A334440 Irregular triangle T(n,k) read by rows: row n lists numbers of distinct parts of the n-th integer partition in Abramowitz-Stegun (sum/length/lex) order.

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 1, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 2, 2, 3

Offset: 0

Author

Gus Wiseman, May 05 2020

Keywords

• nonn
• tabf

Comments

The total number of parts, counting duplicates, is A036043. The version for reversed partitions is A103921.

Examples

			Triangle begins:
  0
  1
  1 1
  1 2 1
  1 1 2 2 1
  1 2 2 2 2 2 1
  1 1 2 2 1 3 2 2 2 2 1
  1 2 2 2 2 2 3 2 2 3 2 2 2 2 1
  1 1 2 2 2 2 2 3 3 2 1 3 2 3 2 2 3 2 2 2 2 1
		

Links

• Robert Price, Table of n, a(n) for n = 0..9295 (25 rows).
• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row lengths are A000041.

The number of not necessarily distinct parts is A036043.

The version for reversed partitions is A103921.

Ignoring length (sum/lex) gives A103921 (also).

a(n) is the number of distinct elements in row n of A334301.

The maximum part of the same partition is A334441.

Lexicographically ordered reversed partitions are A026791.

Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.

Partitions in increasing-length colex order (sum/length/colex) are A036037.

Graded reverse-lexicographically ordered partitions are A080577.

Partitions counted by sum and number of distinct parts are A116608.

Graded lexicographically ordered partitions are A193073.

Partitions in colexicographic order (sum/colex) are A211992.

Partitions in dual Abramowitz-Stegun (sum/length/revlex) order are A334439.

Cf. A001221, A049085, A124734, A185974, A296774, A299755, A334028, A334302, A334433, A334434, A334435, A334438.

Programs

Mathematica

Join@@Table[Length/@Union/@Sort[IntegerPartitions[n]],{n,0,10}]

Formula

a(n) = A001221(A334433(n)).