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
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).
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
KEYWORD
nonn
AUTHOR
Vladimir Shevelev and Peter J. C. Moses, Jun 07 2015
STATUS
approved