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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: Nadia Lafreniere

Nadia Lafreniere's wiki page.

Nadia Lafreniere has authored 3 sequences.

A367109 Triangle of number of interval-closed sets T(m,n) in the product of two chains [m]x[n], for m <= n, read by rows.

Original entry on oeis.org

2, 4, 13, 7, 33, 114, 11, 71, 321, 1146, 16, 136, 781, 3449, 12578, 22, 239, 1702, 9115, 39614, 146581, 29, 393, 3403, 21743, 111063, 477097, 1784114, 37, 613, 6349, 47737, 283243, 1398211, 5953639, 22443232, 46, 916, 11191, 97861, 667684, 3754186, 18060127, 76372470, 289721772, 56, 1321, 18811, 189377, 1472692, 9358669
Offset: 1

Views

Author

Nadia Lafreniere, Jan 26 2024

Keywords

Comments

An interval-closed set of a poset is a subset I such that if x and y are in I with x <= z <= y, then z is in I.
Interval-closed sets are also called convex subsets of a poset.

Examples

			The initial rows of the triangle are:
  [1] 2
  [2] 4,  13
  [3] 7,  33,  114
  [4] 11, 71,  321,   1146
  [5] 16, 136, 781,   3449,  12578
  [6] 22, 239, 1702,  9115,  39614,  146581
  [7] 29, 393, 3403,  21743, 111063, 477097,  1784114
  [8] 37, 613, 6349,  47737, 283243, 1398211, 5953639,  22443232
  [9] 46, 916, 11191, 97861, 667684, 3754186, 18060127, 76372470, 289721772
The T(1,1) = 2 through T(3,1) = 7 interval-closed sets:
  {}       {}              {}                             {}
  {[1,1]}  {[1,1]}         {[1,1]}                        {[1,1]}
           {[2,1]}         {[1,2]}                        {[2,1]}
           {[1,1], [2,1]}  {[2,1]}                        {[3,1]}
                           {[2,2]}                        {[1,1], [2,1]}
                           {[1,1], [1,2]}                 {[2,1], [3,1]}
                           {[1,1], [2,1]}                 {[1,1], [2,1], [3,1]}
                           {[1,2], [2,1]}
                           {[1,2], [2,2]}
                           {[2,1], [2,2]}
                           {[1,1], [1,2], [2,1]}
                           {[1,2], [2,1], [2,2]}
                           {[1,1,], [1,2], [2,1], [2,2]}

Links

Crossrefs

Cf. A369313.

Programs

  • SageMath
    ICS_count = 0
x = Posets.ProductOfChains([m, n])
for A in x.antichains_iterator():
    I = x.order_ideal(A)
    Q = x.subposet(set(I).difference(A))
    ICS_count += Q.antichains().cardinality()
ICS_count

Extensions

Corrected by Nadia Lafreniere, Dec 10 2024

A367316 Number of interval-closed sets in the root poset of type A(n).

Original entry on oeis.org

1, 2, 8, 45, 307, 2385, 20362, 186812, 1814156, 18448851, 194918129, 2126727740
Offset: 0

Views

Author

Nadia Lafreniere, Jan 26 2024

Keywords

Comments

An interval-closed set of a poset is a subset I such that if x and y are in I with x <= z <= y, then z is in I.
Interval-closed sets are also called convex subsets of a poset.
The root poset of a root system is the partial order on positive roots where a <= b if b-a is a nonnegative sum of simple roots.

Examples

			For n = 0, the poset is empty, so there is only one subset.For n = 1, the poset has only one element, and both subsets are interval-closed.For n = 2, the poset has three elements, and rank 1. Every subset of a poset of rank at most 1 is interval-closed, and therefore there are a(2) = 8 interval-closed sets.For n = 3, the poset has six elements, and only 45 of the 64 subsets are interval-closed.

Links

Crossrefs

Interval-closed sets are a superset of order ideals. Order ideals of the type A root poset are counted by the Catalan numbers. Cf. A000108
Interval-closed sets for other posets: Cf. A369313, A367109

Programs

  • SageMath
    ICS_count = 0
x = RootSystem(['A',n]).root_poset()
for A in x.antichains_iterator():
    I = x.order_ideal(A)
    Q = x.subposet(set(I).difference(A))
    ICS_count += Q.antichains().cardinality()
ICS_count

A369313 Number of interval-closed sets in the boolean lattice of dimension n.

Original entry on oeis.org

2, 4, 13, 101, 3938, 3257610, 676675164063
Offset: 0

Views

Author

Nadia Lafreniere, Jan 19 2024

Keywords

Comments

An interval-closed set of a poset is a subset I such that if x and y are in I with x <= z <= y, then z is in I.
Interval-closed sets are also called convex subsets of a poset.

Examples

			The a(0) = 2 through a(2) = 13 interval-closed sets:
{}    {}       {}
{{}}  {{}}     {{}}
      {{1}}    {{1}}
      {{}{1}}  {{2}}
               {{12}}
               {{}{1}}
               {{}{2}}
               {{1}{2}}
               {{1}{12}}
               {{2}{12}}
               {{}{1}{2}}
               {{1}{2}{12}}
               {{}{1}{2}{12}}

Links

Crossrefs

Interval-closed sets are a superset of order ideals. Cf. A000372.

Programs

  • SageMath
    ICS_count = 0
x = Posets.BooleanLattice(n)
for A in x.antichains_iterator():
    I = x.order_ideal(A)
    Q = x.subposet(set(I).difference(A))
    ICS_count += Q.antichains().cardinality()
ICS_count

Extensions

a(6) from Christian Sievers, Jan 27 2024

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