|
|
A303204
|
|
Number of permutations p of [n] such that 0p has a nonincreasing jump sequence beginning with ceiling(n/2).
|
|
4
|
|
|
1, 1, 1, 2, 5, 12, 36, 98, 327, 988, 3392, 10872, 38795, 129520, 469662, 1609176, 5935728, 20786804, 77416352, 274792342, 1035050705, 3719296036, 14094000938, 51119572738, 195075365778, 712918642042, 2734475097609, 10055531355652, 38747262233793
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.
|
|
LINKS
|
|
|
FORMULA
|
|
|
MAPLE
|
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(sort([u-j, o+j-1])[], j), j=1..min(t, u))+
add(b(sort([u+j-1, o-j])[], j), j=1..min(t, o)))
end:
a:= n-> `if`(n=0, 1, (j-> b(0, n, j)-b(0, n, j-1))(ceil(n/2))):
seq(a(n), n=0..30);
|
|
MATHEMATICA
|
b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
Sum[b[Sequence @@ Sort[{u-j, o+j-1}], j], {j, Min[t, u]}]+
Sum[b[Sequence @@ Sort[{u+j-1, o-j}], j], {j, Min[t, o]}]];
a[n_] := If[n == 0, 1,
Function[j, b[0, n, j] - b[0, n, j-1]][Ceiling[n/2]]];
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|