Jim Propp writes: << Here's a fun question from Hunter Snevily (snevily@uidaho.edu; not a member of math-fun): Let x1,....xn be n real numbers that sum to 0 (i.e. x1+x2+...+xn=0) For each of the 2^n possible subsets of the xi's we associate the sum of the elements in that set. By def we assign the empty set the value negative infinity. Let ri be the number of subsets of size i that have sum >=0. (note that r0=0 and that rn=1) Claim 1. r0,r1,r2,...,rn is a unimodal sequence. Claim 2. For any 1<=j<=n, Bigsum i=0 to i=j ri >= Bigsum i=1 to i=j binom(n-1,i-1).
Let's call the real numbers a1,...,an so we can use x1,...,xn as the coordinates of n-space. Let's assume that none of the sums of the numbers a1,...,an are exactly 0, except for a1 + ... + an, as given. Here's a geometrical way to look at this question. Let the corners of the unit n-cube be denoted as C_n = {0,1}^n. Let H be an arbitrary open halfspace in R^n whose boundary plane intersects C_n in precisely { (0,0,...,0), (1,1,...,1) }. Then the subsets of the indices {1,2,...,n} of size k correspond to those points S(k,n) of C_n lying on the plane x1+...+xn = k. The possible counts rk, k = 0,...n referred to in the question are the cardinalities #( S(k,n) int H ), k = 0,...,n over possible halfspaces H as above. Maybe someone else will take it from here. --Dan