How to denote sum over partitions?

Question

How does one denote a sum of partitions of an integer when writing an article? For example, I have a formula regarding an integer-valued variable (say q=5), and I need to write an expression of the form:

$$\sum_{p\in P(q)}$$

where $P(q)$ is the set of all partitions of $q$ (for example $\{1+1+1+1+1,1+1+1+2,1+2+2,1+1+3,2+3,4+1,5\}$) and $p$ is an element of the set, a particular partition. Then inside the summand, I need to refer to both the size of the partition (which I suppose I could denote $|p|$) and the contents of the partition. Is there any standard way of doing this?

Answer 1

In general, for sums over structures rather than simple integer ranges you have a few options:

1. Use notation indicating the structure. Here, for example, you could follow George Andrews in using $\lambda \vdash n$ to denote that $\lambda$ is a partition of $n$, and write

   $$\sum_{\lambda\, \vdash \, n}f(\lambda_1, \ldots, \lambda_n, n)$$

   NB I've used \, for spacing in the subscript.

2. As a variant on the previous point, express the constraint algebraically, as suggested in the comments:

   $$\sum_{\lambda_1+\ldots+\lambda_k = n} f(\lambda_1,\ldots,\lambda_k,n)$$

3. Use notation indicating the bound variables and express the constraint in text separately. E.g.

   $$\sum_{\lambda_1,\,\ldots,\,\lambda_n} f(\lambda_1,\ldots,\lambda_n,n)$$ where the sum is over the partitions of $n$

Since the purpose of mathematical writing is communication, I would favour a combination of 1 and 3 (where possible) for most audiences:

$$\sum_{\lambda\, \vdash \, n}f(\lambda_1, \ldots, \lambda_n, n)$$ where the sum is over the partitions of $n$