A333711 Number of permutations of [n] such that the product of the first k elements and the product of the last k elements are multiples of k! for all k in [n].

1, 1, 2, 2, 8, 4, 32, 4, 96, 244, 1400, 20, 3988, 12, 256, 1328, 3107082, 7900, 4352004, 2676, 752280, 4710724, 23591664, 672, 79424164, 51627164, 4705224, 802988332, 25488756038104, 47736592, 1706618983956, 826828

Offset: 0

Author

Alois P. Heinz, Apr 09 2020

Keywords

• nonn

Examples

			a(4) = 8: 1234, 1432, 2134, 2314, 2341, 4132, 4312, 4321.
a(5) = 4: 12345, 14325, 52341, 54321.
a(7) = 4: 1234567, 1654327, 7234561, 7654321.
a(13) = 12: 123456789(10)(11)(12)(13), 143256789(10)(11)(12)(13), 143(10)987652(11)(12)(13), 1(12)(11)256789(10)34(13), 1(12)(11)(10)98765234(13), 1(12)(11)(10)98765432(13), (13)23456789(10)(11)(12)1, (13)43256789(10)(11)(12)1, (13)43(10)987652(11)(12)1, (13)(12)(11)256789(10)341, (13)(12)(11)(10)987652341, (13)(12)(11)(10)987654321.
		

Links

• Wikipedia, Permutation

Crossrefs

Cf. A000142, A333710, A333934.

Programs

Maple

b:= proc(s, n) option remember; (m-> `if`(m=0, 1, `if`(irem(
      mul(h, h=({$1..n} minus s)), (n-m)!)=0 and irem(mul(h,
      h=s), m!)=0, add(b(s minus {j}, n), j=s), 0)))(nops(s))
    end:
a:= n-> b({$1..n}, n):
seq(a(n), n=0..17);

Mathematica

b[s_, n_] := b[s, n] = With[{m = Length[s]}, If[m == 0, 1, If[Mod[ Product[h, {h, Range[n] ~Complement~ s}], (n-m)!] == 0 && Mod[Times@@s, m!] == 0, Sum[b[s ~Complement~ {j}, n], {j, s}], 0]]];
a[n_] := b[Range[n], n];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 27}] (* Jean-François Alcover, Nov 01 2021, after Alois P. Heinz *)