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

%I A036043 #89 Mar 31 2025 11:52:38
%S A036043 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 A036043 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 A036043 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 A036043 Irregular triangle read by rows: row n (n >= 0) gives number of parts in all partitions of n (in Abramowitz and Stegun order).
%C A036043 The sequence of row lengths of this array is p(n) = A000041(n) (partition numbers).
%C A036043 The sequence of row sums is A006128(n).
%C A036043 The number of times k appears in row n is A008284(n,k). - _Franklin T. Adams-Watters_, Jan 12 2006
%C A036043 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 A036043 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 A036043 Also the maximum part in the n-th partition in graded lexicographic order (sum/lex, A193073). - _Gus Wiseman_, May 24 2020
%D A036043 Abramowitz and Stegun, Handbook, p. 831, column labeled "m".
%H A036043 T. D. Noe, <a href="/A036043/b036043.txt">Rows n = 0..25 of irregular triangle, flattened</a>
%H A036043 M. Abramowitz and I. A. Stegun, eds., <a href="https://personal.math.ubc.ca/~cbm/aands/">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [scanned copy], p. 831.
%H A036043 Kevin Brown, <a href="http://www.mathpages.com/home/kmath057.htm">Generalized Birthday Problem (N Items in M Bins)</a>, 1994-2010.
%H A036043 Wolfdieter Lang, <a href="/A036043/a036043.txt">Rows n = 1 ..20.</a>
%H A036043 OEIS Wiki, <a href="http://oeis.org/wiki/Orderings of partitions">Orderings of partitions</a>
%H A036043 Wikiversity, <a href="https://en.wikiversity.org/wiki/Lexicographic_and_colexicographic_order">Lexicographic and colexicographic order</a>
%F A036043 a(n) = A001222(A334433(n)). - _Gus Wiseman_, May 22 2020
%e A036043 0;
%e A036043 1;
%e A036043 1, 2;
%e A036043 1, 2, 3;
%e A036043 1, 2, 2, 3, 4;
%e A036043 1, 2, 2, 3, 3, 4, 5;
%e A036043 1, 2, 2, 2, 3, 3, 3, 4, 4, 5, 6;
%e A036043 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 7;
%p A036043 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 A036043 # alternative implementation based on A119441 by _R. J. Mathar_, Jul 12 2013
%p A036043 A036043 := proc(n,k)
%p A036043     local pi;
%p A036043     pi := ASPrts(n)[k] ;
%p A036043     nops(pi) ;
%p A036043 end proc:
%p A036043 for n from 1 to 10 do
%p A036043     for k from 1 to A000041(n) do
%p A036043         printf("%d,",A036043(n,k)) ;
%p A036043     end do:
%p A036043     printf("\n") ;
%p A036043 end do:
%t A036043 Table[Length/@Sort[IntegerPartitions[n]],{n,0,30}] (* _Gus Wiseman_, May 22 2020 *)
%o A036043 (PARI) A036043(n,k)=#partitions(n)[k] \\ _M. F. Hasler_, Jun 06 2018
%o A036043 (SageMath)
%o A036043 def A036043_row(n):
%o A036043     return [len(p) for k in (0..n) for p in Partitions(n, length=k)]
%o A036043 for n in (0..10): print(A036043_row(n)) # _Peter Luschny_, Nov 02 2019
%Y A036043 Cf. A036037, A036038, A036039, A036040, A036042.
%Y A036043 Row lengths are A000041.
%Y A036043 Partition lengths of A036036 and A334301.
%Y A036043 The version not sorted by length is A049085.
%Y A036043 The generalization to compositions is A124736.
%Y A036043 The Heinz number of the same partition is A334433.
%Y A036043 The number of distinct elements in the same partition is A334440.
%Y A036043 The maximum part of the same partition is A334441.
%Y A036043 Lexicographically ordered reversed partitions are A026791.
%Y A036043 Lexicographically ordered partitions are A193073.
%Y A036043 Cf. A103921, A115623, A124734, A185974, A296150, A334435, A334438, A334439.
%K A036043 nonn,easy,tabf
%O A036043 0,4
%A A036043 _N. J. A. Sloane_
%E A036043 More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001
%E A036043 a(0) inserted by _Franklin T. Adams-Watters_, Jun 24 2014
%E A036043 Incorrect formula deleted by _M. F. Hasler_, Jun 06 2018