A316293 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), k>=0, k<=n<=k*(k+1)/2, read by columns.

1, 1, 2, 1, 5, 8, 5, 1, 16, 50, 79, 69, 34, 9, 1, 65, 314, 872, 1539, 1823, 1494, 856, 339, 89, 14, 1, 326, 2142, 8799, 24818, 50561, 76944, 89546, 80938, 57284, 31771, 13707, 4520, 1103, 188, 20, 1, 1957, 16248, 89273, 355271, 1070455, 2514044, 4705648

Offset: 0

Author

Alois P. Heinz, Jun 28 2018

Keywords

• nonn
• tabf

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.

Examples

			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;
		

Links

• Alois P. Heinz, Columns k = 0..40, flattened

Crossrefs

Row sums give A000142.

Column sums give A316294.

Main diagonal gives A000522.

Cf. A000217, A258829, A291722, A303697, A316292 (same read by rows).

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:
T:= (n, k)-> b(n, 0$2, k) -`if`(k=0, 0, b(n, 0$2, k-1)):
seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..8);

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], {n, k, k(k+1)/2}], {k, 0, 8}] // Flatten (* Jean-François Alcover, Mar 14 2021, after Alois P. Heinz *)