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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. 3
1, 1, 3, 19, 258, 7406, 442668, 54371100, 13585980916, 6859762797636, 6969135518632452, 14209819222900305044, 58061006907633910998660, 474996314819118381967232244, 7776635831062534849079443379908, 254723669580125156112963535996038036 (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..40

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-Fran├žois Alcover, Sep 01 2021, after Alois P. Heinz *)

CROSSREFS

Column sums of A316292 or A316293.

Sequence in context: A261495 A069344 A305562 * A233240 A173799 A003011

Adjacent sequences:  A316291 A316292 A316293 * A316295 A316296 A316297

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jun 28 2018

STATUS

approved

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Last modified August 15 02:47 EDT 2022. Contains 356122 sequences. (Running on oeis4.)