A316294 Total number of permutations p of [k] such that n is the maximum of the partial sums of the signed up-down jump sequence of 0,p summed over all k >= 0.

1, 1, 3, 19, 258, 7406, 442668, 54371100, 13585980916, 6859762797636, 6969135518632452, 14209819222900305044, 58061006907633910998660, 474996314819118381967232244, 7776635831062534849079443379908, 254723669580125156112963535996038036

Offset: 0

Author

Alois P. Heinz, Jun 28 2018

Keywords

• nonn

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..40

Crossrefs

Column sums of A316292 or A316293.

Programs

Maple

b:= proc(u, o, c, k) option remember;
      `if`(c>k, 0, `if`(u+o=0, 1,
       add(b(u-j, o-1+j, c+j, k), j=1..u)+
       add(b(u+j-1, o-j, c-j, k), j=1..o)))
    end:
a:= n-> add(b(k, 0$2, n)-b(k, 0$2, n-1), k=n..n*(n+1)/2):
seq(a(n), n=0..15);

Mathematica

b[u_, o_, c_, k_] := b[u, o, c, k] =
     If[c > k, 0, If[u + o == 0, 1,
     Sum[b[u - j, o - 1 + j, c + j, k], {j, u}] +
     Sum[b[u + j - 1, o - j, c - j, k], {j, o}]]];
a[n_] := Sum[b[k, 0, 0, n] - b[k, 0, 0, n-1], {k, n, n(n+1)/2}];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Sep 01 2021, after Alois P. Heinz *)