I have been working on a project trying to say that the Submonoid membership problem is decidable for different examples, one of those being the Klein Bottle.
I am now left with a problem that I am stuck on and can’t seem to get anywhere with, no matter how hard I try.
Suppose we have a word $u=y^{25}x^p$ and $w_1=y^5x^2$, $w_2=y^3x^7$.
Then it is clear that $25=5a+3b$ for some $a$,$b$ natural numbers, as the rules we get in the complete rewriting system preserve the $y$-exponent sum and so this must be a necessary condition. One set of solutions we get is $a=2$, $b=5$. However, generally, the rules do not create words that commute, and so my question is as follows:
How many different ways are there of writing $y^{25}x^p$?
For example, we could write $w_1^2w_2^5$ or $w_1w_2^5w_1$ or $w_2^2w_1w_2^2w_1w_2$, etc. All of these would give $y^{25}x^p$, but they would possibly give different values of $p$.
I believe this may rely on some combinatorics, but I have tried to find some examples of this, and I have even asked a professor at my university that teaches combinatorics but I have got nowhere!
Any help would be appreciated!