A353741 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with product k, all zeros removed.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 1, 1, 4, 2, 2, 1, 4, 1, 1, 1, 3, 2

Offset: 0

Author

Gus Wiseman, May 20 2022

Keywords

• nonn
• tabf
• less

Comments

Warning: There are certain internal "holes" in A339095 that are removed in this sequence.

Examples

			Triangle begins:
  1
  1
  1 1
  1 1 1
  1 1 1 2
  1 1 1 2 1 1
  1 1 1 2 1 2 2 1
  1 1 1 2 1 2 1 2 1 1 2
  1 1 1 2 1 2 1 3 1 1 3 1 3 1
  1 1 1 2 1 2 1 3 2 1 3 1 1 3 2 2 2 1
  1 1 1 2 1 2 1 3 2 2 3 1 1 4 2 2 1 4 1 1 1 3 2
Row n = 7 counts the following partitions:
  1111111   211111   31111   4111    511   61     7   421    331   52   43
                             22111         3211       2221              322
		

Crossrefs

Row sums are A000041.

Row lengths are A034891.

A partial transpose is A319000.

The full version with zeros is A339095, rank statistic A003963.

A008284 counts partitions by sum, strict A116608.

A225485 counts partitions by frequency depth.

A266477 counts partitions by product of multiplicities, ranked by A005361.

Cf. A002033, A266499, A325242, A325268, A325280, A353503, A353506, A353698.

Programs

Mathematica

DeleteCases[Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,10},{k,1,2^n}],0,2]