On the average number ofk-sets

On the average rank of LYM-sets

Let S be a finite set with some rank function r such that the Whitney numbers w i = I { x e S I r ( x ) = i}l are log-concave. Given k, me N so that Wk-1 < Wk <~ Wk+m, set W = w k + w k + 1 + "'" + Wk + ~" Generalizing a theorem of Kleitman and Milner, we prove that every F ~ S with cardinality IFI/> Whas average rank at least ( k w k + ... + (k + m)wk+m)/W, provided the normalized profile vect...

The Statistical Mechanics ofk

The satissability of random CNF formulae with precisely k variables per clause (\k-SAT") is a popular testbed for the performance of search algorithms. Formulae have M clauses from N variables, randomly negated, keeping the ratio = M=N xed. For k = 2, this model has been proven to have a sharp threshold at = 1 between formulae which are almost aways satissable and formulae which are almost neve...

On Score Sequences ofk-Hypertournaments

Given two nonnegative integers n and k with n ≥ k > 1, a k-hypertournament on n vertices is a pair (V, A), where V is a set of vertices with |V | = n and A is a set of k-tuples of vertices, called arcs, such that for any k-subset S of V , A contains exactly one of the k! k-tuples whose entries belong to S. We show that a nondecreasing sequence (r1, r2, . . . , rn) of nonnegative integers is a l...

on the number of maximum independent sets of graphs

let $g$ be a simple graph. an independent set is a set ofpairwise non-adjacent vertices. the number of vertices in a maximum independent set of $g$ isdenoted by $alpha(g)$. in this paper, we characterize graphs $g$ with $n$ vertices and with maximumnumber of maximum independent sets provided that $alpha(g)leq 2$ or $alpha(g)geq n-3$.

On the average number of sharp crossings of certain Gaussian random polynomials