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A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).

%I A186412 #41 Jul 25 2020 12:15:55
%S A186412 1,3,5,2,9,3,12,2,6,3,20,3,7,4,25,2,6,3,13,5,4,38,3,7,4,14,3,9,5,49,2,
%T A186412 6,3,13,5,4,23,4,10,6,5,69,3,7,4,14,3,9,5,27,5,4,15,7,6,87,2,6,3,13,5,
%U A186412 4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123
%N A186412 Sum of all parts in the n-th region of the set of partitions of j, if 1<=n<=A000041(j).
%C A186412 Also triangle read by rows: T(j,k) = sum of all parts in the k-th region of the last section of the set of partitions of j. See Example section. For more information see A135010. - Omar E. Pol, Nov 26 2011
%C A186412 For the definition of "region" see A206437. - _Omar E. Pol_, Aug 19 2013
%F A186412 a(A000041(n)) = A046746(n).
%e A186412 Contribution from Omar E. Pol, Nov 26 2011 (Start):
%e A186412 Written as a triangle:
%e A186412 1;
%e A186412 3;
%e A186412 5;
%e A186412 2,9;
%e A186412 3,12;
%e A186412 2,6,3,20;
%e A186412 3,7,4,25;
%e A186412 2,6,3,13,5,4,38;
%e A186412 3,7,4,14,3,9,5,49;
%e A186412 2,6,3,13,5,4,23,4,10,6,5,69;
%e A186412 3,7,4,14,3,9,5,27,5,4,15,7,6,87;
%e A186412 2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;
%e A186412 (End)
%e A186412 From _Omar E. Pol_, Aug 18 2013: (Start)
%e A186412 Illustration of initial terms (first seven regions):
%e A186412 .                                             _ _ _ _ _
%e A186412 .                                     _ _ _  |_ _ _ _ _|
%e A186412 .                           _ _ _ _  |_ _ _|       |_ _|
%e A186412 .                     _ _  |_ _ _ _|                 |_|
%e A186412 .             _ _ _  |_ _|     |_ _|                 |_|
%e A186412 .       _ _  |_ _ _|             |_|                 |_|
%e A186412 .   _  |_ _|     |_|             |_|                 |_|
%e A186412 .  |_|   |_|     |_|             |_|                 |_|
%e A186412 .
%e A186412 .   1     3       5     2         9       3          12
%e A186412 .
%e A186412 (End)
%t A186412 lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
%t A186412 A186412 = {}; l = {};
%t A186412 For[j = 1, j <= 50, j++,
%t A186412   mx = Max@lex[j][[j]]; AppendTo[l, mx];
%t A186412   For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
%t A186412   AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];
%t A186412   ];
%t A186412 A186412  (* _Robert Price_, Jul 25 2020 *)
%Y A186412 Row sums of triangle A186114 and of triangle A193870.
%Y A186412 Row j has length A187219(j).
%Y A186412 Row sums give A138879.
%Y A186412 Right border gives A046746, j >= 1.
%Y A186412 Records give A046746, j >= 1.
%Y A186412 Partial sums give A182244.
%Y A186412 Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.
%K A186412 nonn,tabf
%O A186412 1,2
%A A186412 _Omar E. Pol_, Aug 12 2011