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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.
1
2, 4, 13, 7, 33, 114, 11, 71, 321, 1146, 16, 136, 781, 3449, 12578, 22, 239, 1702, 9115, 39614, 146581, 29, 393, 3404, 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
1,1
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.
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.
EXAMPLE
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, 3404, 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]}
PROG
(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
CROSSREFS
Cf. A369313.
Sequence in context: A176990 A109928 A023640 * A064454 A116662 A373602
KEYWORD
nonn,hard,tabl
AUTHOR
Nadia Lafreniere, Jan 26 2024
STATUS
approved