|
|
A334966
|
|
Numbers k such that the k-th composition in standard order (row k of A066099) has weakly decreasing non-adjacent parts.
|
|
3
|
|
|
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 47, 48, 49, 51, 55, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
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.
The complement starts: 14, 26, 28, 29, 30, 44, 46, 50, ...
|
|
LINKS
|
|
|
EXAMPLE
|
The sequence together with the corresponding compositions begins:
0: () 17: (4,1) 37: (3,2,1)
1: (1) 18: (3,2) 38: (3,1,2)
2: (2) 19: (3,1,1) 39: (3,1,1,1)
3: (1,1) 20: (2,3) 40: (2,4)
4: (3) 21: (2,2,1) 41: (2,3,1)
5: (2,1) 22: (2,1,2) 42: (2,2,2)
6: (1,2) 23: (2,1,1,1) 43: (2,2,1,1)
7: (1,1,1) 24: (1,4) 45: (2,1,2,1)
8: (4) 25: (1,3,1) 47: (2,1,1,1,1)
9: (3,1) 27: (1,2,1,1) 48: (1,5)
10: (2,2) 31: (1,1,1,1,1) 49: (1,4,1)
11: (2,1,1) 32: (6) 51: (1,3,1,1)
12: (1,3) 33: (5,1) 55: (1,2,1,1,1)
13: (1,2,1) 34: (4,2) 63: (1,1,1,1,1,1)
15: (1,1,1,1) 35: (4,1,1) 64: (7)
16: (5) 36: (3,3) 65: (6,1)
For example, (2,3,1,2) is such a composition because the non-adjacent pairs are (2,1), (2,2), (3,2), all of which are weakly decreasing, so 166 is in the sequence
|
|
MATHEMATICA
|
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], !MatchQ[stc[#], {___, x_, __, y_, ___}/; y>x]&]
|
|
CROSSREFS
|
The case of normal sequences appears to be A028859.
A version for ordered set partitions is A332872.
These compositions are enumerated by A333148.
The strict case is enumerated by A333150.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|