A373955 Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.

3, 11, 14, 19, 27, 28, 29, 35, 46, 51, 56, 57, 67, 75, 78, 83, 91, 92, 93, 99, 110, 112, 113, 114, 116, 118, 131, 139, 142, 155, 156, 157, 163, 179, 184, 185, 195, 203, 206, 211, 219, 220, 221, 224, 225, 226, 229, 230, 232, 233, 236, 237, 259, 267, 270, 275

Offset: 1

Author

Gus Wiseman, Jun 29 2024

Keywords

• nonn

Comments

Also numbers k such that the excess compression of the k-th integer composition in standard order is 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.

postn of 1 in

Examples

			The terms and corresponding compositions begin:
    3: (1,1)
   11: (2,1,1)
   14: (1,1,2)
   19: (3,1,1)
   27: (1,2,1,1)
   28: (1,1,3)
   29: (1,1,2,1)
   35: (4,1,1)
   46: (2,1,1,2)
   51: (1,3,1,1)
   56: (1,1,4)
   57: (1,1,3,1)
   67: (5,1,1)
   75: (3,2,1,1)
   78: (3,1,1,2)
   83: (2,3,1,1)
   91: (2,1,2,1,1)
   92: (2,1,1,3)
   93: (2,1,1,2,1)
   99: (1,4,1,1)
		

Links

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

Crossrefs

These compositions are counted by A373950.

Positions of ones in A373954.

A003242 counts compressed compositions (or anti-runs).

A114901 counts compositions with no isolated parts.

A116861 counts partitions by compressed sum, by compressed length A116608.

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

A240085 counts compositions with no unique parts.

A333755 counts compositions by compressed length.

A373948 encodes compression using compositions in standard order.

A373949 counts compositions by compression-sum.

A373953 gives compression-sum of standard compositions.

Cf. A106356, A124762, A238130, A238279, A238343, A285981, A333382, A333489, A373951, A373952.

Programs

Mathematica

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