Murali K. Srinivasan

The polytope of degree sequences of hypergraphs

Abstract Let D n ( r ) denote the convex hull of degree sequences of simple r -uniform hypergraphs on the vertex set {1,2,…, n }. The polytope D n (2) is a well-studied object. Its extreme points are the threshold sequences (i.e., degree sequences of threshold graphs) and its facets are given by the Erdos–Gallai inequalities. In this paper we study the polytopes D n ( r ) and obtain some partial information. Our approach also yields new, simple proofs of some basic results on D n (2). Our main results concern the extreme points and facets of D n ( r ). We characterize adjacency of extreme points of D n ( r ) and, in the case r =2, determine the distance between two given vertices in the graph of D n (2). We give a characterization of when a linear inequality determines a facet of D n ( r ) and use it to bound the sizes of the coefficients appearing in the facet defining inequalities; give a new short proof for the facets of D n (2); find an explicit family of Erdos–Gallai type facets of D n ( r ); and describe a simple lifting procedure that produces a facet of D n +1 ( r ) from one of D n ( r ).

A Statistic on Involutions

AbstractWe define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I(n) denote the set of all involutions on [n](={1,2,..., n}) and let F(2n) denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let[ n]q=1+q+⋯+qn-1 and let

An inversion number statistic on set partitions

\left( {_{\text{k}}^{\text{n}} } \right)q

Boolean Packings in Dowling Geometries

denote the q-binomial coefficient. There is a statistic wt on I(n) such that the following results are true.(i) We have the expansion

On quotients of posets, with an application to the q -analog of the hypercube

\left( {_{\text{k}}^{\text{n}} } \right)q = \sum\limits_{\delta \in I(n)} {(q - 1)\left| \delta \right|} \left( {_{k - \left| \delta \right|}^{n - 2\left| \delta \right|} } \right).

Vicinal orders of trees

(ii) An analog of the (strong) Bruhat order on permutations is defined on F(2n) and it is shown that this gives a rank-2