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

0,3

Links

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

Example

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;

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 *)

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.

Sequence in context: A058054 A054588 A084106 * A263691 A160657 A222087

Adjacent sequences: A295984 A295985 A295986 * A295988 A295989 A295990

Keyword

nonn,tabf

Author

Alois P. Heinz, Dec 01 2017

Status

approved