A385814 Triangle read by rows where T(n,k) is the number of integer partitions of n with k maximal proper anti-runs (sequences decreasing by more than 1).

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 3, 2, 3, 1, 1, 1, 0, 3, 4, 2, 3, 1, 1, 1, 0, 4, 5, 4, 3, 3, 1, 1, 1, 0, 5, 5, 6, 5, 3, 3, 1, 1, 1, 0, 6, 8, 7, 6, 6, 3, 3, 1, 1, 1, 0, 7, 9, 10, 8, 7, 6, 3, 3, 1, 1, 1

Offset: 0

Author

Gus Wiseman, Jul 09 2025

Keywords

• nonn
• tabl

Examples

			The partition (8,5,4,2,1) has maximal proper anti-runs ((8,5),(4,2),(1)) so is counted under T(20,3).
The partition (8,5,3,2,2) has maximal proper anti-runs ((8,5,3),(2),(2)) so is also counted under T(20,3).
Row n = 8 counts the following partitions:
  .  8   611  5111  41111  32111   221111  2111111  11111111
     71  521  4211  3221   311111
     62  44   332   2222   22211
     53  431  3311
         422
Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  2  1  1  1
  0  2  2  1  1  1
  0  3  2  3  1  1  1
  0  3  4  2  3  1  1  1
  0  4  5  4  3  3  1  1  1
  0  5  5  6  5  3  3  1  1  1
  0  6  8  7  6  6  3  3  1  1  1
  0  7  9 10  8  7  6  3  3  1  1  1
  0  9 11 13 12  9  8  6  3  3  1  1  1
  0 10 14 16 15 13 10  8  6  3  3  1  1  1
  0 12 19 18 21 17 14 11  8  6  3  3  1  1  1
  0 14 21 26 23 24 19 15 11  8  6  3  3  1  1  1
  0 17 26 31 33 28 26 20 16 11  8  6  3  3  1  1  1
  0 19 32 37 40 39 31 28 21 16 11  8  6  3  3  1  1  1
  0 23 38 47 50 47 45 34 29 22 16 11  8  6  3  3  1  1  1
  0 26 45 57 61 61 54 48 36 30 22 16 11  8  6  3  3  1  1  1
  0 31 53 71 75 76 70 60 51 37 31 22 16 11  8  6  3  3  1  1  1
		

Crossrefs

Row sums are A000041, strict A000009.

Column k = 1 is A003114.

For anti-runs instead of proper anti-runs we have A268193.

The corresponding rank statistic is A356228.

For proper runs instead of proper anti-runs we have A384881.

For subsets instead of partitions we have A384893, runs A034839.

The strict case is A384905.

For runs instead of proper anti-runs we have A385815.

A007690 counts partitions with no singletons (ranks A001694), complement A183558.

A034296 counts flat or gapless partitions, ranks A066311 or A073491.

A047993 counts partitions with max part = length, ranks A106529.

A098859 counts Wilf partitions, complement A336866 (ranks A325992).

A116608 counts partitions by distinct parts.

A116931 counts sparse partitions, ranks A319630.

Cf. A001227, A008284, A089259, A116674, A239455, A325325, A356226, A384880, A384885, A384887, A384906.

Programs

Mathematica

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