Related papers: The f-vector of the descent polytope

The type A_n full root polytope is the convex hull in R^{n+1} of the origin and the points e_i-e_j for 1<= i<j <= n+1. Given a tree T on the vertex set [n+1], the associated root polytope P(T) is the intersection of the full root polytope…

We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input…

Let M denote the maximal function along the polynomial curve p(t)=(t,t^2,...,t^d) in R^d: M(f)=sup_{r>0} (1/2r) \int_{|t|<r} |f(x-p(t))| dt. We show that the L^2-norm of this operator grows at most logarithmically with the parameter d:…

In this paper we study the generating polynomials obtained by enumerating signed simsun permutations by number of the descents. Properties of the polynomials, including the recurrence relations and generating functions are studied.

For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show…

Let $p$ be a prime. Suppose that integers $r$, $e$, $d$ such that $r \ge 2$, $e \ge 0$, $0 \le d \le p$ are given. Let $f(x)=s_0 x^r + s_1 x^{r-1} + \cdots + s_r$ be a generic polynomial of degree $r$ in characteristic $p$. We put…

Given $n$ symmetric Bernoulli variables, what can be said about their correlation matrix viewed as a vector? We show that the set of those vectors $R(\mathcal{B}_n)$ is a polytope and identify its vertices. Those extreme points correspond…

The polytope of integer partitions of $n$ is the convex hull of the corresponding $n$-dimensional integer points. Its vertices are of importance because every partition is their convex combination. Computation shows intriguing features of…

The eigenvectors of the $(N+1)\times (N+1)$ symmetric Pascal matrix $T_N$ are analogs of prolate spheroidal wave functions in the discrete setting. The generating functions of the eigenvectors of $T_N$ are prolate spheroidal functions in…

We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a…

The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of all degree partitions (respectively, degree…

P(n,s) denotes the number of permutations of 1,2,...n that have exactly s sequences. Canfield and Wilf [math.CO/0609704] recently showed that P(n,s) can be written as a sum of s polynomials in n. We determine these polynomials explicitly…

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally):…

We present a new algorithmic approach that can be used to determine whether a given quadruple $(f_0,f_1,f_2,f_3)$ is the f-vector of any convex 4-dimensional polytope. By implementing this approach, we classify the f-vectors of 4-polytopes…

We investigate the coefficients generated by expressing the falling factorial $(xy)_k$ as a linear combination of falling factorial products $(x)_l (y)_m$ for $l,m =1,...,k$. Algebraic and combinatoric properties of these coefficients are…

We study new statistics on permutations that are variations on the descent and the inversion statistics. In particular, we consider the alternating descent set of a permutation sigma = sigma_1sigma_2...sigma_n defined as the set of indices…

We derive formulas for the cd-index and the toric h-vector of a convex polytope P from a sweeping by a hyperplane. These arise from interpreting the corresponding S-shelling of the dual of P. We describe a partition of the faces of the…

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck {\it et al.}\ that all roots $\alpha$ of Ehrhart…

We provide a multidimensional extension of previous results on the existence of polynomial progressions in dense subsets of the primes. Let $A$ be a subset of the prime lattice - the d-fold direct product of the primes - of positive…

Given a self-similar set $\Lambda$ that is the attractor of an iterated function system (IFS) $\{f_1,\dots,f_N\}$, consider the following method for constructing a random subset of $\Lambda$: Let $\mathbf{p}=(p_1,\dots,p_N)$ be a…