where is the number of acyclic orientations,

is an edge of , and and

are got by deleting and contracting the edge . This is (iii’)

in Theorem 1.2 of Richard Stanley’s classic paper on acyclic

orientations. ]]>

Let {(a_i,b_i) : i = 1,..,r} be the pairs of non-adjacent vertices of G; we consider these as ordered pairs (which is equivalent to choosing an orientation of the complement of G). Now let us look at the number N of labellings of G with the numbers 1,..,n, such that for each such pair (a_i,b_i), the label attached to a_i is never greater by exactly 1 than the label attached to b_i. Then N is the number of acyclic orientations. (Each such labelling specifies a total order, and only one order from each equivalence class is picked out.)

Now let J be a subset of {1,..,r}, and let X_J be the set of labellings of the vertices such that the label of a_i is greater by 1 than the label of b_i for all i in J. Then |X_J| = (n-j)! where j=|J|. Now inclusion-exclusion gives your formula.

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