Is there a standard notation to designate the set of all partitions? Let $n$ be a positive integer. We say $\lambda (\in \mathbb{N}^n)$ is a partition of $n$ iff $\lambda_1\geq \cdots \geq \lambda_n \geq 0$ and $\lambda_1+\cdots +\lambda_k=n$. Is there a standard notation to designate the set of all partitions of $n$?

We write:

$$\mathbf\lambda\vdash n$$

to indicate
$$\mathbf\lambda=\\{\lambda_1,\dots,\lambda_k\\}, \;\;\lambda_i\gt0, \lambda_i\in\mathbb{N} $$ and $$\sum_\limits{i=1}^k \lambda_i=n$$

You have $k=n$ and $\lambda_i\ge 0$, but this shouldn't matter.

Capital $\lambda$ is $\Lambda$, so you could use this.