On the f-vectors of r-multichain subdivisions

For a poset P and an integer r >= 1, let P-r be the collection of all r-multichains in P. Corresponding to each strictly increasing map t :[r] -> [2r], there is an order <=(t) ion P-r. Let Delta(G(t)(P-r)) be the clique complex of the graph G(t) associated to P-r and t. In a recent paper [14], it is shown that Delta(G(t)(P-r)) is a subdivision of P for a class of strictly increasing maps. In this paper, we show that all these subdivisions have the same f-vector. We give an explicit description of the transformation matrices from the f- and h-vectors of Delta to the f- and h-vectors of these subdivisions when Pis a poset of faces of Lambda. We study two important subdivisions, namely Cheeger-Muller-Schrader's subdivision and the r-colored barycentric subdivision which fall in our class of r-multichain subdivisions. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

This paper studies two important subdivisions, namely Cheeger-Muller-Schrader's subdivision and the r-colored barycentric subdivision, and proves that these subdivisions have the same f-vector. By analyzing the transformation matrices of the f- and h-vectors, the paper provides an explicit description of the relationship between these subdivisions and the original poset. This research is significant for describing and understanding posets of multichains.