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A384893 Triangle read by rows where T(n,k) is the number of subsets of {1..n} with k maximal anti-runs (increasing by more than 1).

%I A384893 #8 Jun 22 2025 14:37:51
%S A384893 1,1,1,1,2,1,1,4,2,1,1,7,5,2,1,1,12,10,6,2,1,1,20,20,13,7,2,1,1,33,38,
%T A384893 29,16,8,2,1,1,54,71,60,39,19,9,2,1,1,88,130,122,86,50,22,10,2,1,1,
%U A384893 143,235,241,187,116,62,25,11,2,1,1,232,420,468,392,267,150,75,28,12,2,1
%N A384893 Triangle read by rows where T(n,k) is the number of subsets of {1..n} with k maximal anti-runs (increasing by more than 1).
%e A384893 The subset {3,6,7,9,11,12} has maximal anti-runs ((3,6),(7,9,11),(12)), so is counted under T(12,3).
%e A384893 The subset {3,6,7,9,10,12} has maximal anti-runs ((3,6),(7,9),(10,12)), so is counted under T(12,3).
%e A384893 Row n = 5 counts the following subsets:
%e A384893   {}  {1}      {1,2}    {1,2,3}    {1,2,3,4}  {1,2,3,4,5}
%e A384893       {2}      {2,3}    {2,3,4}    {2,3,4,5}
%e A384893       {3}      {3,4}    {3,4,5}
%e A384893       {4}      {4,5}    {1,2,3,5}
%e A384893       {5}      {1,2,4}  {1,2,4,5}
%e A384893       {1,3}    {1,2,5}  {1,3,4,5}
%e A384893       {1,4}    {1,3,4}
%e A384893       {1,5}    {1,4,5}
%e A384893       {2,4}    {2,3,5}
%e A384893       {2,5}    {2,4,5}
%e A384893       {3,5}
%e A384893       {1,3,5}
%e A384893 Triangle begins:
%e A384893    1
%e A384893    1   1
%e A384893    1   2   1
%e A384893    1   4   2   1
%e A384893    1   7   5   2   1
%e A384893    1  12  10   6   2   1
%e A384893    1  20  20  13   7   2   1
%e A384893    1  33  38  29  16   8   2   1
%e A384893    1  54  71  60  39  19   9   2   1
%e A384893    1  88 130 122  86  50  22  10   2   1
%e A384893    1 143 235 241 187 116  62  25  11   2   1
%e A384893    1 232 420 468 392 267 150  75  28  12   2   1
%e A384893    1 376 744 894 806 588 363 188  89  31  13   2   1
%t A384893 Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2!=#1+1&]]==k&]],{n,0,10},{k,0,n}]
%Y A384893 Column k = 1 is A000071.
%Y A384893 Row sums are A000079.
%Y A384893 Column k = 2 is A001629.
%Y A384893 For runs instead of anti-runs we have A034839, for strict partitions A116674.
%Y A384893 The case containing n is A053538.
%Y A384893 For integer partitions instead of subsets we have A268193, strict A384905.
%Y A384893 A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.
%Y A384893 A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.
%Y A384893 Cf. A010027, A384177, A384879, A384889, A384890.
%K A384893 nonn,tabl
%O A384893 0,5
%A A384893 _Gus Wiseman_, Jun 21 2025