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Pseudogenerated two-index sequences. (Psevdoporozhdaemye dvukhindeksnye posledovatel’nosti.) (Russian) Zbl 0759.05003
Moskva: Nedra. 88 p. (1989).
A sequence \(\{P(n,k)\}\) is called pseudogenerated of order \(p\), if two analytic functions \(\psi\) and \(\varphi\), regular in a neighborhood of zero, exist such that 1) \(\psi(0)\neq 0\); 2) \(\varphi(0)=\varphi'(0)=\dots=\varphi^{(p-1)}(0)=0\), \(\varphi^{(p)}(0)\neq 0\); 3) for all nonnegative \(k\) \[ \sum^ \infty_{n=0}P(n,k){z^ n\over n!}=\psi(z){[\varphi(z)]^ k\over k!}. \] It has been shown that most of the two-index sequences occurring in enumeration theory, for instance, the sequence of binomial coefficients, Kronecker \(\delta\)-symbols, and the Stirling sequences of both kinds, are pseudogenerated.
The theory of pseudogenerated sequences is developed, and its applications are given. It is shown that this theory is equivalent to the theory of multiplicative functions on the category of segments of lattices of partitions of finite sets, to the theory of binomial enumeration, and to the theory of pairs of reversible relations of type \(F^ q_ 1\). At the end of the contribution a list of works of the author in mathematics with indication of the corresponding reviews is presented. The book has been published at the expense of the author.
05A10 Factorials, binomial coefficients, combinatorial functions
05A15 Exact enumeration problems, generating functions
05A18 Partitions of sets
05A99 Enumerative combinatorics