A333217 Numbers k such that the k-th composition in standard order covers an initial interval of positive integers.

0, 1, 3, 5, 6, 7, 11, 13, 14, 15, 21, 22, 23, 26, 27, 29, 30, 31, 37, 38, 41, 43, 44, 45, 46, 47, 50, 52, 53, 54, 55, 58, 59, 61, 62, 63, 75, 77, 78, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101, 102, 105, 106, 107, 108, 109, 110, 111, 114, 116, 117, 118

Offset: 1

Author

Gus Wiseman, Mar 15 2020

Keywords

• nonn

Comments

The k-th composition in standard order (row k of 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.

Examples

			The sequence of terms together with the corresponding compositions begins:
    0: ()              37: (3,2,1)           75: (3,2,1,1)
    1: (1)             38: (3,1,2)           77: (3,1,2,1)
    3: (1,1)           41: (2,3,1)           78: (3,1,1,2)
    5: (2,1)           43: (2,2,1,1)         83: (2,3,1,1)
    6: (1,2)           44: (2,1,3)           85: (2,2,2,1)
    7: (1,1,1)         45: (2,1,2,1)         86: (2,2,1,2)
   11: (2,1,1)         46: (2,1,1,2)         87: (2,2,1,1,1)
   13: (1,2,1)         47: (2,1,1,1,1)       89: (2,1,3,1)
   14: (1,1,2)         50: (1,3,2)           90: (2,1,2,2)
   15: (1,1,1,1)       52: (1,2,3)           91: (2,1,2,1,1)
   21: (2,2,1)         53: (1,2,2,1)         92: (2,1,1,3)
   22: (2,1,2)         54: (1,2,1,2)         93: (2,1,1,2,1)
   23: (2,1,1,1)       55: (1,2,1,1,1)       94: (2,1,1,1,2)
   26: (1,2,2)         58: (1,1,2,2)         95: (2,1,1,1,1,1)
   27: (1,2,1,1)       59: (1,1,2,1,1)      101: (1,3,2,1)
   29: (1,1,2,1)       61: (1,1,1,2,1)      102: (1,3,1,2)
   30: (1,1,1,2)       62: (1,1,1,1,2)      105: (1,2,3,1)
   31: (1,1,1,1,1)     63: (1,1,1,1,1,1)    106: (1,2,2,2)
		

Links

• Robert Price, Table of n, a(n) for n = 1..1008

Crossrefs

Sequences covering an initial interval are counted by A000670.

Composition in standard order are A066099.

The case of strictly increasing initial intervals is A164894.

The case of strictly decreasing initial intervals is A246534.

The case of permutations is A333218.

The weakly increasing version is A333379.

The weakly decreasing version is A333380.

Cf. A000120, A029931, A048793, A070939, A225620, A228351, A233564, A272919, A333219, A333220.

Programs

Mathematica

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],normQ[stc[#]]&]