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 1-7 of 7 results.

A174072 Number of permutations of length n with no consecutive triples i,i+2,i+4.

Original entry on oeis.org

1, 1, 2, 6, 24, 114, 674, 4714, 37754, 340404, 3412176, 37631268, 452745470, 5900431012, 82802497682, 1244815252434, 19958707407096, 339960096280062, 6130407887839754, 116675071758609742, 2337186717333367706, 49153251967227002616, 1082860432463176004544
Offset: 0

Views

Author

Isaac Lambert, Mar 06 2010

Keywords

Comments

Note for n<5 there are no such subsequences, so those values are trivially n!.

Examples

			For n=5 (0,2,4,1,3) is an example of a permutation with an i,i+2,i+4 triple. If we look at 0,2,4 as a block, then we have 3! ways to permute the triple with the remaining 1 & 3. Hence a(5) = 5! - 3! = 114.

Links

Crossrefs

Cf. A002628, A174073, A174074, A174075.
First column of A216716.

Programs

  • Maple
    b:= proc(s, x, y) option remember; `if`(s={}, 1, add(
     `if`(x=0 or x-y<>2 or y-j<>2, b(s minus {j}, y, j), 0), j=s))
    end:
a:= n-> b({$1..n}, 0$2):
seq(a(n), n=0..14);  # Alois P. Heinz, Apr 13 2021
  • Mathematica
    b[s_, x_, y_] := b[s, x, y] = If[s == {}, 1, Sum[
     If[x == 0 || x - y != 2 || y - j != 2,
     b[s ~Complement~ {j}, y, j], 0], {j, s}]];
a[n_] := b[Range[n], 0, 0];
Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)

Extensions

a(0)-a(4) and a(10)-a(11) moved from a duplicate entry based on the Dymacek et al. paper on Apr 13 2021
a(12)-a(22) from Alois P. Heinz, Apr 13 2021

A216718 Triangle read by rows: number of circular permutations of [1..n] with k progressions of rise 1, distance 1 and length 3 (n >= 3, k >= 0).

Original entry on oeis.org

1, 1, 1, 5, 0, 1, 20, 3, 0, 1, 102, 14, 3, 0, 1, 627, 72, 17, 3, 0, 1, 4461, 468, 87, 20, 3, 0, 1, 36155, 3453, 582, 103, 23, 3, 0, 1, 328849, 28782, 4395, 704, 120, 26, 3, 0, 1, 3317272, 267831, 37257, 5435, 834, 138, 29, 3, 0, 1
Offset: 2

Views

Author

N. J. A. Sloane, Sep 15 2012

Keywords

Examples

			Triangle begins:
        1;
        1,      1; [this is row n=3]
        5,      0,     1;
       20,      3,     0,    1;
      102,     14,     3,    0,   1;
      627,     72,    17,    3,   0,   1;
     4461,    468,    87,   20,   3,   0,  1;
    36155,   3453,   582,  103,  23,   3,  0, 1;
   328849,  28782,  4395,  704, 120,  26,  3, 0, 1;
  3317272, 267831, 37257, 5435, 834, 138, 29, 3, 0, 1;
  ...

References

  • Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.

Crossrefs

Cf. A216716, A174072, A165961, A180187.

Extensions

a(2,0)=1 added by N. J. A. Sloane, Apr 16 2021

A216719 Triangle read by rows: number of circular permutations of [1..n] with k progressions of rise 1, distance 2 and length 3 (n >= 3, k >= 0).

Original entry on oeis.org

2, 6, 22, 2, 109, 10, 1, 657, 55, 7, 1, 4625, 356, 54, 4, 1, 37186, 2723, 362, 44, 4, 1, 336336, 23300, 2837, 368, 34, 4, 1, 3379058, 220997, 25408, 2967, 330, 35, 4, 1, 37328103, 2308564, 249736, 26964, 3100, 292, 36, 4, 1
Offset: 3

Views

Author

N. J. A. Sloane, Sep 15 2012

Keywords

Examples

			Triangle begins:
         2
         6
        22       2
       109      10      1
       657      55      7     1
      4625     356     54     4    1
     37186    2723    362    44    4   1
    336336   23300   2837   368   34   4  1
   3379058  220997  25408  2967  330  35  4 1
  37328103 2308564 249736 26964 3100 292 36 4 1
  ...

References

  • Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.

Crossrefs

Columns 1..2 are A174074, A216721.
Row sums are A000142(n-1).
Cf. A216716, A216718.

A216722 Triangle read by rows: number of circular permutations of [1..n] with k modular progressions of rise 1, distance 1 and length 3 (n >= 3, 0 <= k <= n).

Original entry on oeis.org

1, 0, 0, 1, 5, 0, 0, 0, 1, 18, 5, 0, 0, 0, 1, 95, 18, 6, 0, 0, 0, 1, 600, 84, 28, 7, 0, 0, 0, 1, 4307, 568, 116, 40, 8, 0, 0, 0, 1, 35168, 4122, 810, 156, 54, 9, 0, 0, 0, 1, 321609, 33910, 5975, 1100, 205, 70, 10, 0, 0, 0, 1
Offset: 3

Views

Author

N. J. A. Sloane, Sep 15 2012

Keywords

Examples

			Triangle begins:
       1,     0,    0,    1;
       5,     0,    0,    0,   1;
      18,     5,    0,    0,   0,  1;
      95,    18,    6,    0,   0,  0,  1;
     600,    84,   28,    7,   0,  0,  0, 1;
    4307,   568,  116,   40,   8,  0,  0, 0, 1;
   35168,  4122,  810,  156,  54,  9,  0, 0, 0, 1;
  321609, 33910, 5975, 1100, 205, 70, 10, 0, 0, 0, 1;
  ...

References

  • Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.

Crossrefs

Columns 1..2 are A165962, A216723.
Row sums are A000142(n-1).
Cf. A216716, A216718, A216719, A165962.

A216724 Triangle read by rows: T(n,k) is the number of permutations of [1..n] with k modular progressions of rise 2, distance 1 and length 3 (n >= 0, k >= 0).

Original entry on oeis.org

1, 1, 2, 3, 3, 24, 0, 100, 15, 0, 5, 594, 108, 18, 0, 4389, 504, 119, 21, 0, 7, 35744, 3520, 960, 64, 32, 0, 325395, 31077, 5238, 927, 207, 27, 0, 9, 3288600, 288300, 42050, 8800, 900, 100, 50, 0, 36489992, 2946141, 409827, 59785, 9174, 1518, 319, 33, 0, 11
Offset: 0

Views

Author

N. J. A. Sloane, Sep 15 2012

Keywords

Examples

			Triangle begins:
        1
        1
        2
        3      3
       24      0
      100     15     0    5
      594    108    18    0
     4389    504   119   21   0   7
    35744   3520   960   64  32   0
   325395  31077  5238  927 207  27  0 9
  3288600 288300 42050 8800 900 100 50 0
  ...

References

  • Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, Congressus Numerantium, Vol. 208 (2011), pp. 147-165.

Links

Crossrefs

Column 1 is A174073.
Row sums are A000142.
Cf. A216716, A216718, A216719, A216722.

Programs

  • Maple
    b:= proc(s, x, y, n) option remember; expand(`if`(s={}, 1, add(
     `if`(x>0 and irem(n+x-y, n)=2 and irem(n+y-j, n)=2, z, 1)*
        b(s minus {j}, y, j, n), j=s)))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..max(0,
         iquo(n-1,2)*2-1)))(b({$1..n}, 0$2, n)):
seq(T(n), n=0..11);  # Alois P. Heinz, Apr 13 2021
  • Mathematica
    b[s_, x_, y_, n_] := b[s, x, y, n] = Expand[If[s == {}, 1, Sum[
     If[x>0 && Mod[n + x - y, n] == 2 && Mod[n + y - j, n] == 2, z, 1]*
     b[s~Complement~{j}, y, j, n], {j, s}]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Max[0,
     Quotient[n - 1, 2]*2 - 1]}]][b[Range[n], 0, 0, n]];
Table[T[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Mar 06 2022, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Apr 13 2021

A216726 Triangle read by rows: number of circular permutations of [1..n] with k modular progressions of rise 1, distance 2 and length 3 (n >= 3, 0 <= k <= n).

Original entry on oeis.org

11, 0, 0, 1, 6, 0, 0, 0, 0, 18, 5, 0, 0, 0, 1, 93, 18, 9, 0, 0, 0, 0, 600, 84, 28, 7, 0, 0, 0, 1, 4320, 512, 192, 0, 16, 0, 0, 0, 0, 35168, 4122, 810, 156, 54, 9, 0, 0, 0, 1, 321630, 34000, 5625, 1400, 200, 0, 25, 0, 0, 0, 0
Offset: 3

Views

Author

N. J. A. Sloane, Sep 15 2012

Keywords

Examples

			Triangle begins:
1 0 0 1
6 0 0 0 0
18 5 0 0 0 1
93 18 9 0 0 0 0
600 84 28 7 0 0 0 1
4320 512 192 0 16 0 0 0 0
35168 4122 810 156 54 9 0 0 0 1
321630 34000 5625 1400 200 0 25 0 0 0 0
...

References

  • Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012.

Crossrefs

Cf. A216716, A216718, A216719, A216722, A216724, A174075.

A343535 Number T(n,k) of permutations of [n] having exactly k consecutive triples j, j+1, j-1; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.

Original entry on oeis.org

1, 1, 2, 5, 1, 20, 4, 102, 18, 626, 92, 2, 4458, 564, 18, 36144, 4032, 144, 328794, 32898, 1182, 6, 3316944, 301248, 10512, 96, 36755520, 3057840, 102240, 1200, 443828184, 34073184, 1085904, 14304, 24, 5800823880, 413484240, 12538080, 174000, 600, 81591320880
Offset: 0

Views

Author

Alois P. Heinz, Apr 18 2021

Keywords

Comments

Terms in column k are multiples of k!.

Examples

			T(4,1) = 4: 1342, 2314, 3421, 4231.
Triangle T(n,k) begins:
              1;
              1;
              2;
              5,           1;
             20,           4;
            102,          18;
            626,          92,          2;
           4458,         564,         18;
          36144,        4032,        144;
         328794,       32898,       1182,        6;
        3316944,      301248,      10512,       96;
       36755520,     3057840,     102240,     1200;
      443828184,    34073184,    1085904,    14304,     24;
     5800823880,   413484240,   12538080,   174000,    600;
    81591320880,  5428157760,  156587040,  2214720,  10800;
  1228888215960, 76651163160, 2105035440, 29777520, 175800, 120;
  ...

Links

Crossrefs

Column k=0 gives A212580.
Row sums give A000142.
Cf. A047921, A123513, A197365, A216716.

Programs

  • Maple
    b:= proc(s, l, t) option remember; `if`(s={}, 1, add((h->
      expand(b(s minus {j}, j, `if`(h=1, 2, 1))*
     `if`(t=2 and h=-2, x, 1)))(j-l), j=s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
               b({$1..n}, -1, 1)):
seq(T(n), n=0..13);
  • Mathematica
    b[s_, l_, t_] := b[s, l, t] = If[s == {}, 1, Sum[Function[h,
     Expand[b[s ~Complement~ {j}, j, If[h == 1, 2, 1]]*
     If[t == 2 && h == -2, x, 1]]][j - l], {j, s}]];
T[n_] := CoefficientList[b[Range[n], -1, 1], x];
T /@ Range[0, 13] // Flatten (* Jean-François Alcover, Apr 26 2021, after Alois P. Heinz *)

Formula

T(3n,n) = n!.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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