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%I #85 Sep 22 2023 05:17:17
%S 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,
%T 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,
%U 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
%N Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).
%C The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).
%C The sequence of row sums is A006128(n).
%C The number of times k appears in row n is A008284(n,k). - _Franklin T. Adams-Watters_, Jan 12 2006
%C 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
%C 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
%C Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - _Gus Wiseman_, May 24 2020
%D Abramowitz and Stegun, Handbook, p. 831, column labeled "m".
%H T. D. Noe, <a href="/A036043/b036043.txt">Rows n = 0..25 of irregular triangle, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://people.math.sfu.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], p. 831.
%H Kevin Brown, <a href="http://www.mathpages.com/home/kmath057.htm">Generalized Birthday Problem (N Items in M Bins)</a>, 1994-2010.
%H Wolfdieter Lang, <a href="/A036043/a036043.txt">Rows n = 1 ..20.</a>
%H OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F a(n) = A001222(A334433(n)). - _Gus Wiseman_, May 22 2020
%e 0;
%e 1;
%e 1, 2;
%e 1, 2, 3;
%e 1, 2, 2, 3, 4;
%e 1, 2, 2, 3, 3, 4, 5;
%e 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
%e 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7;
%p 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
%p # alternative implementation based on A119441 by _R. J. Mathar_, Jul 12 2013
%p A036043 := proc(n,k)
%p local pi;
%p pi := ASPrts(n)[k] ;
%p nops(pi) ;
%p end proc:
%p for n from 1 to 10 do
%p for k from 1 to A000041(n) do
%p printf("%d,",A036043(n,k)) ;
%p end do:
%p printf("\n") ;
%p end do:
%t Table[Length/@Sort[IntegerPartitions[n]],{n,0,30}] (* _Gus Wiseman_, May 22 2020 *)
%o (PARI) A036043(n,k)=#partitions(n)[k] \\ _M. F. Hasler_, Jun 06 2018
%o (SageMath)
%o def A036043_row(n):
%o return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
%o for n in (0..10): print(A036043_row(n)) # _Peter Luschny_, Nov 02 2019
%Y Cf. A036037, A036038, A036039, A036040, A036042.
%Y Row lengths are A000041.
%Y Partition lengths of A036036 and A334301.
%Y The version not sorted by length is A049085.
%Y The generalization to compositions is A124736.
%Y The Heinz number of the same partition is A334433.
%Y The number of distinct elements in the same partition is A334440.
%Y The maximum part of the same partition is A334441.
%Y Lexicographically ordered reversed partitions are A026791.
%Y Lexicographically ordered partitions are A193073.
%Y Cf. A103921, A115623, A124734, A185974, A296150, A334435, A334438, A334439.
%K nonn,easy,tabf
%O 0,4
%A _N. J. A. Sloane_
%E More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
%E a(0) inserted by _Franklin T. Adams-Watters_, Jun 24 2014
%E Incorrect formula deleted by _M. F. Hasler_, Jun 06 2018
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