A242820 Number T(n,k) of permutations of [n] with exactly k occurrences of the consecutive step pattern up, down, down, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows.

1, 1, 2, 6, 24, 116, 4, 672, 48, 4536, 504, 34944, 5376, 302896, 59488, 496, 2916992, 697856, 13952, 30899616, 8720448, 296736, 357080064, 116109312, 5812224, 4470310976, 1645662912, 110697408, 349504, 60269056512, 24776769024, 2114735616, 17730048

Offset: 0

Author

Alois P. Heinz, May 23 2014

Keywords

• nonn
• tabf

Examples

			T(5,1) = 4: (1,5,4,3,2), (2,5,4,3,1), (3,5,4,2,1), (4,5,3,2,1).
Triangle T(n,k) begins:
:  0 :        1;
:  1 :        1;
:  2 :        2;
:  3 :        6;
:  4 :       24;
:  5 :      116,       4;
:  6 :      672,      48;
:  7 :     4536,     504;
:  8 :    34944,    5376;
:  9 :   302896,   59488,    496;
: 10 :  2916992,  697856,  13952;
: 11 : 30899616, 8720448, 296736;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Column k=0 gives A177518.

Row sums give: A000142.

Cf. A242783, A242784, A295987.

Programs

Maple

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, [1, 3, 4, 1][t])*`if`(t=4, x, 1), j=1..u)+
      add(b(u+j-1, o-j, 2), j=1..o)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..15);

Mathematica

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[
     Sum[b[u - j, o + j - 1, {1, 3, 4, 1}[[t]]]*If[t==4, x, 1], {j, 1, u}]+
     Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
T[n_] := CoefficientList[b[n, 0, 1], x];
T /@ Range[0, 15] // Flatten (* Jean-François Alcover, Mar 23 2021, after Alois P. Heinz *)