# Counting linear extensions of unlabeled series parallel structures

I am interested in the problem of counting the number of linear extensions of series-parallel structures. The wikipedia article at http://en.wikipedia.org/wiki/Series-parallel_partial_order pointed me to the following formulae:

L( P ; Q ) = L(P)L(Q)

and

L( P || Q ) = ( (|P| + |Q|)! / (|P|! . |Q|!) ) . L(P) . L(Q)

However, these do not cover the cases where the duplicates are omitted.

For example, for the structure

x || x

the number of linear extensions should be 1, when duplication is omitted, however the formula above would deliver 2.

Does anyone know how the factoring of the duplicates can be integrated into the formula above?

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Normally linear extensions are counted for labelled posets. You are asking about the unlabelled case. Simply take the number of labelled linear extensions and divide by the order of the automorphism group $G$ of the poset, since in the action of $G$ on labelled linear extensions all orbits have size $|G|$. It is easy to compute $|G|$ as a product of factorials arising from parallel connections of isomorphic posets.