A103921 Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order.

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

Offset: 0

Author

Wolfdieter Lang, Mar 24 2005

Keywords

• nonn
• tabf

Comments

T(n, m) is the number of distinct parts of the m-th partition of n in Abramowitz-Stegun order; n >= 0, m = 1..p(n) = A000041(n).

The row length sequence of this table is p(n)=A000041(n) (number of partitions).

In order to count distinct parts of a partition consider the partition as a set instead of a multiset. E.g., n=6: read [1,1,1,3] as {1,3} and count the elements, here 2.

Rows are the same as the rows of A115623, but in reverse order.

From Wolfdieter Lang, Mar 17 2011: (Start)

The number of 1s in row number n, n >= 1, is tau(n)=A000005(n), the number of divisors of n.

For the proof read off the divisors d(n,j), j=1..tau(n), from row number n of table A027750, and translate them to the tau(n) partitions d(n,1)^(n/d(n,1)), d(n,2)^(n/d(n,2)),..., d(n,tau(n))^(n/d(n,tau(n))).

See a comment by Giovanni Resta under A000005. (End)

From Gus Wiseman, May 20 2020: (Start)

The name is correct if integer partitions are read in reverse, so that the parts are weakly increasing. The non-reversed version is A334440.

Also the number of distinct parts of the n-th integer partition in lexicographic order (A193073).

Differs from the number of distinct parts in the n-th integer partition in (sum/length/revlex) order (A334439). For example, (6,2,2) has two distinct elements, while (1,4,5) has three.

(End)

Examples

			Triangle starts:
  0,
  1,
  1, 1,
  1, 2, 1,
  1, 2, 1, 2, 1,
  1, 2, 2, 2, 2, 2, 1,
  1, 2, 2, 1, 2, 3, 1, 2, 2, 2, 1,
  1, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 2, 3, 2, 2, 2, 2, 1,
  1, 2, 2, 2, 2, ...
a(5,4)=2 from the fourth partition of 5 in the mentioned order, i.e., (1^2,3), which has two distinct parts, namely 1 and 3.
		

Links

• Robert Price, Table of n, a(n) for n = 0..9295 (first 25 rows).
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
• Wolfdieter Lang, First 10 rows.
• OEIS Wiki, Orderings of partitions
• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row sums are A000070.

Row lengths are A000041.

The lengths of these partitions are A036043.

The maxima of these partitions are A049085.

The version for non-reversed partitions is A334440.

The version for colex instead of lex is (also) A334440.

Lexicographically ordered reversed partitions are A026791.

Reversed partitions in Abramowitz-Stegun order are A036036.

Reverse-lexicographically ordered partitions are A080577.

Compositions in Abramowitz-Stegun order are A124734.

Cf. A001221, A036037, A112798, A115621, A115623, A185974, A193073, A228531, A334301, A334302, A334433, A334435, A334441.

Programs

Mathematica

Join@@Table[Length/@Union/@Sort[Reverse/@IntegerPartitions[n]],{n,0,8}] (* Gus Wiseman, May 20 2020 *)

Formula

a(n) = A001221(A185974(n)). - Gus Wiseman, May 20 2020

Extensions

Edited by Franklin T. Adams-Watters, May 29 2006