Yes, that would be ideal.

I don't seem to have access to it either, but at least one other paper (<a href="jstor.org/stable/2001573">Berstel and Reutenauer</a>) suggests that this is an open problem. Indeed I have essentially the same motivation as them for asking this question, so I suppose I should've read this paper more carefully.

Part of what I mean is that w should be invariant under any automorphisms of P. Is this necessary / sufficient?

Just a random thought. My guess is that w should be some generalization of the factors of 1/Aut(X) that appear when you compute groupoid cardinality, but I don't really know enough about this stuff to suggest what that generalization should be.

Agreed. These books are very clearly written and motivate the subject well.

My guess is you want this to go under "I wish someone had told me about this when I was younger." I'd have to agree, at least for the first few chapters.

Two quick observations: for lambda = (1, 1, ...), e_k(1, 2, ... n) is an unsigned Stirling number of the first kind, and for lambda = (k), h_k(1, 2, ... n) is a Stirling number of the second kind.

Let me suggest the following strategy, then: to any chain of subspaces in V there is associated a dual chain in V*. If one can show that strict inclusions are sent to strict inclusions, then V and V* have the same dimension.

Thanks! But is it also clear that L' is unambiguous?