On the number of realizations of a Hasse diagram by finite sets

Autor:	Abraham P. Hillman
Rok vydání:	1955
Předmět:	Combinatorics Discrete mathematics Set (abstract data type) Applied Mathematics General Mathematics Direct methods Covering relation Order (group theory) Hasse diagram Dedekind cut Distributive lattice Finite set Mathematics
Zdroj:	Proceedings of the American Mathematical Society. 6:542-548
ISSN:	1088-6826 0002-9939
DOI:	10.1090/s0002-9939-1955-0074380-6
Popis:	R. Dedekind proposed the problem, still unsolved for n>6, of finding the order f(n) of the free distributive lattice with n generators.' It is known that f(n) is the number of families F of subsets of a set of n objects such that no subset in F includes any other.2 A closely related problem is that of finding the number of families of subsets having specified inclusion relations, i.e., a given Hasse diagram.3 For families of two subsets, G. N. Raney has solved this problem in terms of a general n.4 The present paper contains the solution, by other methods, for families of four or less subsets. In the process, we illustrate both recursive and direct methods of finding the numbers of realizations of Hasse diagrams by subsets of a finite set.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::12ecb0e45814b9c8d2d5425290afe8a7 https://doi.org/10.1090/s0002-9939-1955-0074380-6 Zobrazit plný text záznamu