%I #41 May 23 2024 21:14:31

%S 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 21,24,35,36,66,129,256,15,22,25,37,40,67,68,130,257,512,23,26,38,41,

%U 48,69,72,131,132,258,513,1024,27,28,39,42,49,70,73,80,133,136,259,260,514

%N Decimal equivalent of binary encoding of partitions into distinct parts.

%C 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 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 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 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 Alois P. Heinz, <a href="/A118462/b118462.txt">Rows n = 0..42, flattened</a>

%H 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%e Partition 11 is [4,2], which gives binary 1010 (2^(4-1)+2^(2-1)), or 10, so a(11)=10.

%e Triangle begins:

%e 0;

%e 1;

%e 2;

%e 3, 4;

%e 5, 8;

%e 6, 9, 16;

%e 7, 10, 17, 32;

%e 11, 12, 18, 33, 64;

%e 13, 19, 20, 34, 65, 128;

%e 14, 21, 24, 35, 36, 66, 129, 256;

%e 15, 22, 25, 37, 40, 67, 68, 130, 257, 512;

%e ...

%e From _Gus Wiseman_, May 21 2024: (Start)

%e The tetrangle of strict partitions (A118457) begins:

%e (1) (2) (2,1) (3,1) (3,2) (3,2,1) (4,2,1) (4,3,1) (4,3,2)

%e (3) (4) (4,1) (4,2) (4,3) (5,2,1) (5,3,1)

%e (5) (5,1) (5,2) (5,3) (5,4)

%e (6) (6,1) (6,2) (6,2,1)

%e (7) (7,1) (6,3)

%e (8) (7,2)

%e (8,1)

%e (9)

%e (End)

%p b:= proc(n, i) option remember; `if`(n=0, [0], `if`(i<1, [], [seq(

%p map(p->p+2^(i-1)*j, b(n-i*j, i-1))[], j=0..min(1, n/i))]))

%p end:

%p T:= n-> sort(b(n$2))[]:

%p seq(T(n), n=0..14); # _Alois P. Heinz_, Sep 06 2014

%t 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 Table[Total[2^(#-1)]&/@Select[Reverse[IntegerPartitions[n]],UnsameQ@@#&],{n,0,10}] (* _Gus Wiseman_, May 21 2024 *)

%Y Cf. A118463, A118457, A000009 (row lengths).

%Y Cf. A089633 (first column), A000079 (last in each column). - _Franklin T. Adams-Watters_, Mar 16 2009

%Y Cf. A246867.

%Y A variation encoding all partitions is A225620.

%Y Row sums are A372888.

%Y A048793 lists binary indices, sum A029931, length A000120.

%Y Cf. A000041, A019565, A029837, A048675, A272020.

%K base,nonn,tabf,look

%O 0,3

%A _Franklin T. Adams-Watters_, Apr 28 2006