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A054653 Acyclic orientations of the Hamming graph (K_3) x (K_n). 1


%S 1,6,204,19164,3733056,1288391040,712770186240,589563294888960,

%T 692610802412175360,1110893919113884631040,2357555468242103997235200,

%U 6453187419589244410090291200,22305345996450386267133758668800

%N Acyclic orientations of the Hamming graph (K_3) x (K_n).

%C This number is equivalent to the number of plans (i.e. structural solutions) of the open shop problem with n jobs and 3 machines - see problems in scheduling theory.

%D K.B. Athreya, C.R. Pranesachar, N.M. Singhi, On the number of Latin rectangles and chromatic polynomial of L(K_{r,s}), European J. Combin. 1 (1980) 9-17

%D M. Harborth, Structural analysis of shop scheduling problems, PhD thesis, Otto-von-Guericke-Univ. Magdeburg, GCA-Verlag, 1999 (in German)

%H <a href="http://www.math.uni-magdeburg.de/publ/diss/sources/harborth_diss.ps.gz">Structural analysis of shop scheduling problems (PhD thesis in German with English abstract)</a>

%F (-1)^n*(z!n!/(((z-n)!)^3)*Sum[If[a+b+c*n, (-1)^b*2^c*((z-n+a)!)^2/(a!c!)*Bino mial[3z-3n+3a+b+2, b], 0], {c, 0, n}, {b, 0, n}, {a, 0, n}]) with z=-1

%t Table[n!*Evaluate[(-1)^n*FunctionExpand[z!n!/(((z-n)!)^3)*Sum[If[a+b+c*n, (-1 )^b*2^c*((z-n+a)!)^2/(a!c!)*Binomial[3z-3n+3a+b+2, b], 0], {c, 0, n}, {b, 0, n}, {a, 0, n}]]/.z->-1]/n!, {n, 0, 15}]

%Y Cf. A054652, A053870, A054583.

%K nonn,easy

%O 0,2

%A M. Harborth (Martin.Harborth(AT)vt.siemens.de)

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Last modified August 13 19:30 EDT 2020. Contains 336451 sequences. (Running on oeis4.)