On the cardinality of layers in some partially ordered sets

T.V. Andreeva∗ , Yu.S. Semenov∗∗

Russian University of Transport, Moscow, 127994 Russia

E-mail: ∗t-v-andreeva@mail.ru, ∗∗yuri_semenoff@mail.ru

Received August 4, 2020

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DOI: 10.26907/2541-7746.2020.3.269-284

For citation: Andreeva T.V., Semenov Yu.S. On the cardinality of layers in some partially ordered sets. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 3, pp. 269–284. doi: 10.26907/2541-7746.2020.3.269-284. (In Russian)

Abstract

In this paper, we explicitly calculated additional terms of cardinality asymptotics of layers in the n-dimensional k-valued lattice Enk for odd k as n → ∞. The main term had been previously determined by V.B. Alekseev for a class of posets and, particularly, for En. Additionally, we precised the cardinality asymtotics of central layers in Cartesian powers of the non-graded poset given by V.B. Alekseev in the same work and calculated the sums of boundary functionals for the n-dimensional three-valued lattice. The obtained theorems, lemmas, and formulas are of combinatorial interest by themselves. They can also be used for estimating the cardinality of maximal antichain or the number of antichains in posets of a definite class.

Keywords: poset, asymptotics, antichain

References

Alekseev V.B. On the number of k -valued monotone functions. Probl. Kibern., 1974, no. 28, pp. 5–24. (In Russian)

Korshunov A.D. On the number of monotone Boolean functions. Probl. Kibern., 1981, no. 38, pp. 5–108. (In Russian)

Sapozhenko A.A. The number of antichains in multilayered ranked sets. Discrete Math. Appl., 1989, vol. 1, no. 2, pp. 149–169. doi: 10.1515/dma.1991.1.2.149.

Sapozhenko A.A. Problema Dedekinda i metod granichnykh funktsionalov [Dedekind’s Problem and the Method of Boundary Functionals]. Moscow, FIZMATLIT, 2009. 152 p. (In Russian)

Andreeva T.V. Development of the boundary functional method and its applications to combinatorial problems. Mat. Vopr. Kibern., 2004, no. 13, pp. 145–222. (In Russian)

Dwight H.R. Tablitsy integralov i drugie matematicheskie formuly [Tables of Integrals and Other Mathematical Data]. Moscow, Nauka, 1978. 224 p. (In Russian)

 

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