I’m working with integer partitions $ eta$ of $w$, with parts of size at most $n$, so that $eta$ satisfies $sum_{i = 1}^{n} i, eta_i = w$.

I have a set of functions $c_{n,w}(eta)$ on integer partitions in computer algebra in Mathematica for a given $n$ and $w$, and I want to try to reconstruct the form of the functions for general $n$ and $w$. The functions are valued on rational numbers, ie. $c_{n,w}(eta) in mathbb{Q}$.

For a given $n$ and $w$ some of the $c_{n,w}(eta)$ are zero, and I’m trying to work out the zeros first.

For example, for some $n$ and $w$ I have a $mathbb{Z}_2$ symmetry which forces the $c_{n,w}$ to be zero, depending on whether $sum_{i = 1}^{n} eta_i$ is even or odd. So I know that in those cases, I have that

$$

c_{n,w}(eta) sim 1+(-1)^{sum_{i = 1}^{n} eta_i}.$$

or $$

c_{n,w}(eta) sim 1-(-1)^{sum_{i = 1}^{n} eta_i}.$$

I also have some other cases with $c_{n,w}(eta) = 0$, but in those cases it’s not so clear to me to see what the pattern is. I’m looking for suggestions about natural functions $c_{n,w}(eta)$ which have zeros, additional to the ones I gave above.

I’ve also thought of $$c_{n,w}(eta) = sum_{i = 1}^{n} (-1)^i eta_i,$$ and $$c_{n,w}(eta) = 1-(-1)^{f(eta)}$$ for some different functions $f$, maybe such as $f(eta) = sum_{i = 1}^{n} i^a (eta_i)^b$ for $a,b in mathbb{N}.$ Neither of those seem to work for my current problem. I also considered taking the sums up to some $m < n$ in the functional forms I gave above so that I consider only some of the parts of the partition, but that didn’t seem to help either, and I’m not really sure how natural it would be to do that either.

I’m looking for functions constructed of only addition, subtraction, multiplication and integer powers of the componenets of $eta$. (So combinatorial functions like factorial or binomial coefficients would also be fine). In the end I sum the $c_{n,w}(eta)$ over $eta$ and multiply by $prod_{i=1}^{n}a_i^{eta_i}$ to construct a polynomial with a well-defined scaling weight $w$, if that helps at all in suggesting relevant functions.

As an example, I was looking at a case with $w = 9$, $n=6$. In that case, I have all of the $c_{n,w}(eta) = 0$ when $sum_{i = 1}^{n} eta_i$ is even, so I know I need the relevant function above as a factor (I also have the $mathbb{Z}_2$ symmetry in this case). I was surprised to find that there are also two additional zeros, when $eta = (3,0,2,0,0,0)$ and when $eta = (1,1,0,0,0,1)$, so I think I need to multiply in an additional factor which is zero on these two $eta$, non-zero on $eta$ where $sum_{i = 1}^{n} eta_i$ is odd, and which I don’t have any information about when $sum_{i = 1}^{n} eta_i$ is even. Does this give enough information for someone to suggest a relevant function in this particular situation?

I appreciate that this is unlikely to fully specify the function that I’m interested in. My philosophy is to try some of the more simple possible such functions, and see if they match against the form I need in Mathematica. Hopefully this way I’ll be able to reconstruct the full functional form that I’m looking for, and this should make it easier to go back and prove that form later on. (Also I enjoy doing it this way!)

Any references which may be relevant for me to read about this would also be very welcome. I have some basic understanding of integer partitions, but I’ve not worked with them before.