8 0. A generalization of this result may be stated as follows. Theorem 1.8.7 (Theorem 11, [37]) Let A\/it and v\/p be any two skew shapes. Then we have \u2014 5\/ILSV\/p S 0. Given partitions A and ,u, we have the partition v = A U p obtained by listing all the parts of A and p together in weakly decreasing order. We then set sorti(A,p) = (vj,v3,5...) and sort2(A,p) = (v2,46. . Theorem 1.8.8 (Conjecture 2.7, [18]) Given partitions A and p, we have Ssortl(),)Ssort2()L) \u2014 5Ap s 0. 18 CHAPTER 1. INTRODUCTION This result may be generalized to skew shapes as follows. Theorem 1.8.9 (Corollary 12, [37]) Let A\/ti and v\/p be skew shapes. Then we have p,v)\/sorti 5sort2(,)\/sort,p) \u20145A\/LSv\/p s 0. Given a partition A, let A\u201d1 (A Ai+n, Ai+2n, . . .). In this way the parts of A are distributed among the partitions A[i,?] for i = 1,. . . Theorem 1.8.10 (Conjecture 6.4, [41]) For integers 1 m < n and a partition A, we have fl m fJ S[i,n] \u2014 S[i,m] > 0. A generalization of this result to skew shapes can also be given. Theorem 1.8.11 (Theorem 13, [37]) Let A(\u2019)\/i(1), ..., A()\/1i(\u2019 , be n skew shapes, let A = U be the par tition obtained by the decreasing rearrangement of the parts in all and = U. Then we have > 0. Given two partitions A and ji, we can define A V ji and A A ji by A V i := (max{A1, max{A2,Ii2},...) and A A := (min{A1,ii}, min{A2Ji2},. . Then, given skew diagrams A\/1i and v\/p, the operators V and A can be defined on pairs of skew diagrams via (A\/ii) V (v\/p) := (A V v)\/( V p) and (A\/1i) A (v\/p) := (A A v)\/(i A p). Theorem 1.8.12 (Conjecture 4.6, [38]) Let A\/1u and v\/p be any two skew shapes. Then we have SA\/Sii\/p 0. 19 CHAPTER 1. INTRODUCTION Theorem 1.8.6 was first conjectured in [53], Theorem 1.8.8 was first con jectured in [18], Theorem 1.8.10 was first conjectured in [41], and Theo rem 1.8.12 was first conjectured in [38]. In [37], Theorem 1.8.12 was proved and was then used to prove each of Theorem 1.8.7, Theorem 1.8.9, and The orem 1.8.11, which in turn imply each of Theorem 1.8.6, Theorem 1.8.8, and Theorem 1.8.10, respectively. We have seen many operations that construct examples of Schur-positive differences. A seperate problem to investigate involves finding families of skew diagrams for which all instances of Schur-positive differences within these families can be determined. One result of this type occurs in [49], in which the collection of multiplicity- free ribbons is inspected. Using results of [27], multiplicity-free ribbons are classified as those ribbons with at most two rows of length greater than one and at most two columns of length greater than one. Then, among the col lection of multiplicity-free ribbons of a given size, the Hasse diagram which describes all Schur-positive differences and Schur-incomparabilities among these diagrams is explicitly described. Moreover, the Hasse diagram is es sentially a product of two chains. We now begin to inspect another collection of ribbons for which the ques tion of Schur-positivity is completely answered. It is not difficult to see that the skew diagram ).\/i\u2019 is uniquely determined by the row overaps rows1(\/t) and rows2\/t), thus we may identify the skew Schur function using the overlap notation = {rows1\/i)ro2(A\/t)}. In the case when A\/si is a ribbon we have = {c11()_1}, where o is the composition given by the row lengths of )\/1u. In this situation we use the notation := = {c11)_1} and call ra a ribbon Schur function. In [8], it was shown that the collection {rA},H forms a basis of A. In [34], the following theorem was proved, which identifies the Schur-positive differences among this collection. Theorem 1.8.13 (Theorem 3.3, [3]) 20 CHAPTER 1. INTRODUCTION Let A and be partitions of n, then r \u2014 r), 0 if and only if,u -< A and 1(A) = l(,u). More Schur-positive differences were discovered in [34], and are easiest to state in terms of this overlap notation. We begin by considering the following hypothesis on compositions a and r. Hypothesis 1.8.14 Let a and T be compositions such that 1(a) = s > 0 and 1(r) = t> 0, and let a- and \u2018 be sequences of non-negative integers that satisfy the following conditions: 1. The lengths of a- and are s and t respectively; 2. as 1 when s > 0; 3. = 1 when t > 0; 4. \u00f6