Combinatorics of certain higher q,t-Catalan polynomials: chains,joint symmetry,and the Garsia–Haiman formula

The higher q,t-Catalan polynomial $C^{(m)}_{n}(q,t)$ can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of n. This paper proves the equivalence of the two definitions for all m≥1 and all n≤4. We also give a bijective proof of the joint symmetry property $C^{(m)}_{n}(q,t)=C^{(m)}_{n}(t,q)$ for all m≥1 and all n≤4. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in $C^{(m)}_{n}(q,t)$ for all m≥1 and all n≤4. We give analogous results for certain rational-slope q,t-Catalan polynomials.     

Conjectured statistics for the q,t-Catalan numbers

We introduce the distribution function F_n(q,t) of a pair of statistics on Catalan words and conjecture F_n(q,t) equals Garsia and Haiman's q,t-Catalan sequence C_n(q,t), which they defined as a sum of rational functions. We show that F_n,s(q,t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that F_n(q,t)=C_n(q,t) when t=1/q. We also show the partial symmetry relation F_n(q,1)=F_n(1,q). By modifying a proof of Haiman of a q-Lagrange inversion formula based on results of Garsia and Gessel, we obtain a q-analogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word.     

On sizes of complete caps in projective spaces PG(n, q) and arcs in planes PG(2, q)

More than thirty new upper bounds on the smallest size t ₂(2, q) of a complete arc in the plane PG(2, q) are obtained for (169 ≤ q ≤ 839. New upper bounds on the smallest size t ₂(n, q) of the complete cap in the space PG(n, q) are given for n = 3 and 25 ≤ q ≤ 97, q odd; n = 4 and q = 7, 8, 11, 13, 17; n = 5 and q = 5, 7, 8, 9; n = 6 and q = 4, 8. The bounds are obtained by computer search for new small complete arcs and caps. New upper bounds on the largest size m ₂(n, q) of a complete cap in PG(n, q) are given for q = 4, n = 5, 6, and q = 3, n = 7, 8, 9. The new lower bound 534 ≤ m ₂(8, 3) is obtained by finding a complete 534-cap in PG(8, 3). Many new sizes of complete arcs and caps are obtained. The updated tables of upper bounds for t ₂(n, q), n ≥ 2, and of the spectrum of known sizes for complete caps are given. Interesting complete caps in PG(3, q) of large size are described. A proof of the construction of complete caps in PG(3, 2 ^h ) announced in previous papers is given; this is modified from a construction of Segre. In PG(2, q), for q = 17, δ = 4, and q = 19, 27, δ = 3, we give complete ${(\frac{1}{2}(q + 3) + \delta)}$ -arcs other than conics that share ${\frac{1}{2}(q + 3)}$ points with an irreducible conic. It is shown that they are unique up to collineation. In PG(2, q), ${{q \equiv 2}}$ (mod 3) odd, we propose new constructions of ${\frac{1}{2} (q + 7)}$ -arcs and show that they are complete for q ≤ 3701.     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

A new symmetry for Biedenharn's G-functions and classical hypergeometric series

We show that the general bisymmetric polynomials ^m_μG⁽ⁿ⁾_q(γ₁,…, γ_n; δ₁,…, δ_m) are a limiting case of the bisymmetric, invariant polynomials ⁿ_μG⁽ⁿ⁾_q(γ₁,…, γ_n; δ₁,…, δ_n) which characterize U(n) tensor operators 〈p, q,…, q, 0, …, 0〉. By taking suitable limits of a pair of difference equations for ⁿ_μG⁽ⁿ⁾_q(γ; δ) we then deduce “transposition symmetry” for ^m_μG⁽ⁿ⁾_q(γ; δ) from the same symmetry for ⁿ_μG⁽ⁿ⁾_q(γ; δ). As an application of transposition symmetry for ^m_μG⁽ⁿ⁾_q(γ; δ) we derive an elegant, new contiguous relation for classical, well-poised hypergeometric series, and also prove an identity between these series and multiple hypergeometric series well-poised in SU(n).     

Newton Polygons of L Functions Associated with Exponential Sums of Polynomials of Degree Four over Finite Fields

Let F_q be the finite field of q elements with characteristic p and F_q^m its extension of degree m. Fix a nontrivial additive character Ψ of F_p. If f(x₁,…, x_n)∈F_q[x₁,…, x_n] is a polynomial, then one forms the exponential sum S_m(f)=∑_{(x1,…,xn)∈(Fq^m)ⁿ}Ψ(Tr_Fq^m/Fp(f(x₁,…,x_n))). The corresponding L functions are defined by L(f, t)=exp(∑^∞_m=0S_m(f)t^m/m). In this paper, we apply Dwork's method to determine the Newton polygon for the L function L(f(x), t) associated with one variable polynomial f(x) when deg f(x)=4. As an application, we also give an affirmative answer to Wan's conjecture for the case deg f(x)=4.     

A linear-algebra problem from algebraic coding theory

Given an m×n matrix M over E=GF(q^t) and an ordered basis A={z₁,…,z_t} for field E over K=GF(q), expand each entry of M into a t×1 vector of coordinates of this entry relative to A to obtain an mt×n matrix M¹ with entries from the field K. Let r=rank(M) and r¹=rank(M¹). We show that r?r¹?min{rt,n}, and we determine the number b(m,n,r,r¹,q,t) of m×n matrices M of rank r over GF(q^t) with associated mt×n matrix M¹ of rank r¹ over GF (q).     

Blocking Sets in PG(r, q n )

Let $\mathcal S$ be a Desarguesian (n – 1)-spread of a hyperplane Σ of PG(rn, q). Let Ω and ${\bar B}$ be, respectively, an (n – 2)-dimensional subspace of an element of $\mathcal S $ and a minimal blocking set of an ((r – 1)n + 1)-dimensional subspace of PG(rn, q) skew to Ω. Denote by K the cone with vertex Ω and base ${\bar B}$ , and consider the point set B defined by $$B=\left(K\setminus\Sigma\right)\cup \{X\in \mathcal S\, : \, X\cap K\neq \emptyset\}$$ in the Barlotti–Cofman representation of PG(r, q ⁿ ) in PG(rn, q) associated to the (n – 1)-spread $\mathcal S$ . Generalizing the constructions of Mazzocca and Polverino (J Algebraic Combin, 24(1):61–81, 2006), under suitable assumptions on ${\bar B}$ , we prove that B is a minimal blocking set in PG(r, q ⁿ ). In this way, we achieve new classes of minimal blocking sets and we find new sizes of minimal blocking sets in finite projective spaces of non-prime order. In particular, for q a power of 3, we exhibit examples of r-dimensional minimal blocking sets of size q ⁿ⁺² + 1 in PG(r, q ⁿ ), 3 ≤ r ≤ 6 and n ≥ 3, and of size q ⁴ + 1 in PG(r, q ²), 4 ≤ r ≤ 6; actually, in the second case, these blocking sets turn out to be the union of q ³ Baer sublines through a point. Moreover, for q an even power of 3, we construct examples of minimal blocking sets of PG(4, q) of size at least q ² + 2. From these constructions, we also get maximal partial ovoids of the hermitian variety H(4, q ²) of size q ⁴ + 1, for any q a power of 3.     

Oscillatory and asymptotic behavior of solutions of differential equations with deviating arguments

The oscillatory and asymptotic behavior of solutions of a class of nth order nonlinear differential equations, with deviating arguments, of the form (E, δ) L_nx(t) + δq(t) f(x[g₁(t)],…, x[g_m(t)]) = 0, where δ = ± 1 and L₀x(t) = x(t), L_kx(t) = a_k(t)(L_{k ? 1}x(t))^., $\text{k = 1, 2,…, n (}{}^{\mathrm{.}}\text{=}\text{d}\text{dt}\text{)}$, is examined. A classification of solutions of (E, δ) with respect to their behavior as t → ∞ and their oscillatory character is obtained. The comparisons of (E, 1) and (E, ?1) with first and second order equations of the form y^.(t) + c₁(t) f(y[g₁(t)],…, y[g_m(t)]) = 0 and (a_{n ? 1}(t)z^.(t))^. ? c₂(t) f(z[g₁(t)],…, z[g_m(t)]) = 0, respectively, are presented. The obtained results unify, extend and improve some of the results by Graef, Grammatikopoulos and Spikes, Philos and Staikos.     

The supertail of a subspace partition

Let V = V(n, q) be a vector space of dimension n over the finite field with q elements, and let d ₁ < d ₂ < ... < d _m be the dimensions that occur in a subspace partition ${\mathcal{P}}$ of V. Let σ _q (n, t) denote the minimum size of a subspace partition ${\mathcal P}$ of V, in which t is the largest dimension of a subspace. For any integer s, with 1 < s ≤ m, the set of subspaces in ${\mathcal{P}}$ of dimension less than d _s is called the s-supertail of ${\mathcal{P}}$ . The main result is that the number of spaces in an s-supertail is at least σ _q (d _s , d _s?1).     

On the Attainable Order of Collocation Methods for Delay Differential Equations with Proportional Delay

To analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods for the delay differential equation (DDE) y′(t) = by(qt), 0 < q ≤ 1 with y(0) = 1, and the delay Volterra integral equation (DVIE) y(t) = 1 + $\tfrac{b}{q}\int {_0^{qt} }$ y(s) ds with proportional delay qt, 0 < q ≤ 1, our particular interest lies in the approximations (and their orders) at the first mesh point t = h for the collocation solution v(t) of the DDE and the iterated collocation solution u _it(t) of the DVIE to the solution y(t). Recently, H. Brunner proposed the following open problem: “For m ≤ 3, do there exist collocation points c _i = c _i(q), i = 1, 2,..., m in [0,1] such that the rational approximant v(h)is the (m, m)-Padé approximant to y(h)? If these exist, then |v(h) ? y(h)| = O(h ^2m+1) but what is the collocation polynomial M _m(t; q) = K Π _i=1 ^m (t ? c _i) of v(th), t ∈ [0, 1]?” In this paper, we solve this question affirmatively, and give the related results between the collocation solution v(t) of the DDE and the iterated collocation solution u _it(t) of the DVIE. We also answer to Brunner's second open question in the case that one collocation point is fixed at the right end point of the interval.     

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let K _q (n, w, t, d) be the minimum size of a code over Z _q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K _q (n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2 ^m  + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.     

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 (x₁,x₂,…,x_n;y₁,y₂,…,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)=∑_πt^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).     

Translation planes constructed by multiple hyper-regulus replacement

New classes of mutually disjoint hyper-reguli of order q ⁿ , for n > 2, are determined, which are not André hyper-reguli if n > 3. If n is odd, each hyper-regulus permits at least two replacements and if q is odd and (n, q?1) = 1, there are (q?1)/2 mutually disjoint hyper-reguli each of which may be replaced at least two ways. Each of the possible 2^(q-1)/2 translation planes constructed is not André or generalized André. There are also new constructions of mixed subgeometry partitions.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Coefficients of Drinfeld modular forms and Hecke operators

Consider the space of Drinfeld modular forms of fixed weight and type for Γ₀(n)⊂GL₂(Fq[T]). It has a linear form bn, given by the coefficient of tm+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b₁. We give elements in the Hecke annihilator of b₁. Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients bn, where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.     

A contribution to the Zarankiewicz problem

Given positive integers let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m×n that does not contain an all ones submatrix of size s×t. We show that if s?2 and t?2, then for every k=0,…,s-2,

z(m,n,s,t)?(s-k-1)^1/tnm^1-1/t+kn+(t-1)m^1+k/t.     

Projective Varieties Swept Out by Rational Normal Curves

Let X ? ? ⁿ be a complex nondegenerate projective variety of dimension m ≥ 2. For t ≤ n ? m and a general q ∈ ? ⁿ , the linear space L _q spanned by q and t general points of X meets X in a finite set of points. We classify those X ? ? ⁿ for which there exists a point q ∈ ? ⁿ such that L _q meets X in a positive dimensional variety. If this occurs, there exists d ≤ n ? m such that a degree d rational normal curve through d general points of X is contained in X. Examples of this situation are provided. An infinitesimal generalization of part of the main result is also stated.