Geodesic
On the cardinality of layers in some partially ordered sets
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 162 (2020) no. 3, pp. 269-284 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

In this paper, we explicitly calculated additional terms of cardinality asymptotics of layers in the $n$-dimensional $k$-valued lattice $E^n_k$ for odd $k$ as $n\to\infty$. The main term had been previously determined by V. B. Alekseev for a class of posets and, particularly, for $E^n_k$. 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
Mots-clés : antichain. 
@article{UZKU_2020_162_3_a2,
     author = {T. V. Andreeva and Yu. S. Semenov},
     title = {On the cardinality of layers in some partially ordered sets},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
     pages = {269--284},
     year = {2020},
     volume = {162},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2020_162_3_a2/}
}
TY  - JOUR
AU  - T. V. Andreeva
AU  - Yu. S. Semenov
TI  - On the cardinality of layers in some partially ordered sets
JO  - Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
PY  - 2020
SP  - 269
EP  - 284
VL  - 162
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UZKU_2020_162_3_a2/
LA  - ru
ID  - UZKU_2020_162_3_a2
ER  - 
%0 Journal Article
%A T. V. Andreeva
%A Yu. S. Semenov
%T On the cardinality of layers in some partially ordered sets
%J Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki
%D 2020
%P 269-284
%V 162
%N 3
%U http://geodesic.mathdoc.fr/item/UZKU_2020_162_3_a2/
%G ru
%F UZKU_2020_162_3_a2
T. V. Andreeva; Yu. S. Semenov. On the cardinality of layers in some partially ordered sets. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 162 (2020) no. 3, pp. 269-284. http://geodesic.mathdoc.fr/item/UZKU_2020_162_3_a2/

[1] Alekseev V. B., “On the number of $k$-valued monotone functions”, Probl. Kibern., 28, 1974, 5–24 (In Russian)

[2] Korshunov A. D., “On the number of monotone Boolean functions”, Probl. Kibern., 38, 1981, 5–108 (In Russian)

[3] Sapozhenko A. A., “The number of antichains in multilayered ranked sets”, Discrete Math. Appl., 1:2 (1989), 149–169 | DOI | MR

[4] Sapozhenko A. A., Dedekind's Problem and the Method of Boundary Functionals, FIZMATLIT, M., 2009, 152 pp. (In Russian)

[5] Andreeva T. V., “Development of the boundary functional method and its applications to combinatorial problems”, Mat. Vopr. Kibern., 13, 2004, 145–222 (In Russian)

[6] Dwight H. R., Tables of Integrals and Other Mathematical Data, Nauka, M., 1978, 224 pp. (In Russian)