A053871 a(0)=1; a(1)=0; a(n) = 2*(n-1)*(a(n-1) + a(n-2)).

1, 0, 2, 8, 60, 544, 6040, 79008, 1190672, 20314880, 387099936, 8148296320, 187778717632, 4702248334848, 127140703364480, 3691602647581184, 114562300528369920, 3784124901630435328, 132555364873399378432, 4908221631901073295360, 191549525877429961604096

Offset: 0

Author

Cris Moore (moore(AT)santafe.edu) and Christian G. Bower, Mar 29 2000

Keywords

• nonn
• nice
• easy

Comments

Number of deranged matchings of 2n people with partners (of either sex) other than their spouse. 2n objects are initially paired in some way and then are re-paired so that no object is with its original partner (the dancing problem in the article).

Of interest in the "collision problem", where, given a 2-to-1 function f, we are asked for x, y such that f(x)=f(y).

2^n*n!*a(n) = (2n)! b(n) where b(n) are the probabilities that appear in Margolis (2001). One interpretation is in terms of matchings or 1-factors of the complete graph on 2n vertices. The number of these is (2n)!/2^{n}n!. The number of 1-factors being disjoint from (that is, having no edges in common with) a given 1-factor is then a(n); and then b(n) is the probability of picking such a disjoint one factor at random.

Also n!*a(n) = 2^n * c(n) where c(n) = A001499(n). If we define d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0, so d(n,1) are the derangement numbers A000166, then a(n) = d(n,2) (cf. A033030, A088991). On the other hand, taking d*(n,k) = d(n,k)/k^{n}, we have d*(n,k) = (n-1)(d*(n-1,k) + d*(n-2,k)/k), with d*(0,k) = 1 and d*(1,k) = 0 and it is easy to see from Bricard's recurrence for c(n) that c(n)/n! satisfies this for k = 2.

A proof that the description in the first comment as "number of deranged matchings" implies the defining recursion relation: let (x_1, y_1), (x_2, y_2), ..., (x_n, y_n) be the given pairs. In a deranged matching x_1 will be paired with any of the 2(n-1) objects x_2, y_2, x_3, y_3, ..., x_n, y_n. It is sufficient to count only those deranged matchings in which x_1, is matched with x_2. They are of two kinds: (i) y_1 is not matched with y_2; their number is a(n-1); (ii) y_1 is matched with y_2; their number is a(n-2). - Emeric Deutsch, Jan 23 2009

Inverse binomial transform of the odd double factorials (A001147). - David Callan, Aug 25 2009

From Lewis Mammel (l_mammel(AT)att.net), Oct 07 2009: (Start)

The formula is given directly by the Principle of Inclusion and Exclusion.

The first term includes all pairings, the second term excludes all pairings containing each pair, the third term includes all pairings containing each pair of pairs, and so on.

Based on n-a -> n for large n, the ratio a(n)/(2n-1)!! -> exp(-1/2) ~= 0.60653.

We find a(n)/(2n-1)!! for n= 100, 200, 300, 400 ~= 0.6050124904, 0.6057720290, 0.6060250088, 0.6061514604. (End)

This is a subset of the set of pairings of the first 2n integers (A001147) in another way: forbidding pairs of the form (2k,2k+1) for all k as well as the pair (1,2n) (seeing the constraint as circular by opposition to the linear A165968). - Olivier Gérard, Feb 08 2011

a(n) is the n-th central moment of a central chi-squared distribution (with 1 degree of freedom), i.e., a(n) = E[ (Y- E[Y] )^n ] = E[ (X^2 - 1 )^n ] where Y is chi-squared, X is std normal, X~N(0,1), and the expectation operator is E[]. - David Fioramonti, May 11 2016

Exponential self-convolution of a(n)/2^n gives subfactorials (A000166). - Vladimir Reshetnikov, Oct 09 2016

For n > 1, also the number of maximum matchings in the n-cocktail party graph. - Eric W. Weisstein, Jun 15 2017

Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side (cf. A289191, A289269 for nonisomorphic tiles under rotational and rotational and reflectional symmetries). - Marko Riedel, Jun 28 2017

Number of ways n people can hold hands (every hand is holding another hand) where no one is holding their own hand. - Harry Richman, Aug 29 2023

References

• R. Bricard, L'Intermédiaire des Mathématiciens, 8 (1901), 312-313.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..404 (First 100 terms from _T. D. Noe_)
• A. Ayyer, Determinants and Perfect Matchings, arXiv:1106.1465 [math.CO], 2011.
• Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
• Bela Bollobas, The number of 1-factors in 2k-connected graphs, J. Combin. Theory Ser. B 25 (1978), no. 3, 363--366. MR0516268 (80m:05060) - _N. J. A. Sloane_, Mar 26 2012
• James N. Brawner, Dinner, Dancing and Tennis, Anyone?, Mathematics Magazine, Vol. 73, No 1 (2000).
• M. A. Brodie, Avoiding your spouse at a party leads to war, Math. Mag., 75 (2002), 203-208.
• Davi B. Costa, Bogdan A. Dobrescu, and Patrick J. Fox, Chiral Abelian gauge theories with few fermions, arXiv:2001.11991 [hep-ph], 2020.
• Barbara H. Margolius, Dinner-Diner Matching Probabilities
• B. H. Margolius, Avoiding your spouse at a bridge party, Math. Mag., 74 (2001), 33-41.
• B. H. Margolius, The Dinner-Diner Matching Problem, Mathematics Magazine, 76 (2003), 107-118.
• Eric Weisstein's World of Mathematics, Cocktail Party Graph
• Eric Weisstein's World of Mathematics, Independent Edge Set
• Eric Weisstein's World of Mathematics, Matching
• Eric Weisstein's World of Mathematics, Maximum Independent Edge Set

Crossrefs

Cf. A054479, A001147, A001818, A000166, A001499.

See A289191 for when rotational symmetries of the tiles are taken into account. - Marko Riedel, Jun 28 2017

Cf. A165968, number of pairings of 2n things disjoint to a given pairing, and containing a given pair not in the given pairing. It is given by a(n)/(2n-2). - Lewis Mammel (l_mammel(AT)att.net), Oct 07 2009

Programs

Haskell

a053871 n = a053871_list !! n
a053871_list = 1 : 0 : zipWith (*)
   [2,4..] (zipWith (+) a053871_list $ tail a053871_list)
-- Reinhard Zumkeller, Mar 07 2012

Maple

f:= gfun:-rectoproc({a(0) = 1, a(1) = 0, a(n) = 2*(n - 1)*(a(n - 1) + a(n - 2))},a(n),remember):
map(f, [$0..30]); # Robert Israel, May 10 2016

Mathematica

RecurrenceTable[{a[0]==1,a[1]==0,a[n]==2(n-1)(a[n-1]+a[n-2])}, a[n],{n,20}] (* Harvey P. Dale, Sep 15 2011 *)
CoefficientList[Assuming[{Element[x, Reals], x>0}, Series[Sqrt[Pi/2] * (I + Erfi[Sqrt[(1+1/x)/2]]) / (E^((1+x)/(2*x)) * Sqrt[x*(x+1)]), {x, 0, 20}]], x] (* Vaclav Kotesovec, Feb 15 2015 *)
Range[0, 20]! CoefficientList[Series[1/(Exp[x] Sqrt[1 - 2 x]), {x, 0, 20}], x] (* Eric W. Weisstein, Jun 15 2017 *)
Table[(-1)^n HypergeometricPFQ[{1/2, -n}, {}, 2], {n, 20}] (* Eric W. Weisstein, Jun 15 2017 *)
Table[I (-1)^n HypergeometricU[1/2, 3/2 + n, -1/2]/Sqrt[2], {n, 20}] (* Eric W. Weisstein, Dec 31 2017 *)

PARI

a(n)=(-1)^(n+1)*sum(k=0,n,(-1)^k*binomial(n,k)*prod(i=0,k,2*i-1))

Formula

a(n) = A054479(n)/A001147(n).

E.g.f.: 1/(exp(x)*sqrt(1-2x)).

a(n) = (-1)^n*Sum_{k=0..n} (-1)^k*C(n, k)*(2*k-1)!!. - Benoit Cloitre, May 01 2003; corrected by David Fioramonti, May 17 2016

a(n) = Integral_{x>=0} (x-1)^n * (exp(-x/2)/sqrt(2*Pi*x)) dt. - Paul Barry, Apr 12 2010

Conjectured g.f.: T(0)/(1+x), where T(k) = 1 - x*(k+1)/(x*(k+1) - (1+x)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 13 2013

a(n) ~ 2^(n+1/2) * n^n / exp(n+1/2). - Vaclav Kotesovec, Mar 11 2014

G.f.: Sum_{k>=0} (2*k - 1)!!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

Conjecture: if m == n (mod q) for q odd, then (-1)^m*a(m) == (-1)^n*a(n) (mod q). - Harry Richman, Aug 29 2023

Extensions

More terms from James Sellers, Apr 08 2000