A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.

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

Offset: 1

Author

Gus Wiseman, May 16 2022

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 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)
		

Crossrefs

The version for partitions is A001694, counted by A007690.

The version for parts instead of lengths is A022340, counted by A212804.

These compositions are counted by A114901.

A subset of A348612 (counted by A261983).

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.

A011782 counts compositions.

A066099 lists compositions in standard order, reverse A228351.

Statistics of standard compositions:

- Length is A000120, sum A070939.

- Runs are counted by A124767.

- Runs-resistance is A333628.

- Run-lengths are A333769.

Cf. A044813, A128695, A165413, A240085, A244164, A274174, A318928, A333489, A333755, A353402, A353432.

Programs

Mathematica

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