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

0,12

Links

Table of n, a(n) for n=0..77.

Example

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

Mathematica

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

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.

Sequence in context: A051907 A178176 A093569 * A393825 A073091 A125250

Adjacent sequences: A384902 A384903 A384904 * A384906 A384907 A384908

Keyword

nonn,tabl

Author

Gus Wiseman, Jun 21 2025

Status

approved