A052517 Number of ordered pairs of cycles over all n-permutations having two cycles.

0, 0, 2, 6, 22, 100, 548, 3528, 26136, 219168, 2053152, 21257280, 241087680, 2972885760, 39605518080, 566931294720, 8678326003200, 141468564787200, 2446811181158400, 44753976117043200, 863130293635276800

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

a(n) is a function of the harmonic numbers. If we discard the first term and set a(0)=0, a(1)=2..then a(n) = 2n!*h(n) where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Aug 04 2010

a(n+1) is twice the sum over all permutations of the number of its cycles (fixed points included). - Olivier Gérard, Oct 23 2012

In a game where n differently numbered cards are drawn in a random sequence, and a point is earned every time the newest card is either the highest or the lowest yet drawn (the first card gets two points as it is both the highest and the lowest), the expected number of points earned is a(n+1)/n!, for instance if n=3, there are two ways of getting 3 points and four ways of getting 4 points, giving an average of 22/6 = 3 2/3. - Elliott Line, Mar 19 2020

Examples

			a(3)=6 because we have the ordered pairs of cycles: ((1)(23)), ((23)(1)), ((2)(13)), ((13)(2)), ((3)(12)), ((12)(3)). - _Geoffrey Critzer_, Jun 13 2013
G.f. = 2*x^2 + 6*x^3 + 22*x^4 + 100*x^5 + 548*x^6 + 3528*x^7 + ...
		

References

• G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 30.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 83

Crossrefs

Equals 2 * A000254(n+1), n>0.

Equals, for n=>2, the second right hand column of A028421.

Cf. A302827, A304589.

Programs

Magma

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Log(1-x)^2 )); [0,0] cat [Factorial(n+1)*b[n]: n in [1..m-2]]; // G. C. Greubel, May 13 2019

Maple

pairsspec := [S,{S=Prod(B,B),B=Cycle(Z)},labeled]: seq(combstruct[count](pairsspec,size=n), n=0..20); # Typos fixed by Johannes W. Meijer, Oct 16 2009

Mathematica

Flatten[{0,Table[(n+1)!*Sum[1/(k*(n+1-k)),{k,1,n}],{n,0,20}]}] (* Vaclav Kotesovec, Oct 08 2012 *)
With[{m = 25}, CoefficientList[Series[Log[1-x]^2, {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, May 13 2019 *)

PARI

{a(n) = if( n<0, 0, n! * sum(k=1, n-1, 1 / (k * (n - k))))};

Sage

m = 25; T = taylor(log(1-x)^2, x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 13 2019

Formula

E.g.f.: (log(1 - x))^2. - Michael Somos, Feb 05 2004

a(n) ~ 2*(n-1)!*log(n)*(1+gamma/log(n)), where gamma is Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Oct 08 2012

a(n) = Sum_{k=1..n-1} 2*k*|S1(n-1,k)| = 2*|S1(n,2)|. - Olivier Gérard, Oct 23 2012

0 = a(n) * n^2 - a(n+1) * (2*n+1) + a(n+2) for all n in Z. - Michael Somos, Apr 23 2014

0 = a(n)*(a(n+1) - 7*a(n+2) + 6*a(n+3) - a(n+4)) + a(n+1)*(a(n+1) - 6*a(n+2) + 4*a(n+3)) + a(n+2)*(-3*a(n+2)) if n>0. - Michael Somos, Apr 23 2014

For n>=2, a(n) = (n-2)! * Sum_{i=1..n-1} Sum_{j=1..n-1} (i+j)/(i*j). - Pedro Caceres, Feb 14 2021

Extensions

Name improved by Geoffrey Critzer, Jun 13 2013