OFFSET
0,3
COMMENTS
Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). Then a(n) is the number of integer compositions whose run-sums constitute the n-th composition in standard order (graded reverse-lexicographic, A066099).
EXAMPLE
The terms 2^(n - 1) through 2^n - 1 sum to 2^n. As a triangle:
1
1
2 0
2 1 1 0
3 1 2 0 1 1 0 0
2 2 3 0 3 1 1 0 2 1 1 0 0 0 0 0
The a(n) compositions for selected n:
n=1: n=2: n=8: n=32: n=68: n=130:
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(1) (2) (4) (6) (4,3) (6,2)
(1,1) (2,2) (3,3) (2,2,3) (3,3,2)
(1,1,1,1) (2,2,2) (4,1,1,1) (6,1,1)
(1,1,1,1,1,1) (1,1,1,1,3) (3,3,1,1)
(2,2,1,1,1) (2,2,2,1,1)
(1,1,1,1,1,1,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
antirunQ[y_]:=Length[Split[y]]==Length[y];
Table[Length[Select[Tuples[Divisors/@stc[n]], antirunQ]], {n, 0, 30}]
CROSSREFS
First column is 1 followed by A000005.
Row-sums are A011782.
Standard compositions are listed by A066099.
Positions of 0's are A354904.
Positions of first appearances are A354905.
A005811 counts runs in binary expansion.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jun 11 2022
STATUS
approved