This site is supported by donations to The OEIS Foundation.



(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A258673 A total of n married couples, including a mathematician M and his wife, are to be seated at the 2n chairs around a circular table, with no man seated next to his wife. After the ladies are seated at every other chair, M is the first man allowed to choose one of the remaining chairs. The sequence gives the number of ways of seating the other men, with no man seated next to his wife, if M chooses the chair that is 11 seats clockwise from his wife's chair. 8
0, 0, 0, 0, 0, 0, 115, 791, 6204, 55004, 543597, 5922929, 70518904, 910711192, 12678337943, 189252400475, 3015217932052, 51067619064756, 916176426421297, 17355904144767765, 346195850534324608, 7252654441500343712, 159210363453691696379, 3654550890669607979359 (list; graph; refs; listen; history; text; internal format)



This is a variation of the classic ménage problem (cf. A000179).

It is known [Riordan, ch. 8, ex. 7(b)] that, after the ladies are seated at every other chair, the number U_n of ways of seating the men in the ménage problem has asymptotic expansion U_n ~ e^(-2)*n!*(1 + sum{k>=1}(-1)^k/(k!(n-1)_k)), where (n)_k = n*(n-1)*...*(n-k+1).

Therefore, it is natural to conjecture that a(n) ~ e^(-2)*n!/(n-2)*(1 + sum{k>=1}(-1)^k/(k!(n-1)_k)).

In the general case, M chooses a chair at an odd distance d>=3 clockwise from his wife. See the corresponding general formula below.


I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124.

E. Lucas, Théorie des nombres, Paris, 1891, 491-496.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, chs. 7, 8.


Table of n, a(n) for n=1..24.

Peter J. C. Moses, Seatings for 7 couples

Vladimir Shevelev, Peter J. C. Moses, The ménage problem with a known mathematician, arXiv:1101.5321 [math.CO], 2011, 2015.

Vladimir Shevelev and Peter J. C. Moses, Alice and Bob go to dinner:A variation on menage, INTEGERS, Vol. 16(2016), #A72.

J. Touchard, Sur un problème de permutations, C.R. Acad. Sci. Paris, 198 (1934), 631-633.


For n<=6, a(n)=0; otherwise a(n) = Sum_{0<=k<=n-1}(-1)^k*(n-k-1)! * Sum_{max(k-n+6, 0)<=j<=min(k,5)} binomial(10-j, j)*binomial(2*n-k+j-12, k-j).

In the general case (see comment), let r=(d+3)/2 and denote the solution by A(r,n). Then A(r,n) is given by the formula

A(r,n)=0 for n<=(d+1)/2; otherwise A(r,n) = Sum_{0<=k<=n-1} ((-1)^k)*(n-k-1)! * Sum_{max(r+k-n-1, 0)<=j<=min(k,r-2)} binomial(2r-j-4, j)*binomial(2(n-r) - k +j +2, k-j).

Note that, if n is even, then 2*sum{r=3,...,(n+2)/2)A(r,n) = A000179(n); if n is odd, then 2*sum{r=3,...,(n+1)/2))A(r,n)+ A((n+3)/2, n) = A000179(n).


a[d_, n_]:=If[n<=#-1, 0, Sum[((-1)^k)*(n-k-1)!Sum[Binomial[2#-j-4, j]*Binomial[2(n-#)-k+j+2, k-j], {j, Max[#+k-n-1, 0], Min[k, #-2]}], {k, 0, n-1}]]&[(d+3)/2];

Map[a[11, #]&, Range[20]] (* Peter J. C. Moses, Jun 07 2015 *)


Cf. A000179, A258664, A258665, A258666, A258667, A259673.

Sequence in context: A154070 A251215 A218324 * A287430 A256889 A256354

Adjacent sequences:  A258670 A258671 A258672 * A258674 A258675 A258676




Vladimir Shevelev and Peter J. C. Moses, Jun 07 2015



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 23 05:50 EDT 2018. Contains 316519 sequences. (Running on oeis4.)