By Gottfried Wilhelm Leibniz

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Voxn has negative real roots. Show that Yo' vl' ... ,vn is log-concave. Possible hint: This result has a combinatorial proof. Let {-r1' ... ,-rn} be the roots of v(x). Define the weight w(A) of a subset A c [n] to be 53 w(A) = n ieA r i' Show that v~ L w(A)w(B), (A,B) IAI=IBI =n-k and L w(C)w(D). ,vk+l to (A, B) for vk2 which preserves the weight. [3] Let Sn(x) be the polynomial of degree n n+1 Sn(x) L k=1 S(n+1, k) xk-I. 1) of Chapter 1, show that Sn(x) = (x + 1) Sn_1 (x) + x S'n_1 (x). (b) Now show that Sn(x) has n distinct negative roots following these steps.

I)? Write a program which gives the number of maximal chains each poset has. What are your conclusions? [2] How many maximal chains do En and I'n have? Qn have? [3C] Write a program to count the number of edges in the Hasse diagram of G'n. Formulate a conjecture, based upon an appropriate combination of Bell numbers. (Hint: Try (Bn+2 - a·Bn+l + b·B n) 12, for appropriate positive integers a and b. ' A. = nrn. Formulate and prove as many conjectures as you can. 3 may be useful. [2] Suppose Yo' vl' ...

Clearly these chains are disjoint We now return to the Littlewood-Offord problem. The idea is to decompose the n 2 vectors w A into M(n) blocks {B j}, 1:5; i :5; M(n), such that the distance between any two vectors within a block Bj is ~ 1. Then any sphere S of diameter 1 could contain, at most, one vector from each B j. The blocks {BJ are built inductively just as the chains {Cj} were built. Suppose blocks {B j }, 1:5; i:5; M(n), have been chosen for {vi' .. , ,vn}. We define new blocks for for the vectors {vI' ...

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