A182181 Total number of parts in the section model of partitions of A135010 with n regions.

1, 3, 6, 7, 12, 13, 20, 21, 23, 24, 35, 36, 38, 39, 54, 55, 57, 58, 62, 63, 64, 86, 87, 89, 90, 94, 95, 97, 98, 128, 129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192, 193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275

Offset: 1

Author

Omar E. Pol, Apr 23 2012

Keywords

• nonn
• tabf

Examples

			The first four regions of the section model of partitions are [1],[2, 1],[3, 1, 1],[2]. We can see that there are seven parts so a(4) = 7.
Written as a triangle begins:
    1;
    3;
    6;
    7,  12;
   13,  20;
   21,  23,  24,  35;
   36,  38,  39,  54;
   55,  57,  58,  62,  63,  64,  86;
   87,  89,  90,  94,  95,  97,  98, 128;
  129, 131, 132, 136, 137, 138, 145, 146, 148, 149, 150, 192;
  193, 195, 196, 200, 201, 203, 204, 212, 213, 214, 217, 218, 219, 275;
  ...
From _Omar E. Pol_, Oct 20 2014: (Start)
Illustration of initial terms:
.                                                _ _ _ _ _
.                                      _ _ _    |_ _ _    |
.                            _ _ _ _  |_ _ _|_  |_ _ _|_  |
.                    _ _    |_ _    | |_ _    | |_ _    | |
.            _ _ _  |_ _|_  |_ _|_  | |_ _|_  | |_ _|_  | |
.      _ _  |_ _  | |_ _  | |_ _  | | |_ _  | | |_ _  | | |
.  _  |_  | |_  | | |_  | | |_  | | | |_  | | | |_  | | | |
. |_| |_|_| |_|_|_| |_|_|_| |_|_|_|_| |_|_|_|_| |_|_|_|_|_|
.
.  1    3      6       7        12        13         20
.
.                                          _ _ _ _ _ _
.                             _ _ _       |_ _ _      |
.                _ _ _ _     |_ _ _|_     |_ _ _|_    |
.   _ _         |_ _    |    |_ _    |    |_ _    |   |
.  |_ _|_ _ _   |_ _|_ _|_   |_ _|_ _|_   |_ _|_ _|_  |
.  |_ _ _    |  |_ _ _    |  |_ _ _    |  |_ _ _    | |
.  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  |  |_ _ _|_  | |
.  |_ _    | |  |_ _    | |  |_ _    | |  |_ _    | | |
.  |_ _|_  | |  |_ _|_  | |  |_ _|_  | |  |_ _|_  | | |
.  |_ _  | | |  |_ _  | | |  |_ _  | | |  |_ _  | | | |
.  |_  | | | |  |_  | | | |  |_  | | | |  |_  | | | | |
.  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|  |_|_|_|_|_|_|
.
.       21           23           24            35
(End)
		

Links

• Robert Price, Table of n, a(n) for n = 1..204226, rows 1-50.
• Omar E. Pol, Illustration of the seven regions of 5

Crossrefs

Partial sums of A194446.

Row j has length A187219(j).

Right border gives A006128.

For the definition of "region" see A206437.

Cf. A000041, A135010, A138121, A141285, A182244, A182276, A182727, A186114, A210990.

Programs

Mathematica

lex[n_]:=DeleteCases[Sort@PadRight[Reverse /@ IntegerPartitions@n], x_ /; x==0,2];
reg = {}; l = {};
For[j = 1, j <= 56, j++,
  mx = Max@lex[j][[j]]; AppendTo[l, mx];
  For[i = j, i > 0, i--, If[l[[i]] > mx, Break[]]];
  AppendTo[reg, j - i];
  ];
Accumulate@reg  (* Robert Price, Apr 22 2020, revised Jul 25 2020 *)

Formula

a(A000041(n)) = A006128(n), n >= 1.

a(A000041(n)) = A182727(A000041(n)). - Omar E. Pol, May 24 2012