|| Chain Partition Analysis
Abstract: We introduce and study a hybrid of (restricted) partition functions from combinatorial number theory and zeta polynomials from the theory of partially ordered sets (posets), giving rise to a concept that in a sense is dual to Stanley's P-partitions.
Given a finite poset P and an order preserving map f from P into the positive integers, a chain partition of an integer n is a sum of evaluations of f along a multichain in P. A special case is when P is itself a chain; then chain partitions are precisely partitions with parts in the range of f. On the other extreme, if P is an antichain then chain partitions are compositions with parts in the range of f.
We develop an enumerative theory of chain partitions, unifying and combining concepts such as flag f-vectors (which yield the Dehn-Sommerville relations for Eulerian posets) and Hilbert functions of Stanley-Reisner rings (realizing abstract simplicial complexes as posets).
This is joint work with Raman Sanyal (Frankfurt).