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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.

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    $\begingroup$ Not that I know, but I wouldn't use $\lambda !$. Unless you explicitly define the factorial of a sequence that way, I would hesitate to say that, especially given that the factorial has less-than-standard notation as functions go. $\endgroup$ – Carl Schildkraut Sep 14 '16 at 3:40
  • $\begingroup$ I think the more straightforward notation is $\prod_k \lambda_k$ $\endgroup$ – Masacroso Sep 14 '16 at 6:13

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