Characterizing profiles of \(k\)-families in additive Macaulay posets
The famous theorem of Kruskal and Katona gives a complete characterization of \(f\)-vectors of simplicial complexes. A similar result for \textit{semisimplicial complexes} has been proved by F. S. Macaulay in 1927. In 1969 G. F. Clements and B. Lindström proved a ``\textit{generalized Macaulay theorem}'' which contains both the Kruskal-Katona and the Macaulay theorem as special cases. Later, further generalizations led the author to the notion of \textit{additive Macaulay posets} [Order 14, 39-46 (1997; Zbl 0884.05090)]. Here the author investigates some properties of these posets. Namely, a subset of a poset is a \textsl{\(k\)-family} if there is no chain consisting of \(k+1\) of its elements. A subset of a ranked poset consisting of \(p_i\) elements of rank \(i\) is said to have \textsl{profile} \((p_0, p_1,\ldots)\). A characterization is given for profiles of \(k\)-families in additive Macaulay posets.
scientific article; zbMATH DE number 1093105