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A353427
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Numbers k such that the k-th composition in standard order has all run-lengths > 1.
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7
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0, 3, 7, 10, 15, 31, 36, 42, 43, 58, 63, 87, 122, 127, 136, 147, 170, 171, 175, 228, 234, 235, 250, 255, 292, 295, 343, 351, 471, 484, 490, 491, 506, 511, 528, 547, 586, 591, 676, 682, 683, 687, 698, 703, 904, 915, 938, 939, 943, 983, 996, 1002, 1003, 1018
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The terms and corresponding compositions begin:
0: ()
3: (1,1)
7: (1,1,1)
10: (2,2)
15: (1,1,1,1)
31: (1,1,1,1,1)
36: (3,3)
42: (2,2,2)
43: (2,2,1,1)
58: (1,1,2,2)
63: (1,1,1,1,1,1)
87: (2,2,1,1,1)
122: (1,1,1,2,2)
127: (1,1,1,1,1,1,1)
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !MemberQ[Length/@Split[stc[#]], 1]&]
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CROSSREFS
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The version for parts instead of lengths is A022340, counted by A212804.
These compositions are counted by A114901.
The case of all run-lengths = 2 is A351011.
The case of all run-lengths > 2 is counted by A353400.
A005811 counts runs in binary expansion.
Statistics of standard compositions:
Cf. A044813, A128695, A165413, A240085, A244164, A274174, A318928, A333489, A333755, A353402, A353432.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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