A theorem of logconcavity or inequalities for elementary symmetric polynomialsPhilippe PitteloudMIT
September 27,

ABSTRACT


Many integer sequences arising from combinatorics turn out to be unimodal, or even logconcave. The aim of this talk is to present such an example. Recall that a sequence (f_{i})_{i=>0} of nonnegative integers is called unimodal if there exists an index h => 0 such that f_{i} <= f_{i+1} for i < h and f_{i} => f_{i+1} for i => h. The sequence is logconcave if f_{i}^{2} => f_{i+1} f_{i1} for i => 1, which implies the unimodality of (f_{i}) if this sequence has no internal zeros. An integer basis is a strictly increasing sequence B = (b_{i})_{i=>0} of positive integers with the following property: there exists a sequence (c_{i})_{i=>1} of positive integers (called multiplicity sequence) such that every positive integer n can be written uniquely in the form
For instance, the most common integer bases are the powers of a fixed integer b => 2, (ie b_{i} := b^{i}); the associated multiplicity sequences are constant, given by c_{i} = b1 for all i => 1. Set θ_{B}(m,l) := # { n in {0,1,...,m1}: n has l nonzero digits when written in basis B}. The result is: (θ_{B}(m,l))_{l=>0} is strongly logconcave, for any positive integer m and any integer basis B. We will see that this result is indeed an inequality for elementary symmetric polynomials, and we will specialize it to obtain inequalities for sums of binomial coefficients, sums of Stirling numbers of the first kind, or sums of qanalogues of these numbers. I will end with a few words about a proof of this theorem. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

