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A216716
Triangle read by rows: number of permutations of [1..n] with k progressions of rise 2, distance 1 and length 3 (n >= 0, k >= 0).
8
1, 1, 2, 6, 24, 114, 6, 674, 44, 2, 4714, 294, 30, 2, 37754, 2272, 276, 16, 2, 340404, 20006, 2236, 216, 16, 2, 3412176, 193896, 20354, 2200, 156, 16, 2, 37631268, 2056012, 206696, 20738, 1908, 160, 16, 2, 452745470, 23744752, 2273420, 215024, 21136, 1616, 164, 16, 2
OFFSET
0,3
LINKS
K. J. Parsons, Arithmetic progressions in permutations, Thesis, Washington and Lee University, 2011
Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, 2012. [broken link]
EXAMPLE
Triangle begins:
1
1
2
6 [this is for n=3]
24
114 6
674 44 2
4714 294 30 2
37754 2272 276 16 2
340404 20006 2236 216 16 2
3412176 193896 20354 2200 156 16 2
37631268 2056012 206696 20738 1908 160 16 2
...
MAPLE
b:= proc(s, x, y) option remember; expand(`if`(s={}, 1, add(
`if`(x>0 and x-y=2 and y-j=2, z, 1)*b(s minus {j}, y, j), j=s)))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b({$1..n}, 0$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Apr 13 2021
MATHEMATICA
b[s_, x_, y_] := b[s, x, y] = Expand[If[s == {}, 1, Sum[
If[x > 0 && x - y == 2 && y - j == 2, z, 1]*
b[s ~Complement~ {j}, y, j], {j, s}]]];
T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0,
Exponent[p, z]}]][b[Range[n], 0, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Mar 02 2022, after Alois P. Heinz *)
CROSSREFS
Row sums give A000142.
Column k=0 gives A174072.
Sequence in context: A152328 A152325 A152329 * A192088 A245233 A228907
KEYWORD
nonn,tabf
AUTHOR
N. J. A. Sloane, Sep 15 2012
EXTENSIONS
More terms from Alois P. Heinz, Apr 13 2021
STATUS
approved