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A118462
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Decimal equivalent of binary encoding of partitions into distinct parts.
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10
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0, 1, 2, 3, 4, 5, 8, 6, 9, 16, 7, 10, 17, 32, 11, 12, 18, 33, 64, 13, 19, 20, 34, 65, 128, 14, 21, 24, 35, 36, 66, 129, 256, 15, 22, 25, 37, 40, 67, 68, 130, 257, 512, 23, 26, 38, 41, 48, 69, 72, 131, 132, 258, 513, 1024, 27, 28, 39, 42, 49, 70, 73, 80, 133, 136, 259, 260, 514
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OFFSET
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0,3
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COMMENTS
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A part of size k in the partition makes the 2^(k-1) bit of the number be 1. The partitions of n are in reverse Mathematica ordering, so that each row is in ascending order. This is a permutation of the nonnegative integers.
The sequence is the concatenation of the sets: e_n={j>=0: A029931(j)=n}, n=0,1,...: e_0={0}, e_1={1}, e_2={2}, e_3={3,4}, e_4={5,8}, e_5={6,9,16}, e_6={7,10,17,32}, e_7={11,12,18.33.64}, ... . - Vladimir Shevelev, Mar 16 2009
This permutation of the nonnegative integers A001477 has fixed points 0, 1, 2, 3, 4, 5, 325, 562, 800, 4449, ... and inverse permutation A118463. - Alois P. Heinz, Sep 06 2014
Row n lists in increasing order the binary ranks of all strict integer partitions of n, where the binary rank of a partition y is given by Sum_i 2^(y_i-1). - Gus Wiseman, May 21 2024
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LINKS
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EXAMPLE
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Partition 11 is [4,2], which gives binary 1010 (2^(4-1)+2^(2-1)), or 10, so a(11)=10.
Triangle begins:
0;
1;
2;
3, 4;
5, 8;
6, 9, 16;
7, 10, 17, 32;
11, 12, 18, 33, 64;
13, 19, 20, 34, 65, 128;
14, 21, 24, 35, 36, 66, 129, 256;
15, 22, 25, 37, 40, 67, 68, 130, 257, 512;
...
The tetrangle of strict partitions (A118457) begins:
(1) (2) (2,1) (3,1) (3,2) (3,2,1) (4,2,1) (4,3,1) (4,3,2)
(3) (4) (4,1) (4,2) (4,3) (5,2,1) (5,3,1)
(5) (5,1) (5,2) (5,3) (5,4)
(6) (6,1) (6,2) (6,2,1)
(7) (7,1) (6,3)
(8) (7,2)
(8,1)
(9)
(End)
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MAPLE
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b:= proc(n, i) option remember; `if`(n=0, [0], `if`(i<1, [], [seq(
map(p->p+2^(i-1)*j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
end:
T:= n-> sort(b(n$2))[]:
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[n==0, {0}, If[i<1, {}, Flatten[Table[b[n-i*j, i-1 ] + 2^(i-1)*j, {j, 0, Min[1, n/i]}]]]]; T[n_] := Sort[b[n, n]]; Table[ T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 27 2015, after Alois P. Heinz *)
Table[Total[2^(#-1)]&/@Select[Reverse[IntegerPartitions[n]], UnsameQ@@#&], {n, 0, 10}] (* Gus Wiseman, May 21 2024 *)
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CROSSREFS
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A variation encoding all partitions is A225620.
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KEYWORD
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AUTHOR
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STATUS
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approved
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