On some formulas in the problem of enumeration of finite labeled topologies
Khalid Sh. Al’Dzhabri, В.И. Родионов · Discrete Mathematics Algorithms and Applications · 2022
Among the unsolved problems of enumeration of graphs, one of the most difficult is the problem of enumeration of transitive digraphs, which is equivalent to the problem of enumeration of finite partial orders and is equivalent to the problem of enumeration of finite labeled [Formula: see text]-topologies. The sequence [Formula: see text], where [Formula: see text] is the number of all labeled [Formula: see text]-topologies defined on an n-set, has been studied from different points of view. In the previous work of the second author, a formula was obtained in which the number [Formula: see text] is represented as a linear combination of numbers [Formula: see text] (where the sequences [Formula: see text] are compositions of the number n). Recurrent relations between individual numbers [Formula: see text] were obtained. In this paper, new recursive formulas between separate numbers [Formula: see text] are obtained.