upper triangular matrices on the variety $\ms{X}$ of complete quadrics. First, we show that $b$ is equal to the number of standard Young tableaux on skew-diagrams. Then, we relate $b$ to certain values of a modified Hermite polynomial. Third, we relate $b$ to a certain cell decomposition on $\ms{X}$ previously studied by De Concini, Springer, and Strickland. Using these, we give asymptotic estimates for $b$ as the dimension of the quadrics increases.
Let $G$ be a 3-partite graph with $k$ vertices in each part and suppose that between any two parts, there is no cycle of length four. Fischer and Matouusek asked for the maximum number of triangles in such a graph. A simple construction involving arbitrary projective planes shows that there is such a graph with $(1 - o(1)) k^{3/2} $ triangles, and a double counting argument shows that one cannot have more than $(1+o(1)) k^{7/4} $ triangles. Using affine planes defined by specific planar polynomials over finite fields, we improve the lower bound to $(1 - o(1)) k^{5/3}$.
Recently, Davies, Jenssen, Perkins, and Roberts gave a very nice proof of the result (due, in various parts, to Kahn, Galvin-Tetali, and Zhao) that the independence polynomial of a $d$-regular graph is maximized by disjoint copies of $K_{d,d}$. Their proof uses linear programming bounds on the distribution of a cleverly chosen random variable. In this paper, we use this method to give lower bounds on the independence polynomial of regular graphs. We also give new bounds on the number of independent sets in triangle-free regular graphs.
Log-concavity and almost log-convexity of the cycle index polynomials were proved by Bender and Canfield [J. Combin. Theory Ser. A 74 (1996)]. Schirmacher [J. Combin. Theory Ser. A 85 (1999)] extended them to $q$-log-concavity and almost $q$-log-convexity. Motivated by these, we consider the stronger properties total positivity from the Toeplitz matrix and Hankel matrix. By using exponential Riordan array methods, we give some criteria for total positivity of the triangular matrix of coefficients of the generalized cycle index polynomials, the Toeplitz matrix and Hankel matrix of the polynomia…
Motivated by the classical Eulerian number, descent and excedance numbers in the hyperoctahedral groups, an triangular array from staircase tableaux and so on, we study a triangular array $[\mathcal {T}_{n,k}]_{n,k\ge 0}$ satisfying the recurrence relation: \begin{equation*} \mathcal {T}_{n,k}=λ(a_0n+a_1k+a_2)\mathcal {T}_{n-1,k}+(b_0n+b_1k+b_2)\mathcal {T}_{n-1,k-1}+\frac{cd}λ(n-k+1)\mathcal {T}_{n-1,k-2} \end{equation*} with $\mathcal {T}_{0,0}=1$ and $\mathcal {T}_{n,k}=0$ unless $0\le k\le n$. We derive a functional transformation for its row-generating function $\mathcal{T}_n(x)$ from the…
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. Motzkin path) is a polynomial function in the length of the shorter base of the trapezoid. We also study the rational roots of the polynomials and formulate several conjectures and derive some partial results. Lastly, we deduce a c…
The number of Monotone Triangles with bottom row k1 < k2 < ... < kn is given by a polynomial alpha(n; k1,...,kn) in n variables. The evaluation of this polynomial at weakly decreasing sequences k1 >= k2 >= ... >= kn turns out to be interpretable as signed enumeration of new combinatorial objects called Decreasing Monotone Triangles. There exist surprising connections between the two classes of objects -- in particular it is shown that alpha(n; 1,2,...,n) = alpha(2n; n,n,n-1,n-1,...,1,1). In perfect analogy to the correspondence between Monotone Triangles and Alternating Sign Matrices, the set …
In this paper we calculate the Ehrhart's polynomial associated with a 2-dimensional regular polytope (i.e. equilateral triangles) in $\mathbb Z^3$. The polynomial takes a relatively simple form in terms of the coordinates of the vertices of the polytope and it depends heavily on the value $d$ and its divisors, where $d=\sqrt{\frac{a^2+b^2+c^2}{3}}$ and $(a,b,c)$ ($\gcd(a,b,c)=1$) is a vector with integer coordinates normal to the plane containing the triangle.
A common theme in many extremal problems in graph theory is the relation between local and global properties of graphs. One of the most celebrated results of this type is the Ruzsa-Szemerédi triangle removal lemma, which states that if a graph is $\varepsilon$-far from being triangle free, then most subsets of vertices of size $C(\varepsilon)$ are not triangle free. Unfortunately, the best known upper bound on $C(\varepsilon)$ is given by a tower-type function, and it is known that $C(\varepsilon)$ is not polynomial in $\varepsilon^{-1}$. The triangle removal lemma has been extended to many ot…
The Ehrhart polynomial of a lattice polygon P is completely determined by the pair (b(P),i(P)) where b(P) equals the number of lattice points on the boundary and i(P) equals the number of interior lattice points. All possible pairs (b(P),i(P)) are completely described by a theorem due to Scott. In this note, we describe the shape of the set of pairs (b(T),i(T)) for lattice triangles T by finding infinitely many new Scott-type inequalities.
In this paper, we show that Alexander polynomials for any 2-bridge knots are specializations of cluster variables. A key tool is an ancestral triangle which appeared in both quantum topology and hyperbolic geometry in different ways.
In this paper, we give a polynomial time algorithm which determines if a given graph containing a triangle and no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists. In previous work, we gave a polynomial time algorithm for three-coloring triangle-free graphs with no induced seven-vertex path. Combined, our work shows that three-coloring a graph with no induced seven-vertex path can be done in polynomial time.
It is well-known that the binomial transformation preserves the log-concavity property and log-convexity property. Let $\binom{a+n}{b+k}$ be the binomial coefficients and $\binom{n,k}{j}$ be defined by $(b_0+b_1x+\cdots+b_kx^{k})^n:=\sum_{j=0}^{kn}\binom{n,k}{j}x^j,$ where the sequence $(b_i)_{0\leq i\leq k}$ is log-concave. In this paper, we prove that the linear transformation $$y_n(q)=\sum_{k=0}^n\binom{a+n}{b+k}x_k(q)$$ preserves the strong $q$-log-concavity property for any fixed nonnegative integers $a$ and $b$, which strengthens and gives a simple proof of results of Ehrenborg and Stein…
The weighted $\mathcal{T}$-free $2$-matching problem is the following problem: given an undirected graph $G$, a weight function on its edge set, and a set $\mathcal{T}$ of triangles in $G$, find a maximum weight $2$-matching containing no triangle in $\mathcal{T}$. When $\mathcal{T}$ is the set of all triangles in $G$, this problem is known as the weighted triangle-free $2$-matching problem, which is a long-standing open problem. A main contribution of this paper is to give a first polynomial-time algorithm for the weighted $\mathcal{T}$-free $2$-matching problem under the assumption that $\ma…
In the theory of finite order knot invariants, the universal $\text{sl}_2$ weight system maps the chord diagrams to polynomials in a single variable with integer coefficients. In this paper, we define a family of polynomials that generalize the Kreweras triangle (known to refine the normalized median Genocchi numbers), and we show how it appears in this weight system.
The Maximum Weight Independent Set Problem (WIS) is a well-known NP-hard problem. A popular way to study WIS is to detect graph classes for which WIS can be solved in polynomial time, with particular reference to hereditary graph classes, i.e., defined by a hereditary graph property or equivalently by forbidding one or more induced subgraphs. Given two graphs $G$ and $H$, $G+H$ denotes the disjoint union of $G$ and $H$. This manuscript shows that (i) WIS can be solved for ($P_4+P_4$, Triangle)-free graphs in polynomial time, where a $P_4$ is an induced path of four vertices and a Triangle is a…
A polynomial triangle is an array whose inputs are the coefficients in integral powers of a polynomial. Although polynomial coefficients have appeared in several works, there is no systematic treatise on this topic. In this paper we plan to fill this gap. We describe some aspects of these arrays, which generalize similar properties of the binomial coefficients. Some combinatorial models enumerated by polynomial coefficients, including lattice paths model, spin chain model and scores in a drawing game, are introduced. Several known binomial identities are then extended. In addition, we calculat…
We calculate moments of the so-called Kesten distribution by means of the expansion of the denominator of the density of this distribution and then integrate all summands with respect to the semicircle distribution. By comparing this expression with the formulae for the moments of Kesten's distribution obtained by other means, we find identities involving polynomials whose power coefficients are closely related to Catalan numbers, Catalan triangles, binomial coefficients. Finally, as applications of these identities we obtain various interesting relations between the aforementioned numbers, al…
A "truncation" of Pascal's triangle is a triangular array of numbers that satisfies the usual Pascal recurrence but with a boundary condition that declares some terminal set of numbers along each row of the array to be zero. Presented here is a family of natural truncations of Pascal's triangle that generalize a kind of Catalan triangle. The numbers in each array are realized as differences of binomial coefficients, as counts of certain lattice paths and tableaux, and as entries of representing matrices for certain linear transformations of polynomial spaces. Lucas's theorem is applied to dete…
In this paper, we present some criteria for the $2$-$q$-log-convexity and $3$-$q$-log-convexity of combinatorial sequences, which can be regarded as the first column of certain infinite triangular array $[A_{n,k}(q)]_{n,k\geq0}$ of polynomials in $q$ with nonnegative coefficients satisfying the recurrence relation $$A_{n,k}(q)=A_{n-1,k-1}(q)+g_k(q)A_{n-1,k}(q)+h_{k+1}(q)A_{n-1,k+1}(q).$$ Those criterions can also be presented by continued fractions and generating functions. These allow a unified treatment of the $2$-$q$-log-convexity of alternating Eulerian polynomials, $2$-log-convexity of Eu…
In this note, we formulate and prove branching rules of simple polynomial modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. Our branching rules depend on the conjugacy class of the Borel subalgebra. A Gelfand-Tsetlin basis of a polynomial module associated to each Borel subalgebra is obtained in terms of generalized semistandard tableaux.
We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are co…
In this paper, we give a polynomial time algorithm which determines if a given triangle-free graph with no induced seven-vertex path is 3-colorable, and gives an explicit coloring if one exists.
In this paper, we show the existence of a polynomial time graph isomorphism algorithm for all graphs excluding graphs that are locally trianglefree. This particular class of graphs allows to divide the graph into neighbourhood sub-graph where each of induced sub-graph (neighbourhood) has at least 2 vertices. We construct all possible permutations for each induced sub-graph using a search tree. We construct automorphisms of subgraphs based on these permutations. Finally, we decide isomorphism through automorphisms . The author expects that the solution, present in this paper, may lead to a fast…
Intersection graphs of geometric objects have been extensively studied, both due to their interesting structure and their numerous applications; prominent examples include interval graphs and permutation graphs. In this paper we study a natural graph class that generalizes both interval and permutation graphs, namely \emph{simple-triangle} graphs. Simple-triangle graphs - also known as \emph{PI} graphs (for Point-Interval) - are the intersection graphs of triangles that are defined by a point on a line $L_{1}$ and an interval on a parallel line $L_{2}$. They lie naturally between permutation a…
To any combinatorial triangulation $T$ of a square, there is an associated polynomial relation $p_T$ among the areas of the triangles of $T$. With the goal of understanding this polynomial, we consider polynomials obtained from $p_T$ by choosing $l$ of its variables and specializing $p_T$ to these variables by zeroing out the remaining variables. We show that for fixed $l$, the set ${\mathcal E}_l$ of integer polynomials that appear as irreducible factors of such specializations is finite. We compute this area encyclopedia ${\mathcal E}_l$ for $l\leq 4$. We also show that in any dissection of …
We study a family of infinite-type Coxeter groups defined by the avoidance of certain rank 3 parabolic subgroups. For this family, rationally smooth elements can be detected by looking at only a few coefficients of the Poincaré polynomial. We also prove a factorization theorem for the Poincaré polynomial of rationally smooth elements. As an application, we show that a large class of infinite-type Coxeter groups have only finitely many rationally smooth elements. Explicit enumerations and descriptions of these elements are given in special cases.
The main perpose of this paper is to sudy the roots of a familly of polynomials that arise from a linear recurrences associated to Pascal's triangle and their zero attractor, using an analytical methods based on conformal mappings.
In a recent paper, Cristofaro-Gardiner--Li--Stanley [CGLS15] constructed examples of irrational triangles whose Ehrhart functions (i.e. lattice-point count) are polynomials when restricted to positive integer dilation factors. This is very surprising because the Ehrhart functions of rational polygons are usually only quasi-polynomials. We demonstrate that most of their triangles can also be obtained by a simple cut-and-paste procedure that allows us to build new examples with more sides. Our examples might potentially have applications in the theory of symplectic embeddings.
