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%I #35 Dec 03 2018 05:05:32
%S 0,1,7,216,10956,803400,83003040,11579823360,2080493573760,
%T 469031859192960,129727461014726400,43176116371928601600,
%U 17025803126147196057600,7850538273249476117913600
%N Number of essentially different seating arrangements for n couples around a circular table with 2n seats avoiding spouses being neighbors and avoiding clusters of 3 persons with equal gender.
%C Arrangements that differ only by rotation or reflection are excluded by the following conditions: Seat number 1 is assigned to person (a). Person (a)'s spouse (A) can only take seats with numbers <=(n+1). If (A) gets seat n+1 (i.e. sits exactly opposite to her/his spouse) then person (B) can only take seats with numbers <= n.
%H M. A. Alekseyev, Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations. Lecture Notes in Computer Science 9843 (2016), 151-162. doi:<a href="http://doi.org/10.1007/978-3-319-44543-4_12">10.1007/978-3-319-44543-4_12</a>; arXiv:<a href="http://arxiv.org/abs/1510.07926">1510.07926</a> [math.CO], 2015-2016.
%F See Alekseyev (2016) and the PARI code for the formula.
%F a(n) = A258338(n) / (4*n).
%e a(2)=1 because the only valid arrangement is aBAb.
%e a(3)=7 because the only valid arrangements under the given conditions are: abAcBC, aBAcbC, aBcAbC, aBcACb, acAbCB, acBAbC, aCAbcB.
%t a[1] = 0;
%t a[n_] := (n-1)!/4 Sum[(-1)^j(n-j)! SeriesCoefficient[ SeriesCoefficient[Tr[ MatrixPower[{{0, 1, 0, y^2, 0, 0}, {z y^2, 0, 1, 0, y^2, 0}, {z y^2, 0, 0, 0, y^2, 0}, {0, 1, 0, 0, 0, z}, {0, 1, 0, y^2, 0, z}, {0, 0, 1, 0, y^2, 0}}, 2n]], {y, 0, 2n}] , {z, 0, j}], {j, 0, n}];
%t Array[a, 14] (* _Jean-François Alcover_, Dec 03 2018, from PARI *)
%o (PARI) { a(n) = if(n<=1, 0, (-1)^n*(n-1)!*2^(n-1) + n! * polcoeff( polcoeff( [0, 2*y*z^3 + z^2, -3*y*z^5 - 4*z^4 + ((-2*y^2 - 1)/y)*z^3, 6*y*z^7 + (4*y^2 + 11)*z^6 + ((8*y^2 + 4)/y)*z^5 + 3*z^4] * sum(j=0,n-1, j! * [0, 0, 0, -z^6 + z^4; 1, 0, 0, ((y^2 + 1)/y)*z^5 - 2*z^4 + ((-y^2 - 1)/y)*z^3; 0, 1, 0, ((2*y^2 + 2)/y)*z^3 + z^2; 0, 0, 1, -2*z^2]^(n+j) ) * [1,0,0,0]~, 2*n,z), 0,y) / 2 ); }
%Y Cf. A114938, A137729, A137730, A137737, A137749, A258338.
%K nonn,nice
%O 1,3
%A _Hugo Pfoertner_, Jan 08 2006
%E a(4)-a(7) corrected, formula and further term provided by _Max Alekseyev_, Feb 15 2008
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