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A094047 Number of seating arrangements of n couples around a round table (up to rotations) so that each person sits between two people of the opposite sex and no couple is seated together. 13
0, 0, 2, 12, 312, 9600, 416880, 23879520, 1749363840, 159591720960, 17747520940800, 2363738855385600, 371511874881100800, 68045361697964851200, 14367543450324474009600, 3464541314885011705344000 (list; graph; refs; listen; history; text; internal format)



Also, the number of Hamiltonian directed circuits in the crown graph of order n.

Or the number of those 3Xn Latin rectangles (cf. A000186) the second row of which is a full cycle. [From Vladimir Shevelev, Mar 22 2010]


V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr.Mat.(J. of the Akademy of Sciences of Russia) 4(1992),91-110. [From Vladimir Shevelev, Mar 22 2010]


Table of n, a(n) for n=1..16.

Max Alekseyev, On enumeration of seating arrangements of couples around a circular table, Automatic Sequences Workshop, 2015

H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]

Eric Weisstein's World of Mathematics, Crown Graph

Eric Weisstein's World of Mathematics, Hamiltonian Cycle


For n>1, a(n) = (-1)^n * 2 * (n-1)! + n! * SUM[j=0..n-1] (-1)^j * (n-j-1)! * binomial(2*n-j-1,j). - Max Alekseyev, Feb 10 2008

a(n) = A059375(n) / (2*n) = A000179(n) * (n-1)!.

Conjecture: a(n) +(-n^2+2*n-3)*a(n-1) -(n-2)*(n^2-3*n+5)*a(n-2) -3*(n-2)*(n-3)*a(n-3) +(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Nov 02 2015

Conjecture: (-n+2)*a(n) +(n-1)*(n^2-3*n+3)*a(n-1) +(n-2)*(n-1)*(n^2-3*n+3)*a(n-2) +(n-2)*(n-3)*(n-1)^2*a(n-3)=0. - R. J. Mathar, Nov 02 2015


A094047 := proc(n)

    if n < 3 then



        (-1)^n*2*(n-1)!+n!*add( (-1)^j*(n-j-1)!*binomial(2*n-j-1, j), j=0..n-1) ;

    end if;

end proc: # R. J. Mathar, Nov 02 2015


Join[{0}, Table[(-1)^n 2(n-1)!+n!Sum[(-1)^j (n-j-1)!Binomial[2n-j-1, j], {j, 0, n-1}], {n, 2, 20}]] (* Harvey P. Dale, Mar 07 2012 *)


Cf. A059375 (rotations are counted as different)

Cf. A114939, A137729.

Sequence in context: A012422 A122767 A260321 * A091472 A156518 A012727

Adjacent sequences:  A094044 A094045 A094046 * A094048 A094049 A094050




Matthijs Coster, Apr 29 2004


Better definition from Joel B. Lewis, Jun 30 2007

Formula and further terms from Max Alekseyev, Feb 10 2008



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Last modified November 29 21:36 EST 2015. Contains 264663 sequences.