A374249 Numbers k such that the k-th composition in standard order has its equal parts contiguous.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 26, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 50, 52, 56, 58, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 78, 79, 80, 81, 83, 84, 85

Offset: 1

Author

Gus Wiseman, Jul 13 2024

Keywords

• nonn

Comments

These are compositions avoiding the patterns (1,2,1) and (2,1,2).

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 together with their standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   5: (2,1)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  11: (2,1,1)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  16: (5)
See A374253 for the complement: 13, 22, 25, 27, 29, ...
		

Links

• Gus Wiseman, Statistics, classes, and transformations of standard compositions

Crossrefs

The strict (also anti-run) case is A233564, counted by A032020.

Compositions of this type are counted by A274174.

Permutations of prime indices of this type are counted by A333175.

The complement is A374253 (anti-run A374254), counted by A335548.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

A066099 lists compositions in standard order.

A124767 counts runs in standard compositions, anti-runs A333381.

A333755 counts compositions by number of runs.

A335454 counts patterns matched by standard compositions.

A335462 counts (1,2,1)- and (2,1,2)-matching permutations of prime indices.

- A335470 counts (1,2,1)-matching compositions, ranks A335466.

- A335471 counts (1,2,1)-avoiding compositions, ranks A335467.

- A335472 counts (2,1,2)-matching compositions, ranks A335468.

- A335473 counts (2,1,2)-avoiding compositions, ranks A335469.

Cf. A106356, A124762, A238130, A238279, A261982, A272919, A333382, A335450, A335460, A335524, A335525.

Programs

Mathematica

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

Formula

Equals A335467 /\ A335469.