On the cardinality of layers in some partially ordered sets

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.