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.

0, 0, 0, 0, 0, 0, 115, 791, 6204, 55004, 543597, 5922929, 70518904, 910711192, 12678337943, 189252400475, 3015217932052, 51067619064756, 916176426421297, 17355904144767765, 346195850534324608, 7252654441500343712, 159210363453691696379, 3654550890669607979359

Offset

1,7

Comments

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.

References

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

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

Links

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

I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy]

Peter J. C. Moses, Seatings for 7 couples.

E. Lucas, Sur le problème des ménages, Théorie des nombres, Paris, 1891, 491-496.

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.

Formula

For n <= 6, a(n)=0; otherwise a(n) = Sum_{k=0..n-1} (-1)^k*(n-k-1)! * Sum_{j=max(k-n+6, 0)..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_{k=0..n-1} ((-1)^k)*(n-k-1)! * Sum_{j=max(r+k-n-1, 0)..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).

Mathematica

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 *)

Crossrefs

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

Sequence in context: A154070 A251215 A218324 * A287430 A256889 A256354

Adjacent sequences: A258670 A258671 A258672 * A258674 A258675 A258676

Keyword

nonn

Author

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

Status

approved