This thesis consists of six papers related to combinatorics and commutative algebra.

In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.

In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.

In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.

In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the generic initial ideal.

In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.

In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.