A345342 Number of permutations of [2n] having n cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i.

1, 1, 11, 195, 5033, 171465, 7264499, 368258891, 21740278417, 1465044247953, 110975742044635, 9334724676616339, 863320991981279033, 87072657503374176985, 9511213780859395685955, 1118615909510940858978075, 140933163945864346869845025, 18937018020284359019138011425

Offset: 0

Author

Alois P. Heinz, Jun 14 2021

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..332
• Wikipedia, Permutation

Crossrefs

Cf. A344855.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*
      b(n-j)*binomial(n-1, j-1)*ceil(2^(j-2))), j=1..n))
    end:
a:= n-> coeff(b(2*n), x, n):
seq(a(n), n=0..18);

Mathematica

b[n_] := b[n] = If[n == 0, 1, Sum[Expand[x b[n-j] Binomial[n-1, j-1]* Ceiling[2^(j-2)]], {j, n}]];
a[n_] := Coefficient[b[2n], x, n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Aug 25 2021, after Alois P. Heinz *)
a[0] := 1; a[n_] := Sum[Binomial[2*n, n + k + 1]*StirlingS2[n + k + 1, k + 1], {k, 0, n}]; Flatten[Table[a[n] , {n, 0, 17}]] (* Detlef Meya, Jan 18 2024 *)

Formula

a(n) = A344855(2n,n).

a(n) ~ c * (1 + exp(2))^n * (n-1)!, where c = sqrt((exp(2) + 1)/(exp(2) - 1))/(2*Pi) = 0.1823720711148962856100934464088354177502714116352616187167... - Vaclav Kotesovec, Jul 15 2021, updated Mar 17 2024

a(0) = 1; a(n) = Sum_{k=0..n} binomial(2*n, n + k + 1)*Stirling2(n + k + 1, k + 1). - Detlef Meya, Jan 18 2024