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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.

%I A367109 #18 Dec 10 2024 13:13:34
%S A367109 2,4,13,7,33,114,11,71,321,1146,16,136,781,3449,12578,22,239,1702,
%T A367109 9115,39614,146581,29,393,3403,21743,111063,477097,1784114,37,613,
%U A367109 6349,47737,283243,1398211,5953639,22443232,46,916,11191,97861,667684,3754186,18060127,76372470,289721772,56,1321,18811,189377,1472692,9358669
%N 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.
%C A367109 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.
%C A367109 Interval-closed sets are also called convex subsets of a poset.
%H A367109 Jennifer Elder, Nadia Lafrenière, Erin McNicholas, Jessica Striker, and Amanda Welch, <a href="http://arxiv.org/abs/2307.08520">Toggling, rowmotion, and homomesy on interval-closed sets</a>, arXiv:2307.08520 [math.CO], 2023.
%e A367109 The initial rows of the triangle are:
%e A367109   [1] 2
%e A367109   [2] 4,  13
%e A367109   [3] 7,  33,  114
%e A367109   [4] 11, 71,  321,   1146
%e A367109   [5] 16, 136, 781,   3449,  12578
%e A367109   [6] 22, 239, 1702,  9115,  39614,  146581
%e A367109   [7] 29, 393, 3403,  21743, 111063, 477097,  1784114
%e A367109   [8] 37, 613, 6349,  47737, 283243, 1398211, 5953639,  22443232
%e A367109   [9] 46, 916, 11191, 97861, 667684, 3754186, 18060127, 76372470, 289721772
%e A367109 The T(1,1) = 2 through T(3,1) = 7 interval-closed sets:
%e A367109   {}       {}              {}                             {}
%e A367109   {[1,1]}  {[1,1]}         {[1,1]}                        {[1,1]}
%e A367109            {[2,1]}         {[1,2]}                        {[2,1]}
%e A367109            {[1,1], [2,1]}  {[2,1]}                        {[3,1]}
%e A367109                            {[2,2]}                        {[1,1], [2,1]}
%e A367109                            {[1,1], [1,2]}                 {[2,1], [3,1]}
%e A367109                            {[1,1], [2,1]}                 {[1,1], [2,1], [3,1]}
%e A367109                            {[1,2], [2,1]}
%e A367109                            {[1,2], [2,2]}
%e A367109                            {[2,1], [2,2]}
%e A367109                            {[1,1], [1,2], [2,1]}
%e A367109                            {[1,2], [2,1], [2,2]}
%e A367109                            {[1,1,], [1,2], [2,1], [2,2]}
%o A367109 (SageMath)
%o A367109 ICS_count = 0
%o A367109 x = Posets.ProductOfChains([m, n])
%o A367109 for A in x.antichains_iterator():
%o A367109     I = x.order_ideal(A)
%o A367109     Q = x.subposet(set(I).difference(A))
%o A367109     ICS_count += Q.antichains().cardinality()
%o A367109 ICS_count
%Y A367109 Cf. A369313.
%K A367109 nonn,hard,tabl
%O A367109 1,1
%A A367109 _Nadia Lafreniere_, Jan 26 2024
%E A367109 Corrected by _Nadia Lafreniere_, Dec 10 2024