|
|
A353431
|
|
Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.
|
|
9
|
|
|
0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
First differs from A353696 (the consecutive version) in having 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are subsequence but not a consecutive subsequence.
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.
|
|
LINKS
|
|
|
EXAMPLE
|
The initial terms, their binary expansions, and the corresponding standard compositions:
0: 0 ()
1: 1 (1)
2: 10 (2)
4: 100 (3)
8: 1000 (4)
10: 1010 (2,2)
16: 10000 (5)
32: 100000 (6)
43: 101011 (2,2,1,1)
58: 111010 (1,1,2,2)
64: 1000000 (7)
128: 10000000 (8)
256: 100000000 (9)
292: 100100100 (3,3,3)
349: 101011101 (2,2,1,1,2,1)
442: 110111010 (1,2,1,1,2,2)
512: 1000000000 (10)
586: 1001001010 (3,3,2,2)
676: 1010100100 (2,2,3,3)
697: 1010111001 (2,2,1,1,3,1)
|
|
MATHEMATICA
|
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
rorQ[y_]:=Length[y]<=1||MemberQ[Subsets[y], Length/@Split[y]]&& rorQ[Length/@Split[y]];
Select[Range[0, 100], rorQ[stc[#]]&]
|
|
CROSSREFS
|
The non-recursive version for partitions is A325755, counted by A325702.
These compositions are counted by A353391.
A005811 counts runs in binary expansion.
Statistics of standard compositions:
Classes of standard compositions:
Cf. A032020, A044813, A114640, A165413, A181819, A329739, A318928, A325705, A333224, A353427, A353403.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|