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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 (list; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Table of n, a(n) for n = 0..500

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-Fran├žois Alcover, Aug 30 2021, after Alois P. Heinz *)

CROSSREFS

Row sums and main diagonal (shifted) of A291684.

Cf. A288910, A288911, A288912.

Sequence in context: A149957 A148393 A148394 * A268571 A001428 A055726

Adjacent sequences:  A291682 A291683 A291684 * A291686 A291687 A291688

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 29 2017

STATUS

approved

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Last modified August 9 21:08 EDT 2022. Contains 356026 sequences. (Running on oeis4.)