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).

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

Author

N. J. A. Sloane, Sep 15 2012

Keywords

• nonn
• tabf

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

• Alois P. Heinz, Rows n = 0..18, flattened

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