A295974 -id:A295974 - cp's OEIS frontend

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

Alois P. Heinz, Dec 01 2017

			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;

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

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.

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);

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 *)

A296054 Number of permutations of [n] with an equal number of occurrences of the consecutive step patterns 010 and 101, where 1=up and 0=down.

Original entry on oeis.org

1, 1, 2, 6, 14, 84, 344, 2714, 15850, 158664, 1254536, 15066332, 151035364, 2091499222, 25438986270, 398610118170, 5712650790562, 99963273184972, 1649446030193764, 31890910904182000, 594935710367215600, 12604160312187654888, 262094375885982582488

Offset: 0

Alois P. Heinz, Dec 03 2017

nonn

a(n) * sqrt(n) / n! tends to 1.064409... - Vaclav Kotesovec, Aug 30 2021

			a(4) = 14: 1234, 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3214, 3421, 4123, 4213, 4312, 4321 with step patterns 111, 110, 110, 100, 011, 110, 100, 011, 001, 100, 011, 001, 001, 000, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A295974, A295987.

b:= proc(u, o, t, h, c) option remember; `if`(2*abs(c)-1>u+o, 0,
            `if`(u+o=0, 1, `if`(t=0, add(b(u-j, j-1, 1$2, 0), j=1..u),
       add(b(u-j, o+j-1, [1, 3, 1][t], 2, c+`if`(h=3, 1, 0)), j=1..u)+
       add(b(u+j-1, o-j, 2, [1, 3, 1][h], c-`if`(t=3, 1, 0)), j=1..o))))
    end:
a:= n-> b(n, 0$4):
seq(a(n), n=0..30);

b[u_, o_, t_, h_, c_] := b[u, o, t, h, c] = If[2 Abs[c] - 1 > u+o, 0,
     If[u+o == 0, 1, If[t == 0, Sum[b[u-j, j-1, 1, 1, 0], {j, u}],
     Sum[b[u-j, o+j-1, {1, 3, 1}[[t]], 2, c+If[h == 3, 1, 0]], {j, u}]+
     Sum[b[u+j-1, o-j, 2, {1, 3, 1}[[h]], c-If[t == 3, 1, 0]], {j, o}]]]];
a[n_] := b[n, 0, 0, 0, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

cp's OEIS Frontend

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.

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