Let $n$ be a natural number. Consider a set $\Lambda_n$ of partitions of $n$ into a sum of natural numbers, like $n = \lambda_1 + \cdots +\lambda_k$ (A set of small lambdas representing a partition is considered to be unordered).

There is a natural partial order on $\Lambda_n$. Namely, we say that $\lambda \ge \lambda'$ if $\lambda$ is a refinement of $\lambda'$. For example for $n=3$ the partition $1+1+1$ is a refinement of the partition $2+1$, so $1+1+1 \ge 2+1$.

From the other hand we know, that elements of $\Lambda_n$ are in 1-to-1 correspondence with a set of elements of a certain basis of the centre of a complex group algebra of the symmetric group $S_n$. This centre has a nice structure of a commutative Frobenius algebra.

Is there a way to find out whether two partitions, represented as basis elements of the above mentioned algebra are comparable?

**Edit:** May be I was not very precise. I mean the following. Consider, we have a *concrete* commutative Frobenius algebra with a *concrete* basis. Now we want to describe some partial order on this basis in the inner terms of the algebra (product, trace) only. That problem arose in the following context. I wanted 1) to describe all possible partitions majorated by given 3 partitions, 2) to introduce an analogue of this partial order on an arbitrary finite group.

Why to use algebra formalism ? In fact I am working on the problem of counting certain combinatorial objects with weights, that can be computed as traces of monomials in the algebra. So, it is just a wild hope that this problems can be related.