A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).

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

Offset

0,4

Comments

The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).

The sequence of row sums is A006128(n).

The number of times k appears in row n is A008284(n,k). - Franklin T. Adams-Watters, Jan 12 2006

The next level (row) gets created from each node by adding one or two more nodes. If a single node is added, its value is one more than the value of its parent. If two nodes are added, the first is equal in value to the parent and the value of the second is one more than the value of the parent. See A128628. - Alford Arnold, Mar 27 2007

The 1's in the (flattened) sequence mark the start of a new row, the value that precedes the 1 equals the row number minus one. (I.e., the 1 preceded by a 0 is the start of row 1, the 1 preceded by a 6 is the start of row 7, etc.) - M. F. Hasler, Jun 06 2018

Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - Gus Wiseman, May 24 2020

References

Abramowitz and Stegun, Handbook, p. 831, column labeled "m".

Links

T. D. Noe, Rows n = 0..25 of irregular triangle, 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], p. 831.

Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.

Wolfdieter Lang, Rows n = 1 ..20.

OEIS Wiki, Orderings of partitions

Wikiversity, Lexicographic and colexicographic order

Formula

a(n) = A001222(A334433(n)). - Gus Wiseman, May 22 2020

Example

0;
1;
1, 2;
1, 2, 3;
1, 2, 2, 3, 4;
1, 2, 2, 3, 3, 4, 5;
1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7;

Maple

with(combinat): nmax:=9: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(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: 0, seq(seq(Q(n, j),  j=1..y(n)), n=1..nmax); # Johannes W. Meijer, Jun 21 2010, revised Nov 29 2012
# alternative implementation based on A119441 by R. J. Mathar, Jul 12 2013
A036043 := proc(n, k)
    local pi;
    pi := ASPrts(n)[k] ;
    nops(pi) ;
end proc:
for n from 1 to 10 do
    for k from 1 to A000041(n) do
        printf("%d, ", A036043(n, k)) ;
    end do:
    printf("\n") ;
end do:

Mathematica

Table[Length/@Sort[IntegerPartitions[n]], {n, 0, 30}] (* Gus Wiseman, May 22 2020 *)

Prog

(PARI) A036043(n, k)=#partitions(n)[k] \\ M. F. Hasler, Jun 06 2018
(SageMath)
def A036043_row(n):
    return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
for n in (0..10): print(A036043_row(n)) # Peter Luschny, Nov 02 2019

Crossrefs

Cf. A036037, A036038, A036039, A036040, A036042.

Row lengths are A000041.

Partition lengths of A036036 and A334301.

The version not sorted by length is A049085.

The generalization to compositions is A124736.

The Heinz number of the same partition is A334433.

The number of distinct elements in the same partition is A334440.

The maximum part of the same partition is A334441.

Lexicographically ordered reversed partitions are A026791.

Lexicographically ordered partitions are A193073.

Cf. A103921, A115623, A124734, A185974, A296150, A334435, A334438, A334439.

Sequence in context: A269970 A333518 A252230 * A333486 A128628 A238966

Adjacent sequences: A036040 A036041 A036042 * A036044 A036045 A036046

Keyword

nonn,easy,tabf

Author

N. J. A. Sloane

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

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

Incorrect formula deleted by M. F. Hasler, Jun 06 2018

Status

approved