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

2,2

Links

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

Wikipedia, Permutation

Example

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).

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 *)

Crossrefs

Column k=2 of A285849.

Cf. A285917.

Sequence in context: A192368 A355539 A323686 * A138748 A097899 A223505

Adjacent sequences: A285850 A285851 A285852 * A285854 A285855 A285856

Keyword

nonn

Author

Alois P. Heinz, Apr 27 2017

Status

approved