A295987 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step patterns 010 or 101, where 1=up and 0=down; triangle T(n,k), n >= 0, k = max(0, n-3), read by rows.

1, 1, 2, 6, 14, 10, 52, 36, 32, 204, 254, 140, 122, 1010, 1368, 1498, 620, 544, 5466, 9704, 9858, 9358, 3164, 2770, 34090, 67908, 90988, 72120, 63786, 18116, 15872, 233026, 545962, 762816, 839678, 560658, 470262, 115356, 101042, 1765836, 4604360, 7458522

Offset: 0

Author

Alois P. Heinz, Dec 01 2017

Keywords

• nonn
• tabf

Examples

			Triangle T(n,k) begins:
:      1;
:      1;
:      2;
:      6;
:     14,     10;
:     52,     36,     32;
:    204,    254,    140,    122;
:   1010,   1368,   1498,    620,    544;
:   5466,   9704,   9858,   9358,   3164,   2770;
:  34090,  67908,  90988,  72120,  63786,  18116,  15872;
: 233026, 545962, 762816, 839678, 560658, 470262, 115356, 101042;
		

Links

• Alois P. Heinz, Rows n = 0..143, flattened

Crossrefs

Column k=0 gives A295974.

Last elements of rows for n>3 give: A001250, A260786, 2*A000111.

Row sums give A000142.

Cf. A227884, A230695, A230797, A231384, A232933, A242783, A242819, A242820, A296054.

Programs

Maple

b:= proc(u, o, t, h) option remember; expand(
           `if`(u+o=0, 1, `if`(t=0, add(b(u-j, j-1, 1$2), j=1..u),
       add(`if`(h=3, x, 1)*b(u-j, o+j-1, [1, 3, 1][t], 2), j=1..u)+
       add(`if`(t=3, x, 1)*b(u+j-1, o-j, 2, [1, 3, 1][h]), j=1..o))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=0..12);

Mathematica

b[u_, o_, t_, h_] := b[u, o, t, h] = Expand[If[u + o == 0, 1, If[t == 0, Sum[b[u - j, j - 1, 1, 1], {j, 1, u}], Sum[If[h == 3, x, 1]*b[u - j, o + j - 1, {1, 3, 1}[[t]], 2], {j, 1, u}] + Sum[If[t == 3, x, 1]*b[u + j - 1, o - j, 2, {1, 3, 1}[[h]]], {j, 1, o}]]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 07 2018, from Maple *)