This thesis is built around a Hopf monoid structure on the linearized vector speciesPO of partial orders. In the third chapter, we present a cancellation-free antipode formula for PO, which we generalize in chapter 4 to the setting of A-species, where A is any simplicial hyperplane arrangement. In chapter 5, we study a polynomial invariant which is defined for any poset p, and whose value at 1 is the number of linear extensions of the poset. We prove that the polynomial invariant studied in [9] is a reparameterization of our polynomial invariant whenever p is (2+2)-free, from which it follows that the two polynomials have the same value at -1 whenever p is (2+2)-free. We use our antipode formula to prove a formula for the value of the polynomial invariant at -1 for all posets p which do not contain a (2+2)-subposet. In doing so, we define a class of simplicial complexes of independent interest. Chapter 6 is concerned with these simplicial complexes, and with proving a result about iteration of the nerve construction on finite simplicial complexes up to stabilization which we call the final nerve construction. This allows us to show that all interval simplicial complexes are homotopy equivalent to either balls or spheres, and to present a correspondence between pairs of finite simplicial complexes which are stable under the final nerve construction and equivalence classes of a certain family of (0,1)-matrices.