A118462 Decimal equivalent of binary encoding of partitions into distinct parts.

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

Offset: 0

Author

Franklin T. Adams-Watters, Apr 28 2006

Keywords

• base
• nonn
• tabf
• look

Comments

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

Examples

			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;
  ...
From _Gus Wiseman_, May 21 2024: (Start)
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)
		

Links

• Alois P. Heinz, Rows n = 0..42, flattened
• V. Shevelev, 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), arXiv:0903.1743 [math.NT], 2009. [From _Vladimir Shevelev_, Mar 17 2009]
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

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

Cf. A089633 (first column), A000079 (last in each column). - Franklin T. Adams-Watters, Mar 16 2009

Cf. A246867.

A variation encoding all partitions is A225620.

Row sums are A372888.

A048793 lists binary indices, sum A029931, length A000120.

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

Programs

Maple

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))[]:
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 06 2014

Mathematica

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 *)