A384905 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with k maximal anti-runs (decreasing by more than 1).

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 5, 2, 1, 0, 0, 0, 0, 0, 0, 0, 6, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 7, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Author

Gus Wiseman, Jun 21 2025

Keywords

• nonn
• tabl

Examples

			The T(10,2) = 3 strict partitions with 2 maximal anti-runs are: (7,2,1), (5,4,1), (5,3,2).
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  2  0  0  0
  0  2  1  0  0  0
  0  3  0  1  0  0  0
  0  3  2  0  0  0  0  0
  0  4  2  0  0  0  0  0  0
  0  5  2  1  0  0  0  0  0  0
  0  6  3  0  1  0  0  0  0  0  0
  0  7  4  1  0  0  0  0  0  0  0  0
  0  9  3  3  0  0  0  0  0  0  0  0  0
		

Crossrefs

Row sums are A000009.

Column k = 1 is A003114.

For subsets instead of strict integer partitions see A053538, A119900, A210034.

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

This is the strict case of A268193.

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, A384886, A384889, A384890.

Programs

Mathematica

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