proposed

approved

editing

proposed

Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337602, A337603, A337604.

allocated for Gus WisemanNumbers 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

1,1

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.

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

Intersection of A014311 and A233564.

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)

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;

Select[Range[0, 100], Length[stc[#]]==3&&UnsameQ@@stc[#]&]

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, A337602, A337603, A337604.

allocated

nonn

Gus Wiseman, Sep 07 2020

approved

editing

allocated for Gus Wiseman

allocated

approved