OFFSET
1,2
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 reverse-alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.
EXAMPLE
The terms and corresponding compositions begin:
0: ()
9: (3,1)
130: (6,2)
135: (5,1,1,1)
141: (4,1,2,1)
153: (3,1,3,1)
177: (2,1,4,1)
193: (1,6,1)
225: (1,1,5,1)
2052: (9,3)
2059: (8,2,1,1)
2062: (8,1,1,2)
2069: (7,2,2,1)
2074: (7,1,2,2)
2079: (7,1,1,1,1,1)
2089: (6,2,3,1)
2098: (6,1,3,2)
2103: (6,1,2,1,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[ Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Select[Range[0, 1000], Total[stc[#]]==-2*sats[stc[#]]&]
CROSSREFS
These compositions are counted by A224274 up to 0's.
The unreversed version is A349154.
A003242 counts Carlitz compositions.
A011782 counts compositions.
Cf. A000070, A000346, A001250, A001700, A008549, A027306, A058622, A088218, A114121, A120452, A262977, A294175, A345917.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Heinz number is given by A333219.
Classes of standard compositions:
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 22 2021
STATUS
approved