In article <email@example.com>, <firstname.lastname@example.org> wrote:
>> Maybe you meant Motzkin *trees*.
>No, I did mean paths...
Okay, whoops - I'd forgotten all about that!
>But of course there are loads of combinatorial interpretations, >including the trees you mention (which was actually the way I found the >link between Leinster and Fiore's construction and the Motzkin >numbers.)
Sorry, I even forgot your role in this. It winds up that trees are more useful than paths, because if we have any polynomial fixed-point equation with natural number coefficients, like
X = 349X^7 + 3X^4 + 2
we can interpret it as defining a set of "colored trees". To do this, we interpret the equal sign as an isomorphism.
For example, the above equation defines the set X of planar, rooted, finite trees where the root is colored in one of 2 ways, and each node either has 4 branches coming out of it (in which case this node is colored in one of 3 ways) or 7 branches coming out of it (in which case it's colored in one of 349 ways).
By the work of Schanuel, Gates, Fiore and Leinster, it makes sense to assign a cardinality to this set which is a root of the above polynomial.
A good example is the "Golden Object" discovered by Robin Houston:
The golden number is usually defined by the equation
G^2 = G + 1 (1)
which is not of the fixed-point form described above. However, Houston made the substitution
G = H + 2 (2)
and notes that
H = H^2 + 4H + 1 (3)
is of the desired fixed-point form. So, H is a set of colored trees and G is the disjoint union of H and the 2-element set.
So, we can think of G as a set of "colored trees" of a slightly more general type. (Houston gives another interpretation.)
Anyway, using this sort of trick we might able to cook up a category of colored trees containing the category of finite sets, but also objects having cardinality equal to any algebraic number! This would be a candidate for a categorified version of the algebraic closure of Q.