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A291685
Number of permutations p of [n] such that 0p has a nonincreasing jump sequence.
5
1, 1, 2, 5, 16, 52, 189, 683, 2621, 10061, 40031, 159201, 650880, 2657089, 11062682, 46065143, 194595138, 822215099, 3513875245, 15021070567, 64785349064, 279575206629, 1214958544538, 5283266426743, 23106210465665, 101120747493793, 444614706427665
OFFSET
0,3
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
EXAMPLE
a(3) = 5 = 6 - 1 counts all permutations of {1,2,3} except 132 with jump sequence 1, 2, 1.
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
add(b(u-j, o+j-1, j), j=1..min(t, u))+
add(b(u+j-1, o-j, j), j=1..min(t, o)))
end:
a:= n-> b(0, n$2):
seq(a(n), n=0..30);
MATHEMATICA
b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1,
Sum[b[u-j, o+j-1, j], {j, Min[t, u]}]+
Sum[b[u+j-1, o-j, j], {j, Min[t, o]}]];
a[n_] := b[0, n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)
CROSSREFS
Row sums and main diagonal (shifted) of A291684.
Sequence in context: A149957 A148393 A148394 * A268571 A001428 A055726
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Aug 29 2017
STATUS
approved