OFFSET
0,4
COMMENTS
A strict composition of n is a finite sequence of distinct positive integers summing to n.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either.
The case starting with an increase (or decrease, it doesn't matter in the enumeration) is counted by A129838.
LINKS
FORMULA
EXAMPLE
The a(1) = 1 through a(7) = 11 compositions:
(1) (2) (3) (4) (5) (6) (7)
(1,2) (1,3) (1,4) (1,5) (1,6)
(2,1) (3,1) (2,3) (2,4) (2,5)
(3,2) (4,2) (3,4)
(4,1) (5,1) (4,3)
(1,3,2) (5,2)
(2,1,3) (6,1)
(2,3,1) (1,4,2)
(3,1,2) (2,1,4)
(2,4,1)
(4,1,2)
MAPLE
g:= proc(u, o) option remember;
`if`(u+o=0, 1, add(g(o-1+j, u-j), j=1..u))
end:
b:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 2, 0), b(n-k, k)+b(n-k, k-1)))
end:
a:= n-> add(b(n, k)*g(k, 0), k=0..floor((sqrt(8*n+1)-1)/2))-1:
seq(a(n), n=0..46); # Alois P. Heinz, Dec 22 2021
MATHEMATICA
wigQ[y_]:=Or[Length[y]==0, Length[Split[y]]==Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&], wigQ]], {n, 0, 15}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 21 2021
STATUS
approved