OFFSET
0,2
COMMENTS
The fork-join structure is a modeling structure, commonly seen for example in parallel computing, usually represented as a DAG (or poset). It has an initial "fork" vertex that spawns a number of m independent children vertices (the width) whose output edges are connected to a final "join" vertex. More generally, we can have a number n of these DAGs, each one with m+2 vertices.
When the width is 4 (i.e., m=4), these fork-join DAGs can be depicted as follows (we omit the first column for n=0 because the graph is empty in this case):
n | 1 | 2 | 3
------------------------------------------------------------
| o | o o | o o o
| /| |\ | /| |\ /| |\ | /| |\ /| |\ /| |\
| o o o o | o o o o o o o o | o o o o o o o o o o o o
| \| |/ | \| |/ \| |/ | \| |/ \| |/ \| |/
| o | o o | o o o
LINKS
Wikipedia, Fork-join model.
FORMULA
a(n) = (6n)!/30^n.
EXAMPLE
a(1) = 24 is the number of linear extensions of one fork-join DAG of width 4.
MATHEMATICA
a[n_] := (6n)!/30^n
Table[a[n], {n, 0, 8}]
CROSSREFS
KEYWORD
nonn
AUTHOR
José E. Solsona, Apr 24 2023
STATUS
approved