The Plancherel measure on the partitions of $n$

For an integer $n\ge 0$ let $Y_n$ be the set of partitions $\lambda ⊢n$ of Partitions, English diagrams, and conjugation. This set is finite: a partition of $n$ has at most $n$ parts and every part is at most $n$, so $Y_n\subseteq \{({\lambda }_1,\dots ,{\lambda }_k):k\le n,n\ge {\lambda }_1\ge \cdots \ge {\lambda }_k\ge 1\}$, a subset of the finite set $\{1,\dots ,n{\}}^{\le n}$.

For $\lambda ⊢n$ put $f^{\lambda }:={\dim }_{\mathbb{C}}{S}^{\lambda }$, the dimension of the complex Specht module, so that $f^{\lambda }$ equals the number of standard $\lambda$-tableaux (Standard polytabloids form a basis of a complex Specht module, The hook length formula); in particular $f^{\lambda }$ is a positive integer, because the standard polytabloids form a basis, and $f^{\lambda }=n!\mathrm{/}{\prod }_{x\in [\lambda ]}h(x)$ with the empty product $1$ for $\lambda =\mathrm{\varnothing }$, so that $f^{\mathrm{\varnothing }}=1$.

The Plancherel measure of order $n$ is the function $P_n(\lambda ):=\frac{(f^{\lambda }{)}^2}{n!},\lambda \in Y_n,$ with $n!$ the factorial of The factorial $n!$ and the falling factorial $n^{\underset{‾}{k}}$, defined by recursion in $\mathbb{N}$. The conventions $0!=1$ and $f^{\mathrm{\varnothing }}=1$ give $P_0(\mathrm{\varnothing })=1\mathrm{/}1=1$. For every $\lambda$ the value $P_n(\lambda )$ is a well-defined nonnegative real number, being a quotient of a nonnegative integer by the positive integer $n!$. Thus $P_n$ is a function on the finite set $Y_n$. No choice principle is used: every quantity involved is finite. Normalization, ${\sum }_{\lambda ⊢n}{P}_n(\lambda )=1$, is not part of this definition and is proved separately in The Plancherel weights sum to one.