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%I A118462 #41 May 23 2024 21:14:31
%S A118462 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,
%T A118462 21,24,35,36,66,129,256,15,22,25,37,40,67,68,130,257,512,23,26,38,41,
%U A118462 48,69,72,131,132,258,513,1024,27,28,39,42,49,70,73,80,133,136,259,260,514
%N A118462 Decimal equivalent of binary encoding of partitions into distinct parts.
%C A118462 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.
%C A118462 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
%C A118462 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
%C A118462 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
%H A118462 Alois P. Heinz, <a href="/A118462/b118462.txt">Rows n = 0..42, flattened</a>
%H A118462 V. Shevelev, <a href="http://arXiv.org/abs/0903.1743">A recursion for divisor function over divisors belonging to a prescribed finite sequence of positive integers and a solution of the Lahiri problem for divisor function sigma_x(n)</a>, arXiv:0903.1743 [math.NT], 2009. [From _Vladimir Shevelev_, Mar 17 2009]
%H A118462 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%e A118462 Partition 11 is [4,2], which gives binary 1010 (2^(4-1)+2^(2-1)), or 10, so a(11)=10.
%e A118462 Triangle begins:
%e A118462    0;
%e A118462    1;
%e A118462    2;
%e A118462    3,  4;
%e A118462    5,  8;
%e A118462    6,  9, 16;
%e A118462    7, 10, 17, 32;
%e A118462   11, 12, 18, 33, 64;
%e A118462   13, 19, 20, 34, 65, 128;
%e A118462   14, 21, 24, 35, 36,  66, 129, 256;
%e A118462   15, 22, 25, 37, 40,  67,  68, 130, 257, 512;
%e A118462   ...
%e A118462 From _Gus Wiseman_, May 21 2024: (Start)
%e A118462 The tetrangle of strict partitions (A118457) begins:
%e A118462   (1)  (2)  (2,1)  (3,1)  (3,2)  (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)
%e A118462             (3)    (4)    (4,1)  (4,2)    (4,3)    (5,2,1)  (5,3,1)
%e A118462                           (5)    (5,1)    (5,2)    (5,3)    (5,4)
%e A118462                                  (6)      (6,1)    (6,2)    (6,2,1)
%e A118462                                           (7)      (7,1)    (6,3)
%e A118462                                                    (8)      (7,2)
%e A118462                                                             (8,1)
%e A118462                                                             (9)
%e A118462 (End)
%p A118462 b:= proc(n, i) option remember; `if`(n=0, [0], `if`(i<1, [], [seq(
%p A118462       map(p->p+2^(i-1)*j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))
%p A118462     end:
%p A118462 T:= n-> sort(b(n$2))[]:
%p A118462 seq(T(n), n=0..14);  # _Alois P. Heinz_, Sep 06 2014
%t A118462 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_ *)
%t A118462 Table[Total[2^(#-1)]&/@Select[Reverse[IntegerPartitions[n]],UnsameQ@@#&],{n,0,10}] (* _Gus Wiseman_, May 21 2024 *)
%Y A118462 Cf. A118463, A118457, A000009 (row lengths).
%Y A118462 Cf. A089633 (first column), A000079 (last in each column). - _Franklin T. Adams-Watters_, Mar 16 2009
%Y A118462 Cf. A246867.
%Y A118462 A variation encoding all partitions is A225620.
%Y A118462 Row sums are A372888.
%Y A118462 A048793 lists binary indices, sum A029931, length A000120.
%Y A118462 Cf. A000041, A019565, A029837, A048675, A272020.
%K A118462 base,nonn,tabf,look
%O A118462 0,3
%A A118462 _Franklin T. Adams-Watters_, Apr 28 2006

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