A258665 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 5 seats clockwise from his wife's chair.

0, 0, 0, 1, 5, 20, 116, 791, 6203, 55000, 543576, 5922813, 70518113, 910704988, 12678282940, 189251856883, 3015212009143, 51067548545968, 916175515710896, 17355891466436025, 346195661281979133, 7252651426282955236, 159210312386078554436, 3654549974493252076175

Offset

1,5

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)).

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.

Peter J. C. Moses, Seatings for 6 couples.

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

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

a(n) = Sum_{0<=k<=n-1} (-1)^k*(n-k-1)! * Sum_{max(k-n+3, 0)<=j<=min(k,2)} binomial(4-j, j)*binomial(2*n-k+j-6, k-j).

Mathematica

enumerateSeatings[pairs_, d_]:=If[d==1||d>=2pairs-1||EvenQ[d], {},
Map[#[[1]]&, DeleteCases[Map[{#, Differences[#]}&[Riffle[Range[pairs], #]]&, Map[Insert[#, 1, (d+1)/2]&, Permutations[#, {Length[#]}]&[Rest[Range[pairs]]]]], {{___}, {___, 0, ___}}]]];
enumerateSeatings[6, 5]
a[pairs_, d_]:=If[pairs<=#-1||EvenQ[d]||d==1, 0, Sum[((-1)^k)*(pairs-k-1)!Sum[Binomial[2#-j-4, j]*Binomial[2(pairs-#)-k+j+2, k-j], {j, Max[#+k-pairs-1, 0], Min[k, #-2]}], {k, 0, pairs-1}]]&[(d+3)/2];
Table[a[n, 5], {n, 15}] (* Peter J. C. Moses, Jun 13 2015 *)

Prog

(PARI) a(n) = sum(k=0, n-1, (-1)^k*(n-k-1)! * sum(j=max(k-n+3, 0), min(k, 2), binomial(4-j, j)*binomial(2*n-k+j-6, k-j))); \\ Michel Marcus, Jun 13 2015

Crossrefs

Cf. A000179, A258664, A258666, A258667, A258673, A259212.

Sequence in context: A319489 A207972 A117736 * A028944 A332710 A352743

Adjacent sequences: A258662 A258663 A258664 * A258666 A258667 A258668

Keyword

nonn

Author

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

Status

approved