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

0,11

Comments

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

Links

Table of n, a(n) for n=0..90.

Example

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

Mathematica

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

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.

Sequence in context: A331284 A331591 A003649 * A287170 A216784 A256067

Adjacent sequences: A353738 A353739 A353740 * A353742 A353743 A353744

Keyword

nonn,tabf,less

Author

Gus Wiseman, May 20 2022

Status

approved