Positive Integer Coefficients Research Articles

A combinatorial proof of a formula for the Lucas-Narayana polynomials

In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for 1≤k≤n and for n≥1. In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers Nn,k,F, providing a combinatorial proof of the conjecture that these are positive integers for n≥1.

Necessary and sufficient conditions for convergence of integer continued fractions

Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, it is unknown precisely which continued fractions with integer coefficients (not necessarily positive) converge. Here we present a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions the test specifies whether the limit is rational or irrational. An attractive way to visualise integer continued fractions is to model them as paths on the Farey graph, which is a graph embedded in the hyperbolic plane that induces a tessellation of the hyperbolic plane by ideal triangles. With this geometric representation of continued fractions our test for convergence can be interpreted in a particularly elegant manner, giving deeper insight into the nature of continued fraction convergence.

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of $\rho$-Removed Uniform Matroids

Let $\rho$ be a non-negative integer. A $\rho$-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of $\rho$ disjoint bases. We present a combinatorial formula for Kazhdan–Lusztig polynomials of $\rho$-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.

A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain

We study the connection between the three-color model and the polynomials $q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for $q_n(z)$ in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We prove the weaker statement that $q_n(z+1)$ and $(z+1)^{n(n+1)}q_n(1/(z+1))$ have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.

Arbitrarily long gaps between the values of positive‐definite cubic and biquadratic diagonal forms

For s = 3 , 4 , we prove the existence of arbitrarily long sequences of consecutive integers none of which is a sum of s nonnegative sth powers. More generally, we study the existence of gaps between the values ⩽ N of diagonal forms of degree s in s variables with positive integer coefficients. We find: (1) gaps of size ≫ log N ( log log N ) 2 when s = 3 ; (2) gaps of size ≫ log log log N log log log log N if s = 4 and the form, up to permutation of the variables, is not equal to a ( c 1 x 1 ) 4 + b ( c 2 x 2 ) 4 + 4 a ( c 3 x 3 ) 4 + 4 b ( c 4 x 4 ) 4 .

Laurent Positivity of Quantized Canonical Bases for Quantum Cluster Varieties from Surfaces

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichm\"uller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an ideal triangulation of $S$, each quantum basis element is a Laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation, with coefficients being Laurent polynomials in $q$ with integer coefficients. We show that these coefficients are Laurent polynomials in $q$ with positive integer coefficients. Our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it, and may also lead to other positivity results, as well as categorification. A key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on $S$. For this problem we introduce a certain graph on $S$, which is interesting in its own right.

Hecke relations in rational conformal field theory

We define Hecke operators on vector-valued modular forms of the type that appear as characters of rational conformal field theories (RCFTs). These operators extend the previously studied Galois symmetry of the modular representation and fusion algebra of RCFTs to a relation between RCFT characters. We apply our results to derive a number of relations between characters of known RCFTs with different central charges and also explore the relation between Hecke operators and RCFT characters as solutions to modular linear differential equations. We show that Hecke operators can be used to construct an infinite set of possible characters for RCFTs with two independent characters and increasing central charge. These characters have multiplicity one for the vacuum representation, positive integer coefficients in their q expansions, and are associated to a two-dimensional representation of the modular group which leads to non-negative integer fusion coefficients as determined by the Verlinde formula.

Convergence to Zero of Exponential Sums with Positive Integer Coefficients and Approximation by Sums of Shifts of a Single Function on the Line

We prove that there is a sequence of trigonometric polynomials with positive integer coefficients, which converges to zero almost everywhere. We also prove that there is a function f: ℝ → ℝ such that the sums of its shifts are dense in all real spaces L p (ℝ) for 2 ≤ p < ∞ and also in the real space C0(R).

SL$_2 (\mathbb Z)$-tilings of the torus, Coxeter–Conway friezes and Farey triangulations

The notion of SL$\_2$-tiling is a generalization of that of classical Coxeter–Conway frieze pattern. We classify doubly antiperiodic SL$\_2$-tilings that contain a rectangular domain of positive integers. Every such SL$\_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2 \times 2$-matrix with positive integer coeffi cients. We relate this notion to triangulated $n$-gons in the Farey graph.

Statistics on Lattice Walks and q-Lassalle Numbers

This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

On Hurwitz Stable Polynomials with Integer Coefficients

Let $$H(N)$$ denote the set of all polynomials with positive integer coefficients which have their zeros in the open left half-plane. We are looking for polynomials in $$H(N)$$ whose largest coefficients are as small as possible and also for polynomials in $$H(N)$$ with minimal sum of the coefficients. Let $$h(N)$$ and $$s(N)$$ denote these minimal values. Using Fekete’s subadditive lemma we show that the $$N$$ th square roots of $$h(N)$$ and $$s(N)$$ have a limit as $$N$$ goes to infinity and that these two limits coincide. We also derive tight bounds for the common value of the limits.

On Products of Irreducible Characters and Products of Conjugacy Classes in Finite Groups

Let G a finite group. For conjugacy classes A and B of G, the subset AB of G is a union of conjugacy classes of G, and for irreducible complex characters χ and ϕ of G, the product χϕ is a linear combination of irreducible characters of G with positive integer coefficients. In this article we give the structure of finite groups G whenever C 2 contains a central conjugacy class or does not contain a central conjugacy class of G, for all conjugacy classes C of G. Also in the case of (C −1) m C n ⊆ Z(G) the structure of G is given. Similar results when C is replaced by an irreducible character χ of G are discussed.

A combinatorial proof of a result for permutation pairs

In this paper, a direct combinatorial proof is given of a result on permutation pairs originally due to Carlitz, Scoville, and Vaughan and later extended. It concerns showing that the series expansion of the reciprocal of a certain multiply exponential generating function has positive integer coefficients. The arguments may then be applied to related problems, one of which concerns the reciprocal of the exponential series for Fibonacci numbers.

Chirp transforms and Chirp series

Motivated by the recent work on the non-harmonic Fourier atoms initiated by T. Qian and the non-harmonic Fourier series which originated from the celebrated work of Paley and Wiener, we introduce an integral version of the non-harmonic Fourier series, called Chirp transform. As an integral transform with kernel e i ϕ ( t ) θ ( ω ) , the Chirp transform is an unitary isometry from L 2 ( R , d ϕ ) onto L 2 ( R , d θ ) and it can be explicitly defined in terms of generalized Hermite polynomials. The corresponding Chirp series take e i n θ ( t ) as a basis which in some sense is dual to the theory of non-harmonic Fourier series which take e i λ n t as a basis. The Chirp version of the Shannon sampling theorem and the Poisson summation formula are also considered by dealing with sampling points which may non-equally distributed. Since the Chirp transform interchanges weighted derivatives into multiplications, it plays a role in solving certain differential equations with variable coefficients. In addition, we extend T. Qian's theorem on the characterization of a measure to be a linear combination of a number of harmonic measures on the unit disc with positive integer coefficients to that with positive rational coefficients.

A Formula for Plucker Coordinates Associated with a Planar Network

For a planar directed graph G, Postnikov's boundary measurement map sends positive weight functions on the edges of G onto the appropriate totally non-negative Grassmann cell. We establish an explicit formula for Postnikov's map by expressing each Plucker coordinate as a ratio of two combinatorially defined polynomials in the edge weights, with positive integer coefficients. In the non-planar setting, we show that a similar formula holds for special choices of Plucker coordinates.

On Plücker coordinates of a perfectly oriented planar network

Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.

Skew Divided Difference Operators and Schubert Polynomials

We study an action of the skew divided difference operators on the Schubert polynomials and give an explicit formula for structural constants for the Schubert polynomials in terms of certain weighted paths in the Bruhat order on the symmetric group. We also prove that, under certain assumptions, the skew divided difference operators transform the Schubert polynomials into polynomials with positive integer coefficients.

Hilbert series, Howe duality, and branching rules.

Let lambda be a partition, with l parts, and let F(lambda) be the irreducible finite dimensional representation of GL(m) associated to lambda when l < or = m. The Littlewood Restriction Rule describes how F(lambda) decomposes when restricted to the orthogonal group O(m) or to the symplectic group Sp(m2) under the condition that l < or = m2. In this paper, this result is extended to all partitions lambda. Our method combines resolutions of unitary highest weight modules by generalized Verma modules with reciprocity laws from the theory of dual pairs in classical invariant theory. Corollaries include determination of the Gelfand-Kirillov dimension of any unitary highest weight representation occurring in a dual pair setting, and the determination of their Hilbert series (as a graded module for p(-)). Let L be a unitary highest weight representation of sp(n, R), so*(2n), or u(p, q). When the highest weight of L plus rho satisfies a partial dominance condition called quasi-dominance, we associate to L a reductive Lie algebra g(L) and a graded finite dimensional representation B(L) of g(L). The representation B(L) will have a Hilbert series P(q) that is a polynomial in q with positive integer coefficients. Let delta(L) = delta be the Gelfand-Kirillov dimension of L and set c(L) equal to the ratio of the dimensions of the zeroth levels in the gradings of L and B(L). Then the Hilbert series of L may be expressed in the form H(L)(q)=c(L) P(q)(1-q)(delta). In the easiest example of the correspondence L --> B(L), the two components of the Weil representation of the symplectic group correspond to the two spin representations of an orthogonal group.

Constructing set systems with prescribed intersection sizes

Let f be an n-variable polynomial with positive integer coefficients, and let H={H 1,H 2,…,H m} be a set system on the n-element universe. We define a set system f( H)={G 1,G 2,…,G m} and prove that f( H i 1 ∩ H i 2 ∩⋯∩ H i k )=| G i 1 ∩ G i 2 ∩⋯∩ G i k |, for any 1⩽ k⩽ m, where f( H i 1 ∩ H i 2 ∩⋯∩ H i k ) denotes the value of f on the characteristic vector of H i 1 ∩ H i 2 ∩⋯∩ H i k . The construction of f( H) is a straightforward polynomial-time algorithm from the set system H and the polynomial f. In this paper we use this algorithm for constructing set systems with prescribed intersection sizes modulo an integer. As a by-product of our method, some upper bounds on the number of sets in set systems with prescribed intersection sizes are extended.

A proof of the q, t-Catalan positivity conjecture

We present here a proof that a certain rational function C n ( q, t) which has come to be known as the “ q, t- Catalan” is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. The precise form of the conjecture is given in Garsia and Haiman (J. Algebraic Combin. 5(3) (1996) 191), where it is further conjectured that C n ( q, t) is the Hilbert series of the diagonal harmonic alternants in the variables ( x 1, x 2,…, x n ; y 1, y 2,…, y n ). Since C n ( q, t) evaluates to the Catalan number at t= q=1, it has also been an open problem to find a pair of statistics a( π), b( π) on Dyck paths π in the n × n square yielding C n ( q, t)=∑ π t a( π) q b( π) . Our proof is based on a recursion for C n ( q, t) suggested by a pair of statistics a( π), b( π) recently proposed by Haglund. Thus, one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic machinery developed in Bergeron et al. (Methods and Applications of Analysis, Vol. VII(3), 1999, p. 363).