A114939 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.

0, 1, 7, 216, 10956, 803400, 83003040, 11579823360, 2080493573760, 469031859192960, 129727461014726400, 43176116371928601600, 17025803126147196057600, 7850538273249476117913600

Offset: 1

Author

Hugo Pfoertner, Jan 08 2006

Keywords

• nonn
• nice

Comments

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.

Examples

			a(2)=1 because the only valid arrangement is aBAb.
a(3)=7 because the only valid arrangements under the given conditions are: abAcBC, aBAcbC, aBcAbC, aBcACb, acAbCB, acBAbC, aCAbcB.
		

Links

• M. A. Alekseyev, Weighted de Bruijn Graphs for the Menage Problem and Its Generalizations. Lecture Notes in Computer Science 9843 (2016), 151-162. doi:10.1007/978-3-319-44543-4_12; arXiv:1510.07926 [math.CO], 2015-2016.

Crossrefs

Cf. A114938, A137729, A137730, A137737, A137749, A258338.

Programs

Mathematica

a[1] = 0;
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}];
Array[a, 14] (* Jean-François Alcover, Dec 03 2018, from PARI *)

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 ); }

Formula

See Alekseyev (2016) and the PARI code for the formula.

a(n) = A258338(n) / (4*n).

Extensions

a(4)-a(7) corrected, formula and further term provided by Max Alekseyev, Feb 15 2008