A285853 Number of permutations of [n] with two ordered cycles such that equal-sized cycles are ordered with increasing least elements.

1, 6, 19, 100, 508, 3528, 24876, 219168, 1980576, 21257280, 234434880, 2972885760, 38715943680, 566931294720, 8514866707200, 141468564787200, 2407290355814400, 44753976117043200, 850965783594393600, 17505896073523200000, 367844990453821440000

Offset: 2

Author

Alois P. Heinz, Apr 27 2017

Keywords

• nonn

Examples

			a(2) = 1: (1)(2).
a(3) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
a(4) = 19: (123)(4), (4)(123), (132)(4), (4)(132), (124)(3), (3)(124), (142)(3), (3)(142), (134)(2), (2)(134), (143)(2), (2)(143), (1)(234), (234)(1), (1)(243), (243)(1),  (12)(34), (13)(24), (14)(23).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 2..450
• Wikipedia, Permutation

Crossrefs

Column k=2 of A285849.

Cf. A285917.

Programs

Maple

a:= n-> 2*add(binomial(n, k)*(k-1)!*(n-k-1)!, k=1..n/2)-
        `if`(n::even, 3/2*binomial(n, n/2)*(n/2-1)!^2, 0):
seq(a(n), n=2..25);
# second Maple program:
a:= proc(n) option remember; `if`(n<5, [0, 1, 6, 19][n],
     ((2*n-1)*(n-1)*a(n-1)+(n-2)*(2*n^2-5*n-1)*a(n-2)
      -(n-3)^2*((2*n^2-5*n+4)*a(n-3)+(n-4)^2*a(n-4)))/(2*n))
    end:
seq(a(n), n=2..25);

Mathematica

Table[(n-1)!*(2*HarmonicNumber[n] - (3 + (-1)^n)/n), {n, 2, 25}] (* Vaclav Kotesovec, Apr 29 2017 *)