OFFSET
0,3
COMMENTS
Up to sign, same as A124754.
The reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
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.
EXAMPLE
The sequence of nonnegative integers together with the corresponding standard compositions and their reverse-alternating sums begins:
0: () -> 0 15: (1111) -> 0 30: (1112) -> 1
1: (1) -> 1 16: (5) -> 5 31: (11111) -> 1
2: (2) -> 2 17: (41) -> -3 32: (6) -> 6
3: (11) -> 0 18: (32) -> -1 33: (51) -> -4
4: (3) -> 3 19: (311) -> 3 34: (42) -> -2
5: (21) -> -1 20: (23) -> 1 35: (411) -> 4
6: (12) -> 1 21: (221) -> 1 36: (33) -> 0
7: (111) -> 1 22: (212) -> 3 37: (321) -> 2
8: (4) -> 4 23: (2111) -> -1 38: (312) -> 4
9: (31) -> -2 24: (14) -> 3 39: (3111) -> -2
10: (22) -> 0 25: (131) -> -1 40: (24) -> 2
11: (211) -> 2 26: (122) -> 1 41: (231) -> 0
12: (13) -> 2 27: (1211) -> 1 42: (222) -> 2
13: (121) -> 0 28: (113) -> 3 43: (2211) -> 0
14: (112) -> 2 29: (1121) -> -1 44: (213) -> 4
Triangle begins (row lengths A011782):
0
1
2 0
3 -1 1 1
4 -2 0 2 2 0 2 0
5 -3 -1 3 1 1 3 -1 3 -1 1 1 3 -1 1 1
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]]
Table[sats[stc[n]], {n, 0, 100}]
CROSSREFS
Up to sign, same as the reverse version A124754.
The version for Heinz numbers of partitions is A344616.
Positions of zeros are A344619.
A116406 counts compositions with alternating sum >= 0.
A344610 counts partitions by sum and positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
All of the following pertain to compositions in standard order:
- The length is A000120.
- Converting to reversed ranking gives A059893.
- The rows are A066099.
- The sum is A070939.
- The runs are counted by A124767.
- The reversed version is A228351.
- Strict compositions are ranked by A233564.
- Constant compositions are ranked by A272919.
- The Heinz number is A333219.
- Anti-run compositions are ranked by A333489.
KEYWORD
sign,tabf
AUTHOR
Gus Wiseman, Jun 03 2021
STATUS
approved