Narrow Results

We introduce a generic identity based on the Ahlswede–Zhang (AZ) function, which can be considered as a template to establish AZ-style identities. Many AZ-style identities can be derived from the generic identity.

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 $T_0$-topologies. The sequence $\{T_0(n)\}$, where $T_0(n)$ is the number of all labeled $T_0$-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 $T_0(n)$ is represented as a linear combination of numbers $W(\overline{p})$ (where the sequences $\overline{p}=(p_1,\dots ,p_k)\vDash n$ are compositions of the number n). Recurrent relations between individual numbers $W(\cdot )$ were obtained. In this paper, new recursive formulas between separate numbers $W(\cdot )$ are obtained.