Getting Results with Negative Thinking

Given a universe of discourse U , a multiset can be thought of as a function M from U to the natural numbers N. In this paper, we define a hybrid set to be any function from the universe U to the integers Z. These sets are called hybrid since they contain elements with either a positive or negative multiplicity. Our goal is to use these hybrid sets as if they were multisets in order to adequately generalize certain combinatorial facts which are true classically only for nonnegative integers. However, the definition above does not tell us much about these hybrid sets; we must define operations on them which provide us with the needed combinatorial structure. In section 1, we will define an analog of a set which can contain either a positive or negative number of elements. We will allow sums to be calculated over an arbitrary hybrid set and in particular over “improper” intervals. This will lead us in section 2 to the calculation of the complementary symmetric functions compn(H) whose argument is a hybrid set of variables which is at the same time a generalization of elementary and complete symmetric functions. In particular, as we will see in section 2.2, the complementary symmetric function generalizes the following two classic results concerning sequences of polynomials with persistant roots, that is to say, having the form pn(x) = (x − a1)(x − a2) · · · (x − an). pn(x) = ∞ ∑ k=0 ek(−a1, . . . ,−an)x n−k (1)