A227884 -id:A227884 - cp's OEIS frontend

A242783 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=up and 0=down; triangle T(n,k), n>=0, read by rows.

1, 1, 2, 5, 1, 21, 3, 70, 50, 450, 270, 4326, 602, 99, 12, 1, 34944, 5376, 209863, 139714, 13303, 1573632, 1366016, 530432, 158720, 21824925, 15302031, 2715243, 74601, 302273664, 161855232, 14872704, 2854894485, 2600075865, 712988175, 59062275

Offset: 0

Alois P. Heinz, May 22 2014

Sum_{k>0} k*T(n,k) = A249249(n).

			T(7,3) = 12 because 12 permutations of {1,2,3,4,5,6,7} have exactly 3 (possibly overlapping) occurrences of the consecutive step pattern up, up, up given by the binary expansion of 7 = 111_2: (1,2,3,4,5,7,6), (1,2,3,4,6,7,5), (1,2,3,5,6,7,4), (1,2,4,5,6,7,3), (1,3,4,5,6,7,2), (2,1,3,4,5,6,7), (2,3,4,5,6,7,1), (3,1,2,4,5,6,7), (4,1,2,3,5,6,7), (5,1,2,3,4,6,7), (6,1,2,3,4,5,7), (7,1,2,3,4,5,6).
Triangle T(n,k) begins:
: n\k :       0        1       2       3  4  ...
+-----+------------------------------------
:  0  :       1;
:  1  :       1;                             [row  1 of A008292]
:  2  :       2;                             [row  2 of A008303]
:  3  :       5,       1;                    [row  3 of A162975]
:  4  :      21,       3;                    [row  4 of A242819]
:  5  :      70,      50;                    [row  5 of A227884]
:  6  :     450,     270;                    [row  6 of A242819]
:  7  :    4326,     602,     99,     12, 1; [row  7 of A220183]
:  8  :   34944,    5376;                    [row  8 of A242820]
:  9  :  209863,  139714,  13303;            [row  9 of A230695]
: 10  : 1573632, 1366016, 530432, 158720;    [row 10 of A230797]

Alois P. Heinz, Rows n = 0..130, flattened

Column k=0-10 give: A242785, A246221, A246222, A246223, A246224, A246225, A246226, A246227, A246228, A246229, A243105.
Row sums give A000142.
Cf. A242784, A249249, A295987, A335308.

T:= proc(n) option remember; local b, k, r, h;
      k:= iquo(n,2,'r'); h:= 2^ilog2(n);
      b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, irem(2*t,   h))*`if`(r=0 and t=k, x, 1), j=1..u)+
      add(b(u+j-1, o-j, irem(2*t+1, h))*`if`(r=1 and t=k, x, 1), j=1..o)))
      end: forget(b);
      (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 0))
    end:
seq(T(n), n=0..15);

T[n_] := T[n] = Module[{b, k, r, h}, {k, r} = QuotientRemainder[n, 2]; h = 2^Floor[Log[2, n]]; b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[ Sum[b[u - j, o + j - 1, Mod[2*t, h]]*If[r == 0 && t == k, x, 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, Mod[2*t + 1, h]]*If[r == 1 && t == k, x, 1], {j, 1, o}]]]; Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

A177477 Number of permutations of 1..n avoiding adjacent step pattern up, down, up.

Original entry on oeis.org

1, 1, 2, 6, 19, 70, 331, 1863, 11637, 81110, 635550, 5495339, 51590494, 524043395, 5743546943, 67478821537, 844983073638, 11240221721390, 158365579448315, 2355375055596386, 36870671943986643, 606008531691619131, 10435226671431973345, 187860338952519968538

Offset: 0

Submitted independently by Signy Olafsdottir (signy06(AT)ru.is), May 09 2010 (9 terms) and R. H. Hardin, May 10 2010 (17 terms)

nonn

Suppose a < b, c < b, and c < d. To avoid abcd means not to have four consecutive letters such that the first letter is less than the second one, the third letter is less than the second one, and the third letter is less than the last one.

Alois P. Heinz, Table of n, a(n) for n = 0..468
Sergey Kitaev, Introduction to partially ordered patterns, Discrete Applied Mathematics 155 (2007), 929-944.

Column k=0 of A227884.
Column k=5 of A242784.
Cf. A227883, A245758.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
       add(b(u-j, o+j-1, [1, 3, 1][t]), j=1..u)+
      `if`(t=3, 0, add(b(u+j-1, o-j, 2), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 10 2020

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1,
     Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]], {j, 1, u}] +
     If[t == 3, 0, Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 08 2022, after Alois P. Heinz *)

a(n) ~ c * d^n * n!, where d = A245758 = 0.7827041801715217018447074977..., c = 2.035127405829990832658061124449458067... . - Vaclav Kotesovec, Aug 22 2014

a(18)-a(23) from Alois P. Heinz, Oct 06 2013

a(0)=1 prepended by Alois P. Heinz, Mar 10 2020

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.

Original entry on oeis.org

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 *)

A227883 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up.

Original entry on oeis.org

0, 0, 0, 0, 5, 50, 328, 2154, 16751, 144840, 1314149, 12735722, 134159743, 1519210786, 18272249418, 233231701166, 3159471128588, 45243728569842, 682183513506619, 10807962134238068, 179606706777512992, 3123700853586733882, 56737351453843424893

Offset: 0

Alois P. Heinz, Oct 25 2013

nonn

			a(4) = 5: 1324, 1423, 2314, 2413, 3412.
a(5) = 50: 12435, 12534, 13245, ..., 52314, 52413, 53412.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..460 (first 195 terms from Alois P. Heinz)

Column k=1 of A227884.

Cf. A177477, A245758.

b:= proc(u, o, t) option remember;
      `if`(t=7, 0, `if`(u+o=0, `if`(t in [4, 5, 6], 1, 0),
      add(b(u-j, o+j-1, [1, 3, 1, 5, 6, 6][t]), j=1..u)+
      add(b(u+j-1, o-j, [2, 2, 4, 4, 7, 4][t]), j=1..o)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);

b[u_, o_, t_] := b[u, o, t] =
    If[t == 7, 0, If[u + o == 0, If[4 <= t <= 6, 1, 0],
    Sum[b[u - j, o + j - 1, {1, 3, 1, 5, 6, 6}[[t]]], {j, 1, u}] +
    Sum[b[u + j - 1, o - j, {2, 2, 4, 4, 7, 4}[[t]]], {j, 1, o}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 20 2020, after Alois P. Heinz *)

a(n) ~ c * d^n * n! * n, where d = A245758 = 0.782704180171521701844707..., c = 0.575076701401064911213333442496869737011... . - Vaclav Kotesovec, Aug 22 2014

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A242783 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern given by the binary expansion of n, where 1=up and 0=down; triangle T(n,k), n>=0, read by rows.

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A177477 Number of permutations of 1..n avoiding adjacent step pattern up, down, up.

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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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A227883 Number of permutations of [n] with exactly one occurrence of the consecutive step pattern up, down, up.

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Author

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Examples

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Crossrefs

Programs

Maple

Mathematica

Formula