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

0,3

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

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

Example

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

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 s<n do
         j:=j+1: s:=s+B(k)[j]: Q(n, k):=j;
      end do:
   od:
od:
T:=0:
for n from 1 to nmax do
   for j from 1 to numbpart(n) do
      T:=T+1:
      a(T):= Q(n, j)
    od;
od:
0, seq(a(n), n=1..T); # Johannes W. Meijer, Jun 21 2010

Mathematica

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

Prog

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

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.

Sequence in context: A052313 A271355 A211230 * A193173 A331581 A227355

Adjacent sequences: A049082 A049083 A049084 * A049086 A049087 A049088

Keyword

nice,nonn,tabf,look

Author

Alford Arnold

Extensions

More terms from Wolfdieter Lang, Apr 28 2005

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

Status

approved