A problem I'm investigating concerns a combinatorial class in which the 'atoms' have irrational sizes. It seems likely that this is something that has been considered before, but I haven't been able to find it in the literature.

As a simple example, consider the bivariate generating function $f(x,y)=(1-x-y)^{-1}$ for sequences of two different types of objects ($X$ and $Y$). Now suppose that the size of an $X$ object is $a$ and the size of a $Y$ object is $b$. If $a$ and $b$ are integers, then $f(z^a,z^b)$ enumerates sequences of a given size, and if $a/b$ is rational, we can get the enumeration by multiplying by the least common denominator.

What is the theory when $a/b$ is irrational? Consider $g(z)=f(z,z^\sqrt{2})$. This function is no longer a formal power series in $z$. However, singularity analysis of $g(z)$ seems to deliver the 'right' answer for the 'growth rate', as might be expected by continuity. On the other hand, the concept of growth rate in this case has to be modified a bit, to be something like $\lim_{t \to \infty} {S_{[t,t+1]}}^{1/t}$, where $S_I$ is the number of sequences whose sizes lie in the interval $I$.

Where can I find this kind of thing discussed?


I received the following from Mireille Bousquet-Mélou:

I have in fact never seen such series in enumeration... when irrational powers are involved, even the notion of such series has to be handled carefully (I think that there could be accumulation points of exponents. This is not the case in your example though).

Such objects are called (I think) transseries. I quick google search leads for instance to this paper (for beginners...): http://arxiv.org/pdf/0801.4877v5.pdf

There is also this book: http://arxiv.org/pdf/1509.02588v1.pdf

I do not promise these references to be of some help, not even relevant!


Some related research: Scott Garrabrant and Igor Pak consider tilings with irrational side lengths in this article.

An open problem here is to decide if the Catalan numbers can be generated from such tilings described in the article.


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