We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume non-central sections of the cross-polytope by hyperplanes which are at a fixed large distance to the origin and minimal volume sections by symmetric slabs of a large fixed width. This parallels recent results about non-central sections of the cube due to

D-finite (or holonomic) functions satisfy linear differential equations with polynomial coefficients. They form a large class of functions that appear in many applications in Mathematics or Physics. It is well-known that these functions are closed under certain operations and these closure properties can be executed algorithmically. Recently, the notion of D-finite functions has been generalized to

We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again

We consider a finite field model of the X-ray transform that integrates functions along lines in dimension 3, within the context of finite fields. The admissibility problem asks for minimal sets of lines for which the restricted transform is invertible. Graph theoretic conditions are known which characterize admissible collections of lines, and these have been counted using a brute force computer program

The m-Tamari lattice of F. Bergeron is an analogue of the classical Tamari order defined on objects counted by Fuss-Catalan numbers, such as m-Dyck paths or (m+1)-ary trees. On another hand, the Tamari order is related to the product in the Loday-Ronco Hopf algebra of planar binary trees. We introduce new combinatorial Hopf algebras based on (m+1)-ary trees, whose structure is described by the m-Tamari

We study the concentration of the norm of a random vector Y uniformly sampled in the centered zero cell of two types of stationary and isotropic random mosaics in Rn for large dimension n. For a stationary and isotropic Poisson-Voronoi mosaic, Y has a radial and log-concave distribution, implying that |Y|/E(|Y|2)12 approaches one for large n. Assuming the cell intensity of the random mosaic scales

Let A be a set of vertices in a graph G. An A-path is a non-trivial path in G that has both of its ends in A. In 1961, Gallai showed that, for any integer k, either G has k disjoint A-paths or there is a set of at most 2(k−1) vertices that hits all of the A-paths. There have been a number of extensions of this result; in each of these extensions we want to find a maximum collection of disjoint “allowable”

We give a new proof of Rado's Theorem that, given a partition (X1,…,Xn) of the ground set of a matroid M, there is an independent set of M containing exactly one element from each of the sets (X1,…,Xn) if and only if for each subset J of {1,…,n} we have r(∪j∈JXj)≥|J|.

We propose an algebraic geometric approach for studying rational solutions of first-order algebraic ordinary difference equations (AOΔEs). For an autonomous first-order AOΔE, we give an upper bound for the degrees of its rational solutions, and thus derive a complete algorithm for computing corresponding rational solutions.

We derive the connection relations between three q-Taylor polynomial bases. We also derive the connection relations between two of these bases and the continuous q-Hermite polynomials, as well as several new generating functions for the continuous q-Hermite polynomials. Our results also lead to some new integral evaluations. Using one of these connection relations, we give a new proof of a general

Crofton's intersection formula states that the (n−j)th intrinsic volume of a compact convex set in Rn can be obtained as an invariant integral of the (k−j)th intrinsic volume of sections with k-planes. This paper discusses the question if the (k−j)th intrinsic volume can be replaced by other functionals, that is, if the measurement function in Crofton's formula is unique. The answer is negative: we

The Loomis-Whitney inequality states that the volume of a convex body is bounded by the product of volumes of its projections onto orthogonal hyperplanes. We provide an extension of both this fact and a generalization of this fact due to Ball to the context of q−concave, 1q−homogeneous measures.

We study the longest increasing subsequence problem for random permutations avoiding the pattern 312 and another pattern τ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern τ is monotone increasing or decreasing, or any pattern of length four

We establish a q-analogue of Sun–Zhao's congruence on harmonic sums. Based on this q-congruence and a q-series identity, we prove a congruence conjecture on sums of central q-binomial coefficients, which was recently proposed by Guo. We also deduce a q-analogue of a congruence due to Apagodu and Zeilberger from Guo's q-congruence.

Let Ω be a bounded convex domain in Rn (n≥2). In this work, we prove that if there exists an integrable function f such that it's Radon transform over (n−1)-dimensional hyperplanes intersecting the domain Ω is a function G depending on the distance to the nearest parallel supporting hyperplane to Ω, then Ω is a ball and f is radial depending on certain assumptions on G. As a consequence we show that

Our main contribution here is the discovery of a new family of standard Young tableaux Tnk which are in bijection with the family Dm,n of Rational Dyck paths for m=k×n±1 (the so called “Fuss” case). Using this family we give a new proof of the invertibility of the sweep map in the Fuss case by means of a very simple explicit algorithm. This new algorithm has running time O(m+n). It is independent of

For n≥3, let r=r(n)≥3 be an integer. A hypergraph is r-uniform if each edge is a set of r vertices, and is said to be linear if two edges intersect in at most one vertex. In this paper, the number of linear r-uniform hypergraphs on n→∞ vertices is determined asymptotically when the number of edges is m(n)=o(r−3n3/2). As one application, we find the probability of linearity for the independent-edge

We derive a formula expressing the joint distribution of the cyclic valley number and excedance number statistics over a fixed conjugacy class of the symmetric group in terms of Eulerian polynomials. Our proof uses a slight extension of Sun and Wang's cyclic valley-hopping action as well as a formula of Brenti. Along the way, we give a new proof for the γ-positivity of the excedance number distribution

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit-hyperplanes is a strongly fractal class when k is at least three

In this paper, by applying a range of classic summation and transformation formulas for basic hypergeometric series, we obtain a three-term identity for partial theta functions. It extends the Andrews–Warnaar partial theta function identity, and also unifies several results on partial theta functions due to Ramanujan, Kim and Lovejoy. We also establish a two-term version of the extension, which can

For any quiver mutation sequence, we define a pair of matrices that describe a fixed point equation of a cluster transformation determined from the mutation sequence. We give an explicit relationship between this pair of matrices and the Jacobian matrix of the cluster transformation. Furthermore, we show that this relationship reduces to a relationship between the pair of matrices and the C-matrix

For a graph G with a positive clustering coefficient C, it is proved that for any positive constant ϵ, the vertex set of G can be partitioned into finitely many parts, say S1,S2,…,Sm, such that all but an ϵ fraction of the triangles in G are contained in the projections of tripartite subgraphs induced by (Si,Sj,Sk) which are ϵ-Δ-regular, where the size m of the partition depends only on ϵ and C. The

What is the product of all odious integers, i.e., of all integers whose binary expansion contains an odd number of 1's? Or more precisely, how to define a product of these integers which is not infinite, but still has a “reasonable” definition? We will answer this question by proving that this product is equal to π1/42φe−γ, where γ and φ are respectively the Euler-Mascheroni and the Flajolet-Martin

In three influential papers in the 1980s and early 1990s, Joe Kung laid the foundations for extremal matroid theory which he envisaged as finding the growth rate of certain classes of matroids along with a characterisation of the extremal matroids in each such class. At the time, he was particularly interested in the minor-closed classes of binary matroids obtained by excluding the cycle matroids of

We introduce techniques for deriving closed form generating functions for enumerating permutations with restricted positions keeping track of various statistics. The method involves evaluating permanents with variables as entries. These are applied to determine the sample size required for a novel sequential importance sampling algorithm for generating random perfect matchings in classes of bipartite

The polynomials ψk(r,x) were introduced by Ramanujan. Berndt, Evans and Wilson obtained a recurrence relation for ψk(r,x). Shor introduced polynomials related to improper edges of a rooted tree, leading to a refinement of Cayley's formula. Zeng realized that the polynomials of Ramanujan coincide with the polynomials of Shor, and that the recurrence relation of Shor coincides with the recurrence relation

Vf-safe delta-matroids have the desirable property of behaving well under certain duality operations. Several important classes of delta-matroids are known to be vf-safe, including the class of ribbon-graphic delta-matroids, which is related to the class of ribbon graphs or embedded graphs in the same way that graphic matroids correspond to graphs. In this paper, we characterize vf-safe delta-matroids

In [30], Tardos studied special delta-matroids obtained from sequences of Higgs lifts; these are the full Higgs lift delta-matroids that we treat and around which all of our results revolve. We give an excluded-minor characterization of the class of full Higgs lift delta-matroids within the class of all delta-matroids, and we give similar characterizations of two other minor-closed classes of delta-matroids

Given an edge-weighted tree T with n leaves, sample the leaves uniformly at random without replacement and let Wk , 2 ≤ k ≤ n, be the length of the subtree spanned by the first k leaves. We consider the question, "Can T be identified (up to isomorphism) by the joint probability distribution of the random vector (W2, …, Wn )?" We show that if T is known a priori to belong to one of various families

In mathematical phylogenetics, given a rooted binary leaf-labeled gene tree topology G and a rooted binary leaf-labeled species tree topology S with the same leaf labels, a coalescent history represents a possible mapping of the list of gene tree coalescences to the associated branches of the species tree on which those coalescences take place. For certain families of ordered pairs (G, S), the number

In this paper we discuss the question of how to decide when a general chemical reaction system is incapable of admitting multiple equilibria, regardless of parameter values such as reaction rate constants, and regardless of the type of chemical kinetics, such as mass-action kinetics, Michaelis-Menten kinetics, etc. Our results relate previously described linear algebraic and graph-theoretic conditions