I believe, and have said in earlier posts here and here on this blog, that the Equivalence Relation Theorem is the modern *pons asinorum*, the bridge which you must cross in order to become a mathematician: it is essential to be able to think about an object whose elements are equivalence classes of an equivalence relation on another object (for example, cosets of an ideal in a ring are the elements of the quotient ring).

This morning, I nearly found my own head on a pole on this bridge.

With Celia Glass and Robert Schumacher, I am looking at acyclic orientations of graphs. There is a simple formula for the number of acyclic orientations of the graph obtained from a complete graph on *n* vertices by deleting *r* pairwise disjoint edges:

.

On first inspection, this is a straightforward inclusion-exclusion formula, and should have a straightforward proof. Maybe it does, but I was unable to find one.

Any acyclic orientation of a graph *G* arises from a total order of the vertices, or in other words (if the vertex set is {1,…,*n*}) a permutation of this set. Call two permutations *equivalent* if they give the same acyclic orientation of *G*. Our job is to count equivalence classes of this relation.

Now inclusion-exclusion works fine for subsets but not so well for equivalence relations. I spent a bit of time trying to get Möbius inversion on the partition lattice to work, without success. Eventually I found a proof. First, you figure out that the size of the equivalence class containing a permutation is 2^{k}, where *k* is the number of “missing edges” mapped to consecutive positions by the permutation. Then we proceed as follows. Write down a formula for the number of permutations in which some given “missing edges” are mapped to consecutive positions. Then use inclusion-exclusion to count the permutations for which these and no more are mapped to consecutive positions. Divide by 2^{k} to get the number of equivalence classes into which these permutations fall. Sum over all subsets of the missing edges to get the total number of equivalence classes. Then a chain of formal manipulations:

- reverse the order of summation;
- gather up terms where the set of missing edges has a given size;
- notice that the Binomial Theorem can be applied;
- now gather up terms where the set in the outer summation has a given size.

The final line gives the required formula.

This description may not be clear enough for you to reconstruct. But am I missing something simple?

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About Peter Cameron

I count all the things that need to be counted.

Can’t you just count the number of total orders on the vertices with the property that non-adjacent vertices are not order-adjacent? This seems to me to give the formula by inclusion-exclusion (with the j-th term in the sum counting the number of total orders in which exactly j pairs of non-adjacent vertices are order-adjacent).

Oh… except that I’d need to take some powers of 2 into account. So perhaps the j-th term should instead count partial orders in which exactly j pairs of non-adjacent vertices are incomparable (and everything else comparable).

Nice pike, er, pic …

I’m afraid that my posts above are rather incoherent. Sorry! Here’s a better argument.

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.

Indeed, as I was walking home with a packet of exam scripts to mark, I came up with exactly your argument. Great minds think alike! But I think that the equivalence relation formulation will be essential for various extensions that I have in mind.

With respect to the total orders. For each equivalence class of permutations there should also be a corresponding set of DAGs — those whose transitive closure is the reflexive reduction of the partial order. A disjoint pair of vertices is eliminated if it can be transitively closed.

This follows from the relation

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.