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 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 car...
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| Format: | Article |
| Language: | English |
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Kazan Federal University
2020-09-01
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| Series: | Учёные записки Казанского университета: Серия Физико-математические науки |
| Subjects: | |
| Online Access: | https://kpfu.ru/uz-eng-phm-2020-3-3.html |
| _version_ | 1826952263912914944 |
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| author | T.V. Andreeva Yu.S. Semenov |
| author_facet | T.V. Andreeva Yu.S. Semenov |
| author_sort | T.V. Andreeva |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-11T01:30:38Z |
| format | Article |
| id | doaj.art-9ce8164463d24a54ac727abe3ad7aa51 |
| institution | Directory Open Access Journal |
| issn | 2541-7746 2500-2198 |
| language | English |
| last_indexed | 2025-02-17T23:03:28Z |
| publishDate | 2020-09-01 |
| publisher | Kazan Federal University |
| record_format | Article |
| series | Учёные записки Казанского университета: Серия Физико-математические науки |
| spelling | doaj.art-9ce8164463d24a54ac727abe3ad7aa512024-12-02T09:16:36ZengKazan Federal UniversityУчёные записки Казанского университета: Серия Физико-математические науки2541-77462500-21982020-09-01162326928410.26907/2541-7746.2020.3.269-284On the cardinality of layers in some partially ordered setsT.V. Andreeva0Yu.S. Semenov1Russian University of Transport, Moscow, 127994 RussiaRussian University of Transport, Moscow, 127994 RussiaIn 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.https://kpfu.ru/uz-eng-phm-2020-3-3.htmlposetasymptoticsantichain |
| spellingShingle | T.V. Andreeva Yu.S. Semenov On the cardinality of layers in some partially ordered sets Учёные записки Казанского университета: Серия Физико-математические науки poset asymptotics antichain |
| title | On the cardinality of layers in some partially ordered sets |
| title_full | On the cardinality of layers in some partially ordered sets |
| title_fullStr | On the cardinality of layers in some partially ordered sets |
| title_full_unstemmed | On the cardinality of layers in some partially ordered sets |
| title_short | On the cardinality of layers in some partially ordered sets |
| title_sort | on the cardinality of layers in some partially ordered sets |
| topic | poset asymptotics antichain |
| url | https://kpfu.ru/uz-eng-phm-2020-3-3.html |
| work_keys_str_mv | AT tvandreeva onthecardinalityoflayersinsomepartiallyorderedsets AT yussemenov onthecardinalityoflayersinsomepartiallyorderedsets |