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A316292 Number T(n,k) of permutations p of [n] such that k is the maximum of the partial sums of the signed up-down jump sequence of 0,p; triangle T(n,k), n>=0, ceiling((sqrt(1+8*n)-1)/2)<=k<=n, read by rows. 6
1, 1, 2, 1, 5, 8, 16, 5, 50, 65, 1, 79, 314, 326, 69, 872, 2142, 1957, 34, 1539, 8799, 16248, 13700, 9, 1823, 24818, 89273, 137356, 109601, 1, 1494, 50561, 355271, 947713, 1287350, 986410, 856, 76944, 1070455, 4923428, 10699558, 13281458, 9864101 (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, Rows n = 0..120, flattened

EXAMPLE

Triangle T(n,k) begins:

: 1;

:    1;

:       2;

:       1, 5;

:          8, 16;

:          5, 50,   65;

:          1, 79,  314,   326;

:             69,  872,  2142,   1957;

:             34, 1539,  8799,  16248,  13700;

:              9, 1823, 24818,  89273, 137356,  109601;

:              1, 1494, 50561, 355271, 947713, 1287350, 986410;

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:

T:= (n, k)-> b(n, 0$2, k) -`if`(k=0, 0, b(n, 0$2, k-1)):

seq(seq(T(n, k), k=ceil((sqrt(1+8*n)-1)/2)..n), n=0..14);

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, 1, u}] +

     Sum[b[u + j - 1, o - j, c - j, k], {j, 1, o}]]];

T[n_, k_] := b[n, 0, 0, k] - If[k == 0, 0, b[n, 0, 0, k - 1]];

Table[Table[T[n, k], {k, Ceiling[(Sqrt[8n+1]-1)/2], n}], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Feb 27 2021, after Alois P. Heinz *)

CROSSREFS

Row sums give A000142.

Column sums give A316294.

Main diagonal gives A000522.

Cf. A002024, A123578, A258829, A291722, A303697, A316293 (same read by columns).

Sequence in context: A193180 A201743 A167816 * A222542 A318052 A026078

Adjacent sequences:  A316289 A316290 A316291 * A316293 A316294 A316295

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Jun 28 2018

STATUS

approved

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Last modified October 2 23:51 EDT 2022. Contains 357230 sequences. (Running on oeis4.)