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).

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, 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, 143, 235, 241, 187, 116, 62, 25, 11, 2, 1, 1, 232, 420, 468, 392, 267, 150, 75, 28, 12, 2, 1

Offset: 0

Author

Gus Wiseman, Jun 21 2025

Keywords

• nonn
• tabl

Examples

			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).
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).
Row n = 5 counts the following subsets:
  {}  {1}      {1,2}    {1,2,3}    {1,2,3,4}  {1,2,3,4,5}
      {2}      {2,3}    {2,3,4}    {2,3,4,5}
      {3}      {3,4}    {3,4,5}
      {4}      {4,5}    {1,2,3,5}
      {5}      {1,2,4}  {1,2,4,5}
      {1,3}    {1,2,5}  {1,3,4,5}
      {1,4}    {1,3,4}
      {1,5}    {1,4,5}
      {2,4}    {2,3,5}
      {2,5}    {2,4,5}
      {3,5}
      {1,3,5}
Triangle begins:
   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  29  16   8   2   1
   1  54  71  60  39  19   9   2   1
   1  88 130 122  86  50  22  10   2   1
   1 143 235 241 187 116  62  25  11   2   1
   1 232 420 468 392 267 150  75  28  12   2   1
   1 376 744 894 806 588 363 188  89  31  13   2   1
		

Crossrefs

Column k = 1 is A000071.

Row sums are A000079.

Column k = 2 is A001629.

For runs instead of anti-runs we have A034839, for strict partitions A116674.

The case containing n is A053538.

For integer partitions instead of subsets we have A268193, strict A384905.

A384175 counts subsets with all distinct lengths of maximal runs, complement A384176.

A384877 gives lengths of maximal anti-runs in binary indices, firsts A384878.

Cf. A010027, A384177, A384879, A384889, A384890.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],Length[Split[#,#2!=#1+1&]]==k&]],{n,0,10},{k,0,n}]