cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Dmitry I. Ignatov

Dmitry I. Ignatov's wiki page.

Dmitry I. Ignatov has authored 21 sequences. Here are the ten most recent ones:

A382350 Number of maximal antichains in the Bruhat order on B_n.

Original entry on oeis.org

2, 5, 215, 24828398365
Offset: 1

Views

Author

Dmitry I. Ignatov, May 30 2025

Keywords

Comments

The number of maximal antichains in the Bruhat order of the Weyl group B_n (the hyperoctahedral group).

Examples

			For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The maximal antichains are {1} and {s1}.
For n=2 the line (Hasse) diagram is below.
      s2*s1*s2*s1
          /   \
    s2*s1*s2  s1*s2*s1
        |   X   |
      s2*s1   s1*s2
        |   X   |
        s2     s1
          \   /
            1
The set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s2*s1*s2, s1*s2*s1}, {s2*s1*s2*s1}}.

References

  • A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.

Links

Crossrefs

Cf. A382346 (antichains), A005900 (the number of join-irreducible elements), A378072 (the size of Dedekind-MacNeille completion)

Extensions

a(4) from Dmitry I. Ignatov, Aug 15 2025

A384959 Number of chains in the Bruhat order of type A_n.

Original entry on oeis.org

4, 36, 4524, 15166380, 2010484649524, 14206021962108887860
Offset: 1

Views

Author

Dmitry I. Ignatov, Jun 13 2025

Keywords

Comments

The number of chains in the Bruhat order of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

References

  • A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.

Links

Crossrefs

Cf. A061710 (maximal chains), A000142 (the order size), A005130 (the size of Dedekind-MacNeille completion), A384061 (number of antichains).

A384687 Number of elements in the Dedekind-MacNeille completion of the Bruhat order on D_n.

Original entry on oeis.org

4, 42, 1292, 114976, 29735760
Offset: 2

Views

Author

Dmitry I. Ignatov, Jun 07 2025

Keywords

Comments

This sequence is the number of elements in the Dedekind-MacNeille completion (completion by cuts) of the Bruhat order of the Weyl group D_n. It is a type D analog of A378072.

Examples

			For n=2 the Bruhat order on D_2 consists of four elements, 1 (identity), s1, s2, and s2*s1. Its completion forms the diamond lattice and coincides with the order.
  s2*s1
   / \
  s1 s2
   \ /
    1

Links

Crossrefs

Cf. A002866 (group D_n order), A005130 (completion for A_n), A378072 (completion for B_n).

A384529 Number of inequivalent sets S (cubic acute n-set), with cardinality A089676(n) >= 3, of points in {0,1}^n in real n-dimensional Euclidean space such that every angle determined by three distinct points in S is acute.

Original entry on oeis.org

0, 0, 2, 5, 18, 3, 64
Offset: 1

Views

Author

Dmitry I. Ignatov, Jun 01 2025

Keywords

Comments

Consider the 2^n points {0,1}^n in real Euclidean space. Then A089676(n) = maximal size of a subset S of these 2^n points such that there is no triple of points P, Q, R in S which subtends a right angle. That is, we are not allowed to have P-Q perpendicular to R-Q. Here we count such inequivalent sets of points under permutation of their coordinates.

Links

Crossrefs

Cf. A089676 (maximal set sizes), A289972 (labeled case).

A382346 Number of antichains in the Bruhat order on B_n.

Original entry on oeis.org

3, 12, 2247
Offset: 1

Views

Author

Dmitry I. Ignatov, May 18 2025

Keywords

Comments

The number of antichains in the Bruhat order of the Weyl group B_n (the hyperoctahedral group).

Examples

			For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The antichains are {}, {1}, and {s1}.
For n=2 the line (Hasse) diagram is below.
      s2*s1*s2*s1
          /   \
    s2*s1*s2  s1*s2*s1
        |   X   |
      s2*s1   s1*s2
        |   X   |
        s2     s1
          \   /
            1
The set of antichains is  {{}, {1}, {s2}, {s2, s1}, {s1}, {s2*s1}, {s2*s1, s1*s2}, {s1*s2}, {s2*s1*s2}, {s2*s1*s2, s1*s2*s1}, {s1*s2*s1}, {s2*s1*s2*s1}}.

References

  • A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.

Links

Crossrefs

Cf. A005900 (the number of join-irreducible elements), A378072 (the size of Dedekind-MacNeille completion).

A383875 Number of pairs in the Bruhat order of type A_n.

Original entry on oeis.org

1, 3, 19, 213, 3781, 98407, 3550919
Offset: 0

Views

Author

Dmitry I. Ignatov, May 18 2025

Keywords

Comments

The number of ordered pairs in the Bruhat poset of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

Examples

			For n=0, the only element is 1 (identity) so a(0)=1.
For n=1 the elements are 1 (identity) and s1. The order relation consists of pairs (1, 1), (1, s1), and (s1, s1). So a(1) = 3.
For n=2 the line (Hasse) diagram is below.
       s1*s2*s1
        /   \
      s2*s1 s1*s2
       |  X  |
       s2    s1
        \   /
          1
The order relation consists of the six reflexive pairs, the eight pairs shown in the diagram as edges, and the five pairs (1, s2*s1), (1, s1*s2), (1, s1*s2*s1), (s1, s1*s2*s1), and (s2, s1*s2*s1). So a(2) = 6+8+5 = 19.

References

  • A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.

Links

Crossrefs

Cf. A000142 (the order size), A002538 (edges in the cover relation), A005130 (the size of Dedekind-MacNeille completion), A384061 (antichains), A384062 (maximal antichains).

Extensions

a(0)=1 prepended by Sara Billey, Jul 02 2025

A384090 Number of ordered pairs in the Bruhat order on B_n.

Original entry on oeis.org

3, 33, 847, 40249, 3089459, 350676009
Offset: 1

Views

Author

Dmitry I. Ignatov, May 19 2025

Keywords

Comments

The number of ordered pairs in the Bruhat order of the Weyl group B_n (the hyperoctahedral group).

Examples

			For n=1 the elements are 1 (identity) and s1. The order relation consists of pairs (1, 1), (1, s1), and (s1, s1). So a(1) = 3.
For n=2 the line (Hasse) diagram is below.
      s2*s1*s2*s1
          /   \
    s2*s1*s2  s1*s2*s1
        |   X   |
      s2*s1   s1*s2
        |   X   |
        s2     s1
          \   /
            1
The order relation is formed by 8 reflexive pairs, 12 pairs shown as edges in the diagram, and 13 pairs taken by transitivity: (1, s2*s1), (1, s1*s2), (1, s2*s1*s2), (1, s1*s2*s1), (1, s2*s1*s2*s1), (s2, s2*s1*s2), (s2, s1*s2*s1), (s2, s2*s1*s2*s1), (s1, s2*s1*s2), (s1, s1*s2*s1), (s1, s2*s1*s2*s1), (s2*s1, s2*s1*s2*s1), (s1*s2, s2*s1*s2*s1). So a(2) = 8+12+13 = 33.

References

  • A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.

Links

Crossrefs

Cf. A005900 (the number of join-irreducible elements), A378072 (the size of Dedekind-MacNeille completion).

A384062 Number of maximal antichains in the Bruhat order of type A_n.

Original entry on oeis.org

2, 4, 43, 183667
Offset: 1

Views

Author

Dmitry I. Ignatov, May 18 2025

Keywords

Comments

The number of maximal antichains in the Bruhat order of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

Examples

			For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The maximal antichains are {1} and {s1}.
For n=2 the line (Hasse) diagram is below.
       s1*s2*s1
        /   \
      s2*s1 s1*s2
       |  X  |
       s2    s1
        \   /
          1
The set of maximal antichains is {{1}, {s2, s1}, {s2*s1, s1*s2}, {s1*s2*s1}}.

References

  • A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
  • V. V. Deodhar, On Bruhat ordering and weight-lattice ordering for a Weyl group, Indagationes Mathematicae, vol. 81, 1 (1978), 423-435.

Crossrefs

Cf. A000142 (the order size), A005130 (the size of Dedekind-MacNeille completion), A384061.

A384061 Number of antichains in the Bruhat order of type A_n.

Original entry on oeis.org

3, 9, 250, 67595432
Offset: 1

Views

Author

Dmitry I. Ignatov, May 18 2025

Keywords

Comments

The number of antichains in the Bruhat order of the Weyl group A_n (isomorphic to the symmetric group S_{n+1}).

Examples

			For n=1 the elements are 1 (identity) and s1, the order contains pair (1, s1). The antichains are {}, {1}, and {s1}.
For n=2 the line (Hasse) diagram is below.
       s1*s2*s1
        /   \
      s2*s1 s1*s2
       |  X  |
       s2    s1
        \   /
          1
The set of antichains is {{}, {1}, {s2}, {s2, s1}, {s1}, {s2*s1}, {s2*s1, s1*s2}, {s1*s2}, {s1*s2*s1}}.

References

  • A. Bjorner, F. Brenti, Combinatorics of Coxeter Groups, Springer, 2009, 27-64.
  • V. V. Deodhar, On Bruhat ordering and weight-lattice ordering for a Weyl group, Indagationes Mathematicae, vol. 81, 1 (1978), 423-435.

Crossrefs

Cf. A000142 (the order size), A005130 (the size of Dedekind-MacNeille completion), A384062.

A367565 Number of reduced contexts on n labeled objects.

Original entry on oeis.org

1, 3, 32, 1863, 1316515, 75868099847
Offset: 1

Views

Author

Dmitry I. Ignatov, Nov 23 2023

Keywords

Comments

Equivalently, number of set systems on n points such that each of the systems obtained from the corresponding closure system on n points by omitting all intersections of other sets in the system and the set {1,...,n}; the systems with all sets shared at least one common element are not allowed.
This is the labeled version of A047684.

Examples

			The a(2)=3 set systems are {{1},{2}}, {{},{1}}, and {{},{2}}. The corresponding formal contexts represented by crosstables are
    1 x.    1 .x    1 ..
    2 .x    2 ..    2 x. .

References

  • B. Ganter and R. Wille, Formal Concept Analysis, Springer-Verlag, 1999, ISBN 3-540-62771-5, p. 24.
  • B. Ganter and S. A. Obiedkov, Conceptual Exploration, Springer 2016, ISBN 978-3-662-49290-1, pages 1-315.

Links

Crossrefs

A047684 (unlabeled version), A102896 (all closure systems).

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