cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

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

Views

Author

Alois P. Heinz, Dec 01 2017

Keywords

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

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 *)

A228408 Number of permutations of [n] with exactly two (possibly overlapping) occurrences of some of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).

Original entry on oeis.org

0, 0, 0, 0, 0, 29, 230, 1537, 11208, 89657, 724755, 6010150, 55305521, 545054759, 5504044595, 59482056555, 690974195737, 8306302563795, 104653460921783, 1401318441726295, 19525683104731681, 282626170020405627, 4296152288224050974, 67974610534037861728
Offset: 0

Views

Author

Alois P. Heinz, Nov 09 2013

Keywords

Comments

The counted patterns are: 1243, 1342, 2341 (=UUD), 1324, 1423, 2314, 2413, 3412 (=UDU), 2134, 3124, 4123 (=DUU).

Examples

			a(5) = 29: 12435, 12534, 13245, 13425, 13524, 14235, 14523, 15234, 21354, 21453, 23145, 23415, 23514, 24135, 24513, 25134, 31254, 31452, 32451, 34125, 34512, 35124, 41253, 41352, 42351, 45123, 51243, 51342, 52341.
a(6) = 230: 123546, 123645, 124365, ..., 651243, 651342, 652341.
a(7) = 1537: 1234657, 1234756, 1235476, ..., 7651243, 7651342, 7652341.

Links

Crossrefs

Column k=2 of A231384.

Programs

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

A231385 Number of permutations of [n] avoiding simultaneously consecutive step patterns UUD, UDU, DUU (U=up, D=down).

Original entry on oeis.org

1, 1, 2, 6, 13, 39, 158, 674, 3304, 19511, 122706, 834131, 6416525, 52909708, 462097526, 4395014406, 44626369587, 476351029850, 5414386451909, 65177788719791, 821378978885730, 10880928171304446, 151423268838929524, 2197946731864495343, 33278572455563069142
Offset: 0

Views

Author

Alois P. Heinz, Nov 08 2013

Keywords

Comments

The avoided patterns are: 1243, 1342, 2341 (=UUD), 1324, 1423, 2314, 2413, 3412 (=UDU), 2134, 3124, 4123 (=DUU).

Examples

			a(4) = 13: 1234, 1432, 2143, 2431, 3142, 3214, 3241, 3421, 4132, 4213, 4231, 4312, 4321.
a(5) = 39: 12345, 14325, 15324, ..., 54231, 54312, 54321.
a(6) = 158: 123456, 143265, 153264, ..., 654231, 654312, 654321.

Links

Crossrefs

Column k=0 of A231384.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t=7, 0, `if`(u+o=0, 1,
      add(b(u+j-1, o-j, [2, 3, 3, 6, 7, 7][t]), j=1..o)+
      add(b(u-j, o+j-1, [4, 5, 7, 4, 4, 5][t]), j=1..u)))
    end:
a:= n-> `if`(n=0, 1, add(b(j-1, n-j, 1), j=1..n)):
seq(a(n), n=0..25);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t == 7, 0, If[u + o == 0, 1,
    Sum[b[u + j - 1, o - j, {2, 3, 3, 6, 7, 7}[[t]]], {j, 1, o}] +
    Sum[b[u - j, o + j - 1, {4, 5, 7, 4, 4, 5}[[t]]], {j, 1, u}]]];
a[n_] := If[n == 0, 1, Sum[b[j - 1, n - j, 1], {j, 1, n}]];
a /@ Range[0, 25] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n!, where d = 0.63140578989563018836..., c = 3.3290259175437715006... . - Vaclav Kotesovec, Aug 28 2014

A231386 Number of permutations of [n] with exactly one occurrence of one of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).

Original entry on oeis.org

0, 0, 0, 0, 11, 52, 233, 1344, 8197, 49846, 351946, 2799536, 22764021, 200196218, 1947350444, 19753229932, 210793513246, 2425636703848, 29307938173409, 369141523106550, 4920501544208343, 68771635812423192, 998694091849893095, 15169308298544690802
Offset: 0

Views

Author

Alois P. Heinz, Nov 08 2013

Keywords

Examples

			a(4) = 11: 1243, 1342, 2341 (=UUD), 1324, 1423, 2314, 2413, 3412 (=UDU), 2134, 3124, 4123 (=DUU).
a(5) = 52: 12354, 12453, 12543, ..., 53124, 53412, 54123.
a(6) = 233: 123465, 123564, 123654, ..., 653124, 653412, 654123.
a(7) = 1344: 1234576, 1234675, 1234765, ..., 7653124, 7653412, 7654123.

Links

Crossrefs

Column k=1 of A231384.

Programs

  • Maple
    b:= proc(u, o, t) option remember; `if`(t=13, 0, `if`(u+o=0,
      `if`(t>6, 1, 0), add(b(u+j-1, o-j,
          [2, 3, 3, 6, 12, 9, 8, 9, 9, 12, 13, 13][t]), j=1..o)+
      add(b(u-j, o+j-1,
          [4, 5, 11, 4, 4, 5, 10, 11, 13, 10, 10, 11][t]), j=1..u)))
    end:
a:= n-> add(b(j-1, n-j, 1), j=1..n):
seq(a(n), n=0..30);
  • Mathematica
    b[u_, o_, t_] := b[u, o, t] = If[t==13, 0, If[u + o == 0, If[t > 6, 1, 0],
  Sum[b[u+j-1, o-j,
    {2, 3, 3, 6, 12, 9, 8, 9, 9, 12, 13, 13}[[t]]], {j, 1, o}] +
  Sum[b[u-j, o+j-1,
    {4, 5, 11, 4, 4, 5, 10, 11, 13, 10, 10, 11}[[t]]], {j, 1, u}]]];
a[n_] := Sum[b[j - 1, n - j, 1], {j, 1, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

Formula

a(n) ~ c * d^n * n! * n, where d = 0.63140578989563018836..., c = 1.015673... . - Vaclav Kotesovec, Aug 28 2014

A231410 Number of permutations of [n] with exactly n-3 (possibly overlapping) occurrences of some of the consecutive step patterns UUD, UDU, DUU (U=up, D=down).

Original entry on oeis.org

6, 11, 29, 99, 367, 1543, 7901, 41759, 241361, 1647843, 11321131, 83279563, 710717285, 6009605795, 53680350389, 549737059971, 5519982252151, 58008028652479, 693065960525741, 8057982367331159, 97381078055591177, 1329697914765988419, 17567989325451095443
Offset: 3

Views

Author

Alois P. Heinz, Nov 08 2013

Keywords

Examples

			a(3) = 6: 123, 132, 213, 231, 312, 321.
a(4) = 11: 1243, 1342, 2341 (UUD), 1324, 1423, 2314, 2413, 3412 (UDU), 2134, 3124, 4123 (DUU).
a(5) = 29: 12435, 12534, 13245, ..., 51243, 51342, 52341.
a(6) = 99: 124356, 125346, 126345, ..., 623514, 624513, 634512.
a(7) = 367: 1243576, 1243675, 1253476, ..., 7346125, 7356124, 7456123.

Links

Crossrefs

Diagonal of A231384.

Programs

  • Maple
    b:= proc(u, o, t, c) option remember;  `if`(u+o add(b(j-1, n-j, 1, n-3), j=1..n):
seq(a(n), n=3..25);
  • Mathematica
    b[u_, o_, t_, c_] := b[u, o, t, c] = If[u + o < c, 0,
     If[u + o == 0, If[c == 0, 1, 0],
     Sum[b[u + j - 1, o - j, {2, 3, 3, 6, 6, 3}[[t]],
          If[5 <= t <= 6, c - 1, c]], {j, 1, o}] +
     Sum[b[u - j, o + j - 1, {4, 5, 5, 4, 4, 5}[[t]],
          If[t == 3, c - 1, c]], {j, 1, u}]]];
a[n_] := Sum[b[j - 1, n - j, 1, n - 3], {j, 1, n}];
a /@ Range[3, 25] (* Jean-François Alcover, Mar 23 2021, after Alois P. Heinz *)
Showing 1-5 of 5 results.

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