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.

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

Author

Nadia Lafreniere, Jan 26 2024

Keywords

• nonn
• hard
• tabl

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

• Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, Toggling, rowmotion, and homomesy on interval-closed sets, arXiv:2307.08520 [math.CO], 2023.

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