A270174 a(n) is the number of different ways to seat a set of n married male-female couples at a straight table so that men and women alternate and every man is separated by at least two men from his wife.

0, 0, 0, 0, 240, 8640, 584640, 40239360, 3493808640, 364941158400, 45683021260800, 6754660222464000, 1166167699041945600, 232618987254682828800, 53114643986227439616000, 13768242163527512973312000, 4021980517038414919532544000, 1315337131173516220415213568000

Offset: 1

Author

Feng Jishe, Mar 12 2016

Keywords

• nonn

Comments

We assume that the chairs are uniform and indistinguishable.

First we arrange the women in alternating seats, in 2*n! ways. Second, we find the number, G_{n} say, of ways of arranging men in the remaining seats such that every husband cannot sit at the left or right next 1, 2, ..., h male's seats from his wife. Note that here h = 2. We give the board B4, where X denotes the seat cannot be set at, where there are h X's in first column, and h+1 X's in first row, ..., 2h X's in the h column, ..., other entries are 1's. Thus the number of different ways to seat a set of n married male-female couples at a straight table is a_{n}=2*n!*G_{n}.

Links

• Feng Jishe, The board B4
• D. Zeilberger, Automatic Enumeration of Generalized Ménage Numbers
• D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Crossrefs

Cf. A267060, A292574.

Formula

a(n) = 2*n! * A292574(n). - Andrew Howroyd, Sep 19 2017

Extensions

a(11)-a(18) from Andrew Howroyd, Sep 19 2017