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.

A216721 Second column of A216719.

Original entry on oeis.org

2, 10, 55, 356, 2723, 23300, 220997, 2308564
Offset: 5

Views

Author

N. J. A. Sloane, Sep 15 2012

Keywords

References

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

Crossrefs

Cf. A174074, A216719.

A174074 Number of circular permutations of length n without consecutive triples i,i+2,i+4.

Original entry on oeis.org

1, 1, 1, 2, 6, 22, 109, 657, 4625, 37186, 336336, 3379058, 37328103, 449669577, 5866178493, 82387080624, 1239364493118, 19881771085788, 338797668091565, 6111688544942463, 116354993433563797, 2331395107113471188, 49042688584011866880, 1080639600739277669092, 24891049832682424745839
Offset: 0

Views

Author

Isaac Lambert, Mar 06 2010

Keywords

Comments

Circular permutations are permutations whose indices are from the ring of integers modulo n.

Examples

			Since a(5)=22, there are (5-1)!-22=2 circular permutations with consecutive triples i,i+2,i+4 in all circular permutations of length 5. They are exactly (0,2,4,1,3) and (0,2,4,3,1).

References

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

Links

Crossrefs

Column 1 of A216719.
Cf. A165961, A174072, A174073, A174075.

Extensions

a(10)-a(15) from Donovan Johnson, Sep 24 2010
a(0)-a(2) prepended and terms a(16) onward added by Max Alekseyev, Feb 04 2024

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.
Showing 1-5 of 5 results.

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

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