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.

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

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-François 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