login
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).
3

%I #24 Mar 06 2022 08:41:38

%S 1,1,2,3,3,24,0,100,15,0,5,594,108,18,0,4389,504,119,21,0,7,35744,

%T 3520,960,64,32,0,325395,31077,5238,927,207,27,0,9,3288600,288300,

%U 42050,8800,900,100,50,0,36489992,2946141,409827,59785,9174,1518,319,33,0,11

%N 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).

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

%H Alois P. Heinz, <a href="/A216724/b216724.txt">Rows n = 0..18, flattened</a>

%e Triangle begins:

%e 1

%e 1

%e 2

%e 3 3

%e 24 0

%e 100 15 0 5

%e 594 108 18 0

%e 4389 504 119 21 0 7

%e 35744 3520 960 64 32 0

%e 325395 31077 5238 927 207 27 0 9

%e 3288600 288300 42050 8800 900 100 50 0

%e ...

%p b:= proc(s, x, y, n) option remember; expand(`if`(s={}, 1, add(

%p `if`(x>0 and irem(n+x-y, n)=2 and irem(n+y-j, n)=2, z, 1)*

%p b(s minus {j}, y, j, n), j=s)))

%p end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..max(0,

%p iquo(n-1,2)*2-1)))(b({$1..n}, 0$2, n)):

%p seq(T(n), n=0..11); # _Alois P. Heinz_, Apr 13 2021

%t b[s_, x_, y_, n_] := b[s, x, y, n] = Expand[If[s == {}, 1, Sum[

%t If[x>0 && Mod[n + x - y, n] == 2 && Mod[n + y - j, n] == 2, z, 1]*

%t b[s~Complement~{j}, y, j, n], {j, s}]]];

%t T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Max[0,

%t Quotient[n - 1, 2]*2 - 1]}]][b[Range[n], 0, 0, n]];

%t Table[T[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Mar 06 2022, after _Alois P. Heinz_ *)

%Y Column 1 is A174073.

%Y Row sums are A000142.

%Y Cf. A216716, A216718, A216719, A216722.

%K nonn,tabf

%O 0,3

%A _N. J. A. Sloane_, Sep 15 2012

%E More terms from _Alois P. Heinz_, Apr 13 2021