A054653 Number of acyclic orientations of the Hamming graph (K_3) x (K_n).

1, 6, 204, 19164, 3733056, 1288391040, 712770186240, 589563294888960, 692610802412175360, 1110893919113884631040, 2357555468242103997235200, 6453187419589244410090291200, 22305345996450386267133758668800

Offset

0,2

Comments

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.

References

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

Links

Table of n, a(n) for n=0..12.

M. Harborth, Structural analysis of shop scheduling problems, (PhD thesis in German with English abstract).

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

Formula

a(n) = (-1)^n*(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}]) with z=-1.

Mathematica

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}]

Crossrefs

Cf. A054652, A053870, A054583.

Sequence in context: A115491 A082405 A183595 * A061540 A173370 A159307

Adjacent sequences: A054650 A054651 A054652 * A054654 A054655 A054656

Keyword

nonn,easy

Author

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

Status

approved