A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.

37, 38, 41, 44, 50, 52, 69, 70, 81, 88, 98, 104, 133, 134, 137, 140, 145, 152, 161, 176, 194, 196, 200, 208, 261, 262, 265, 268, 274, 276, 289, 290, 296, 304, 321, 324, 328, 352, 386, 388, 400, 416, 517, 518, 521, 524, 529, 530, 532, 536, 545, 560, 577, 578

Offset: 1

Author

Gus Wiseman, Sep 07 2020

Keywords

• nonn

Comments

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

Examples

			The sequence together with the corresponding triples begins:
     37: (3,2,1)    140: (4,1,3)    289: (3,5,1)
     38: (3,1,2)    145: (3,4,1)    290: (3,4,2)
     41: (2,3,1)    152: (3,1,4)    296: (3,2,4)
     44: (2,1,3)    161: (2,5,1)    304: (3,1,5)
     50: (1,3,2)    176: (2,1,5)    321: (2,6,1)
     52: (1,2,3)    194: (1,5,2)    324: (2,4,3)
     69: (4,2,1)    196: (1,4,3)    328: (2,3,4)
     70: (4,1,2)    200: (1,3,4)    352: (2,1,6)
     81: (2,4,1)    208: (1,2,5)    386: (1,6,2)
     88: (2,1,4)    261: (6,2,1)    388: (1,5,3)
     98: (1,4,2)    262: (6,1,2)    400: (1,3,5)
    104: (1,2,4)    265: (5,3,1)    416: (1,2,6)
    133: (5,2,1)    268: (5,1,3)    517: (7,2,1)
    134: (5,1,2)    274: (4,3,2)    518: (7,1,2)
    137: (4,3,1)    276: (4,2,3)    521: (6,3,1)
		

Crossrefs

6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.

A007304 is an unordered version.

A014311 is the non-strict version.

A337461 counts the coprime case.

A000217(n - 2) counts 3-part compositions.

A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.

A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.

A014612 ranks 3-part partitions.

Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.

Programs

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Length[stc[#]]==3&&UnsameQ@@stc[#]&]

Formula

These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).

Intersection of A014311 and A233564.