• A
  • A
  • A
  • АБВ
  • АБВ
  • АБВ
  • A
  • A
  • A
  • A
  • A
Обычная версия сайта
  • RU
  • EN
  • HSE University
  • Publications
  • Book chapter
  • A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions
  • RU
  • EN
Расширенный поиск
Высшая школа экономики
Национальный исследовательский университет
Priority areas
  • business informatics
  • economics
  • engineering science
  • humanitarian
  • IT and mathematics
  • law
  • management
  • mathematics
  • sociology
  • state and public administration
by year
  • 2027
  • 2026
  • 2025
  • 2024
  • 2023
  • 2022
  • 2021
  • 2020
  • 2019
  • 2018
  • 2017
  • 2016
  • 2015
  • 2014
  • 2013
  • 2012
  • 2011
  • 2010
  • 2009
  • 2008
  • 2007
  • 2006
  • 2005
  • 2004
  • 2003
  • 2002
  • 2001
  • 2000
  • 1999
  • 1998
  • 1997
  • 1996
  • 1995
  • 1994
  • 1993
  • 1992
  • 1991
  • 1990
  • 1989
  • 1988
  • 1987
  • 1986
  • 1985
  • 1984
  • 1983
  • 1982
  • 1981
  • 1980
  • 1979
  • 1978
  • 1977
  • 1976
  • 1975
  • 1974
  • 1973
  • 1972
  • 1971
  • 1970
  • 1969
  • 1968
  • 1967
  • 1966
  • 1965
  • 1964
  • 1963
  • 1958
  • More
Subject
News
May 25, 2026
HSE Scientists Train Neural Network to 'Hear' Faults in Electric Motors
Researchers at the AI and Digital Science Institute of the HSE Faculty of Computer Science have developed a new method—the Signature-Guided Data Augmentation (SGDA) framework—that achieves 99% accuracy in motor fault detection and 86% accuracy in fault classification. The application of this approach can reduce industrial equipment repair costs, minimise downtime, and improve production safety. The study results have been published in Engineering Applications of Artificial Intelligence.
May 25, 2026
'The Humanities Serve as a Conscience'
Maria Mizernaia studies Soviet literature and the history of book publishing. In this interview for the HSE Young Scientists project, she discusses plans to publish a novel about besieged Leningrad, AI-provoked reflections on what it means to be human, and how novels can help satisfy our dopamine hunger.
May 25, 2026
Is It Possible to Predict a Citys Life Based on the Shape of Its Neighbourhoods?
Is it possible to predict, based on the configuration of streets and buildings, where a café will open or where traffic congestion will occur? Participants in the Spatial Analysis and Modelling of Urban Processes research and study group use open data and machine learning to identify universal patterns. Alexander Sheludkov and Eduard Somov discuss the purpose of comparing cities, the need for new forms of urban statistics, and how open data is transforming approaches to urban studies.

 

Have you spotted a typo?
Highlight it, click Ctrl+Enter and send us a message. Thank you for your help!

Publications
  • Books
  • Articles
  • Chapters of books
  • Working papers
  • Report a publication
  • Research at HSE

?

A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions

P. 56–69.
Ignatov D. I.

Set partitions and partition lattices are well-known objects in combinatorics and play an important role as a search space in many applied problems including ensemble clustering. Searching for antichains in such lattices is similar to that of in Boolean lattices. Counting the number of antichains in Boolean lattices is known as the Dedekind problem. In spite of the known asymptotic for the latter problem, the behaviour of the number of antichains in partition lattices has been paid less attention. In this short paper, we show how to obtain a few first numbers of antichains and maximal antichains in the partition lattices with the help of concept lattices and provide the reader with some related heuristic bounds. The results of our computational experiments confirm the known values and are also recorded in the Online Encyclopaedia of Integer Sequences (see https://oeis.org/A358041).

Language: English
Full text
DOI
Text on another site
Keywords: concept latticesformal concept analysispartition latticesenumerative combinatoricsmaximal antichains
Publication based on the results of:
Models and method for analysis of unstructured data, data mining and recommender systems (2023)

In book

LNAI 14133: 28th International Conference on Conceptual Structures, ICCS 2023, Berlin, Germany, September 11–13, 2023, Proceedings. Graph-Based Representation and Reasoning
Berlin: Springer, 2023.
Similar publications
On the minimum number of maximal distance-k independent sets in trees
Taletskii D., / Series arXiv "math". 2026.
A vertex subset of a graph is called a \textit{distance-$k$ independent set} if the distance between any two of its distinct vertices is at least $k + 1$. For all $n,k \geq 1$, we determine the minimum possible number of inclusion-wise maximal distance-$k$ independent sets among all $n$-vertex trees. It equals~$n$ if $n \leq k ...
Added: May 1, 2026
Is Canfield Right? On the Asymptotic Coefficients for the Maximum Antichain of Partitions and Related Counting Inequalities
Ignatov D. I., , in: 11th International Conference, AIST 2023, Yerevan, Armenia, September 28–30, 2023, Revised Selected Papers. Analysis of Images, Social Networks and Texts. Lecture Notes in Computer Science (LNCS, volume 14486).: Cham: Springer, 2024. P. 349 – 361.
This paper dates back to the asymptotic solutions of Rota’s problem on the size of maximum antichain in the set partition lattice by Canfield and Harper and others. The knowledge of asymptotic coefficients could pave the way to the asymptotic solutions of such problems as (maximal) antichain counting in partition lattices. In addition to our ...
Added: January 23, 2026
Cooperative games with fuzzy characteristic functions on concept lattices
Kemgne M. W., Njionou B. B., Ignatov D. I. et al., International Journal of Approximate Reasoning 2025 Vol. 186 Article 109527
This paper introduces cooperative games with transferable utilities and fuzzy characteristic functions on concept lattices. While previous works have independently addressed games with fuzzy payoffs and games restricted to structured coalition systems such as lattices, our approach combines both perspectives. We consider cooperative settings where coalition formation is constrained by a concept lattice structure, and ...
Added: January 23, 2026
On syntactic concept lattice models for the Lambek calculus and infinitary action logic
Stepan L. Kuznetsov, Journal of Logic and Computation 2026 Vol. 36 No. 1 Article exaf078
The linguistic applications of the Lambek calculus suggest its semantics over algebras of formal languages. A straightforward approach to construct such semantics indeed yields a brilliant completeness theorem (Pentus 1995, Ann. Pure Appl. Logic, 75, 179–213). However, extending the calculus with extra operations ruins completeness. In order to mitigate this issue, Wurm (2017, J. Logic Lang. Inf., ...
Added: January 14, 2026
Syntactic concept lattice models for infinitary action logic
Stepan L. Kuznetsov, , in: Logic, Language, Information, and Computation: 30th International Workshop, WoLLIC 2024, Bern, Switzerland, June 10–13, 2024, ProceedingsVol. 14672: Lecture Notes in Computer Science.: Cham: Springer, 2024. P. 93–107.
Added: June 12, 2024
On the number of independent and k-dominating sets in graphs with average vertex degree at most k
D. S. Taletskii, Sbornik Mathematics 2023 Vol. 214 No. 11 P. 1627–1650
The following conjecture is formulated: if the average vertex degree in a graph is not greater than a positive integer k⩾1, then the number of k-dominating sets in this graph does not exceed the number of its independent sets, and these numbers are equal to each other if and only if the graph is k-regular. This conjecture is proved for k∈{1,2}. ...
Added: March 25, 2024
О количестве независимых и k-доминирующих множеств в графах со средней степенью вершин не более k
Taletskii D., Математический сборник 2023 Т. 214 № 11 С. 133–156
Сформулирована следующая гипотеза: если средняя степень вершин графа не превосходит натурального числа k ⩾ 1, то количество его k-доминирующих множеств не превосходит количества его независимых множеств, при этом равенство возможно, если и только если граф является k-регулярным. Эта гипотеза доказана для случая k ∈ {1,2}. ...
Added: November 2, 2023
FCA4AI 2023 What can FCA do for Artificial Intelligence 2023 Proceedings of the 11th International Workshop "What can FCA do for Artificial Intelligence?" co-located with the 32nd International Joint Conference on Artificial Intelligence (IJCAI 2023) Macao, S.A.R. China; August 20, 2023
CEUR-WS.org, 2023.
Added: September 27, 2023
17th International Conference, ICFCA 2023, Kassel, Germany, July 17–21, 2023, Proceedings. Formal Concept Analysis, (LNCS, volume 13934)
Switzerland: Springer, 2023.
Added: September 27, 2023
On the Number of Maximal Antichains in Boolean Lattices for 𝑛 up to 7
Ignatov D. I., Lobachevskii Journal of Mathematics 2023 No. 44 P. 137–146
We consider two ways how to compute the number of maximal antichains in the Boolean lattice on 𝑛 elements. The first one is based on full direct enumeration, while the second ones relies on concept lattices or Galois lattices (studied in Formal Concept Analysis, an applied branch of lattice theory) and the Dedekind–MacNeille completion of a partial ...
Added: June 13, 2023
Decision Concept Lattice vs. Decision Trees and Random Forests
Dudyrev E., Kuznetsov S., , in: Formal Concept Analysis: 16th International Conference, ICFCA 2021, Strasbourg, France, June 29 – July 2, 2021, Proceedings.: Springer, 2021. Ch. 16 P. 252–260.
Added: September 28, 2021
Towards Stable Significant Subgroup Discovery
Jyoti -., Buzmakov Aleksey, Kailasam S., , in: The 15th International Conference on Concept Lattices and Their Applications CLA2020Issue 2668.: CEUR-WS, 2020. P. 287–292.
Added: July 10, 2021
Formal Concept Analysis: 16th International Conference, ICFCA 2021, Strasbourg, France, June 29 – July 2, 2021, Proceedings
Springer, 2021.
This book constitutes the proceedings of the 16th International Conference on Formal Concept Analysis, ICFCA 2021, held in Strasbourg, France, in June/July 2021. The 14 full papers and 5 short papers presented in this volume were carefully reviewed and selected from 32 submissions. The book also contains four invited contributions in full paper length. The research part ...
Added: July 10, 2021
Proceedings of the Fifthteenth International Conference on Concept Lattices and Their Applications
CEUR-WS.org, 2020.
The CLA conference is an international forum for researchers, practitioners and students dedicated to the practice of Formal Concept Analysis (FCA) and areas closely related to it, including data analysis and mining, information retrieval, knowledge management, knowledge engineering, logic, algebra and lattice theory. The 15th of CLA, CLA 2020, was going to be held in Tallinn, Estonia ...
Added: October 30, 2020
Proceedings of the 9th European Starting AI Researchers' Symposium 2020 co-located with 24th European Conference on Artificial Intelligence (ECAI 2020)
Lang J., CEUR-WS, 2020.
The 9th European Starting AI Researchers’ Symposium (STAIRS) is an international meeting intended to support Starting Researchers (those that have not been awarded a doctoral degree but also researchers with not more than three years of experience after getting their PhD). STAIRS is collocated with the ECAI2020 tutorials and workshops and has an independent program committee. In this edition, ...
Added: October 30, 2020
Adaptive Multi-model Approaches to Pattern Set Mining
Kuznetsov S., Makhalova T., Napoli A., , in: Proceedings of the 9th European Starting AI Researchers' Symposium 2020 co-located with 24th European Conference on Artificial Intelligence (ECAI 2020).: CEUR-WS, 2020. P. 1–8.
Added: October 30, 2020
Eighth International Workshop “What can FCA do for Artificial Intelligence?”
[б.и.], 2020.
The seven preceding editions of the FCA4AI Workshop showed that many researchers working in Artificial Intelligence are deeply interested by a well-founded method for classification and data mining such as Formal Concept Analysis (see https://conceptanalysis. wordpress.com/fca/). FCA4AI was co-located with ECAI 2012 (Montpellier), IJCAI 2013 (Beijing), ECAI 2014 (Prague), IJCAI 2015 (Buenos Aires), ECAI 2016 ...
Added: October 29, 2020
The 15th International Conference on Concept Lattices and Their Applications CLA2020
CEUR-WS, 2020.
Proceedings of the Fifthteenth International Conference on Concept Lattices and Their Applications Tallinn, Estonia, June 29–July 1, 2020. Edited by Francisco J. Valverde-Albacete Martin Trnecka ...
Added: October 29, 2020
Next Priority Concept: A new and generic algorithm computing concepts from complex and heterogeneous data
Kuznetsov S., Demko C., Bertet K. et al., , in: Electronic Procedings Theoretical Computer ScienceVol. 845.: [б.и.], 2020. P. 1–20.
In this article, we present a new data type agnostic algorithm calculating a concept lattice from heterogeneous and complex data. Our NextPriorityConcept algorithm is first introduced and proved in the binary case as an extension of Bordat's algorithm with the notion of strategies to select only some predecessors of each concept, avoiding the generation of ...
Added: October 29, 2020
  • About
  • About
  • Key Figures & Facts
  • Sustainability at HSE University
  • Faculties & Departments
  • International Partnerships
  • Faculty & Staff
  • HSE Buildings
  • HSE University for Persons with Disabilities
  • Public Enquiries
  • Studies
  • Admissions
  • Programme Catalogue
  • Undergraduate
  • Graduate
  • Exchange Programmes
  • Summer University
  • Summer Schools
  • Semester in Moscow
  • Business Internship
  • Research
  • International Laboratories
  • Research Centres
  • Research Projects
  • Monitoring Studies
  • Conferences & Seminars
  • Academic Jobs
  • Yasin (April) International Academic Conference on Economic and Social Development
  • Media & Resources
  • Publications by staff
  • HSE Journals
  • Publishing House
  • iq.hse.ru: commentary by HSE experts
  • Library
  • Economic & Social Data Archive
  • Video
  • HSE Repository of Socio-Economic Information
  • HSE1993–2026
  • Contacts
  • Copyright
  • Privacy Policy
  • Site Map
Edit