A049085 Irregular table T(n,k) = maximal part of the k-th partition of n, when listed in Abramowitz-Stegun order (as in A036043).

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

Offset: 0

Author

Alford Arnold

Keywords

• nice
• nonn
• tabf
• look

Comments

a(0) = 0 by convention. - Franklin T. Adams-Watters, Jun 24 2014

Like A036043 this is important for calculating sequences defined over the numeric partitions, cf. A000041. For example, the triangular array A019575 can be calculated using A036042 and this sequence.

The row sums are A006128. - Johannes W. Meijer, Jun 21 2010

The name is correct if the partitions are read in reverse, so that the parts are weakly increasing. The version for non-reversed partitions is A334441. - Gus Wiseman, May 21 2020

Examples

			Rows:
  [0];
  [1];
  [2,1];
  [3,2,1];
  [4,3,2,2,1];
  [5,4,3,3,2,2,1];
  ...
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.

Links

• Alois P. Heinz, Rows n = 0..26, flattened
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Wolfdieter Lang, First 15 rows.
• OEIS Wiki, Orderings of partitions
• Wikiversity, Lexicographic and colexicographic order

Crossrefs

Row lengths are A000041.

Row sums are A006128.

The length of the partition is A036043.

The number of distinct elements of the partition is A103921.

The Heinz number of the partition is A185974.

The version ignoring length is A194546.

The version for non-reversed partitions is A334441.

Lexicographically ordered reversed partitions are A026791.

Reversed partitions in Abramowitz-Stegun order are A036036.

Reverse-lexicographically ordered partitions are A080577.

Partitions in Abramowitz-Stegun order are A334301.

Cf. A001221, A036037, A036042, A115623, A124734, A193073, A334302, A334433, A334438, A334439, A334440.

Programs

Maple

with(combinat):
nmax:=9:
for n from 1 to nmax do
   y(n):=numbpart(n):
   P(n):=partition(n):
   for k from 1 to y(n) do
      B(k):=P(n)[k]
   od:
   for k from 1 to y(n) do
      s:=0: j:=0:
      while sJohannes W. Meijer, Jun 21 2010

Mathematica

Table[If[n==0,{0},Max/@Sort[Reverse/@IntegerPartitions[n]]],{n,0,8}] (* Gus Wiseman, May 21 2020 *)

PARI

A049085(n,k)=if(n,partitions(n)[k][1],0) \\ M. F. Hasler, Jun 06 2018

Extensions

More terms from Wolfdieter Lang, Apr 28 2005

a(0) inserted by Franklin T. Adams-Watters, Jun 24 2014