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.

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.

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

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}] (* typo corrected by Vaclav Kotesovec, Feb 07 2026 *)

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