A186412 - OEIS

%I #41 Jul 25 2020 12:15:55

%S 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 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 4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123

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

%C 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 For the definition of "region" see A206437. - _Omar E. Pol_, Aug 19 2013

%F a(A000041(n)) = A046746(n).

%e Contribution from Omar E. Pol, Nov 26 2011 (Start):

%e Written as a triangle:

%e 1;

%e 3;

%e 5;

%e 2,9;

%e 3,12;

%e 2,6,3,20;

%e 3,7,4,25;

%e 2,6,3,13,5,4,38;

%e 3,7,4,14,3,9,5,49;

%e 2,6,3,13,5,4,23,4,10,6,5,69;

%e 3,7,4,14,3,9,5,27,5,4,15,7,6,87;

%e 2,6,3,13,5,4,23,4,10,6,5,39,3,9,5,19,4,12,7,6,123;

%e (End)

%e From _Omar E. Pol_, Aug 18 2013: (Start)

%e Illustration of initial terms (first seven regions):

%e .                                             _ _ _ _ _

%e .                                     _ _ _  |_ _ _ _ _|

%e .                           _ _ _ _  |_ _ _|       |_ _|

%e .                     _ _  |_ _ _ _|                 |_|

%e .             _ _ _  |_ _|     |_ _|                 |_|

%e .       _ _  |_ _ _|             |_|                 |_|

%e .   _  |_ _|     |_|             |_|                 |_|

%e .  |_|   |_|     |_|             |_|                 |_|

%e .

%e .   1     3       5     2         9       3          12

%e .

%e (End)

%t lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];

%t A186412 = {}; l = {};

%t For[j = 1, j <= 50, j++,

%t   mx = Max@lex[j][[j]]; AppendTo[l, mx];

%t   For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];

%t   AppendTo[A186412, Total@Take[Reverse[First /@ lex[mx]], j - i]];

%t   ];

%t A186412  (* _Robert Price_, Jul 25 2020 *)

%Y Row sums of triangle A186114 and of triangle A193870.

%Y Row j has length A187219(j).

%Y Row sums give A138879.

%Y Right border gives A046746, j >= 1.

%Y Records give A046746, j >= 1.

%Y Partial sums give A182244.

%Y Cf. A000041, A002865, A066186, A135010, A138121, A138879, A194436, A194437, A194438, A194439, A194446, A194447.

%K nonn,tabf

%O 1,2

%A _Omar E. Pol_, Aug 12 2011