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Many combinatorial proofs rely on induction. When these proofs are formulatedin traditional language, they can be bulky and unmanageable. Coalgebras providea language which can reduce reduce many inductive proofs in graded poset theoryto comprehensible size. As a bonus, the visual form of the resulting recursiveproofs suggests combinatorial interpretations for constants appearing in thelonger arguments. We use the techniques of coalgebras to compute invariants oftoric and affine arrangements as well as of poset products. In additionalchapters we prove structure theorems for acyclic orientations and criticalgroups of graphs.

Homological combinatorics and extensions of the cd-index