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A182181
Total number of parts in the section model of partitions of A135010 with n regions.
10
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
1,2
FORMULA
a(A000041(n)) = A006128(n), n >= 1.
a(A000041(n)) = A182727(A000041(n). - Omar E. Pol, May 24 2012
EXAMPLE
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)
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 *)
CROSSREFS
Partial sums of A194446.
Row j has length A187219(j).
Right border gives A006128.
For the definition of "region" see A206437.
Sequence in context: A038591 A333794 A349214 * A138038 A095029 A028792
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Apr 23 2012
STATUS
approved