Let $\lambda = (\lambda_1, \lambda_2, \ldots, \lambda_k)$ be a partition of $n $, $\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k > 0$.
Is there accepted notation (e.g. in the literature) for the following product-type things?
A simple product of parts: $$ \lambda_1 \lambda_2 \cdots \lambda_k $$ -- I suspect that $\lambda!$ is the most natural notation here.
A product over partial sums (adding next-smallest part each time): $$ \lambda_1 (\lambda_1 + \lambda_2) \cdots (\lambda_1 + \cdots + \lambda_k) $$
A product over 'reverse' partial sums (dropping next-largest part each time): $$ (\lambda_1 +\lambda_2 + \cdots + \lambda_k)(\lambda_2 + \cdots + \lambda_k) \cdots \lambda_k $$
Any references would be much appreciated.