Related papers: Sign variation and descents

In this paper, we find distribution of descents over $(n-3)$- and $(n-4)$-stack-sortable permutations in terms of Eulerian polynomials. Our results generalize the enumeration results by Claesson, Dukes, and Steingr\'{\i}msson on $(n-3)$-…

We compute the expansion of the cohomology class of the permutahedral variety in the basis of Schubert classes. The resulting structure constants $a_w$ are expressed as a sum of \emph{normalized} mixed Eulerian numbers indexed naturally by…

In this paper, we show that the exponent set of nonnegative primitive tensors with order m(\geq 3) and dimension n is {1,2,\ldots, (n-1)^2+1}; and propose some open problems for further research.

Let $X_{m,n}$ be the Segre-Veronese variety $\mathbb{P}^m \times \mathbb{P}^n$ embedded by the morphism given by $\mathcal{O}(1,2)$. In this paper, we provide two functions $\underline{s}(m,n)\le \bar{s}(m,n)$ such that the…

Let $G$ be a $(2,m,n)$-group and let $x$ be the number of distinct primes dividing $\chi$, the Euler characteristic of $G$. We prove, first, that, apart from a finite number of known exceptions, a non-abelian simple composition factor $T$…

Let p(n, k) denote the number of partitions of n into parts less than or equal to k. We show several properties of this function modulo 2. First, we prove that for fixed positive integers k and m, p(n,k) is periodic modulo m. Using this, we…

In this paper we study the class of graphs $G_{m,n}$ that have the same degree sequence as two disjoint cliques $K_m$ and $K_n$, as well as the class $\overline G_{m,n}$ of the complements of such graphs. We establish various properties of…

Let X be a UMD space with type t and cotype q, and let T be a Fourier multiplier operator with a scalar-valued symbol m. If the Mihlin multiplier estimate holds for all partial derivatives of m up to the order n/max(t,q')+1, then T is…

Using a new colored analogue of P-partitions, we prove the existence of a colored Eulerian descent algebra which is a subalgebra of the Mantaci-Reutenauer algebra. This algebra has a basis consisting of formal sums of colored permutations…

We study the $\varrho$-th order variation seminorm of a general Ornstein--Uhlenbeck semigroup $\left(\mathcal H_t\right)_{t>0}$ in $\mathbb R^n$, taken with respect to $t$. We prove that this seminorm defines an operator of weak type…

There is a surprising occurrence of some minus signs in the isomorphisms produced in the well-known technique of dimension shifting in calculating derived functors in homological algebra. We explicitly determine these signs. Getting these…

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $c,d,m$ be positive integers. Let $D$ be a $p$-divisible group of codimension $c$ and dimension $d$ over $k$. Let $\scrD$ be a versal deformation of $D$ over a smooth…

Let $m_{12}$, $m_{13}$, ..., $m_{n-1,n}$ be the slopes of the $\binom{n}{2}$ lines connecting $n$ points in general position in the plane. The ideal $I_n$ of all algebraic relations among the $m_{ij}$ defines a configuration space called…

Let $X$ be a non-degenerate projective irreducible variety of dimension $n \ge 1$, degree $d$, and codimension $e \ge 2$ over an algebraically closed field $\mathbb{K}$ of characteristic $0$. Let $\beta_{p,q} (X)$ be the $(p,q)$-th graded…

This paper contains a theory of elimination and extension to compute varieties symbolically, based on using {\em coordinates} from $(\mathbf{P}^1(\bar{\mathbf{F}}))^n$ and disjoint {\em parts} of varieties (defined by both equality and…

Let X be a smooth proper variety over the quotient field of a Henselian discrete valuation ring with algebraically closed residue field of characteristic p. We show that for any coherent sheaf E on X, the index of X divides the…

For any number $m \equiv 0,1 \, (4)$ we correct the generating function of Hurwitz class number sums $\sum_r H(4n - mr^2)$ to a modular form (or quasimodular form if $m$ is a square) of weight two for the Weil representation attached to a…

We obtain a precise relation between the Chern-Schwartz-MacPherson class of a subvariety of projective space and the Euler characteristics of its general linear sections. In the case of a hypersurface, this leads to simple proofs of…

For every vector $\overline \alpha\in \RR^n$ and for every rational approximation $(\overline p,q)\in \RR^n\times\RR$ we can associate the displacement vector $q\alpha-\overline p$. We focus on algebraic vectors, namely $\overline…

We find the complete branching law for the restriction of certain unitary representations of $O(1,n+1)$ to the subgroups $O(1,m+1)\times O(n-m)$, $0\leq m\leq n$. The unitary representations we consider belong either to the unitary…