Partitions, English diagrams, and conjugation

Partitions. Let $n\ge 0$ be an integer. A partition of $n$ is a finite weakly decreasing sequence $\lambda =({\lambda }_1,\dots ,{\lambda }_k)$ of positive integers with ${\lambda }_1+\cdots +{\lambda }_k=n$. Its entries are the parts of $\lambda$ and $k$ is the number of parts. For $n=0$ the only partition is the empty partition $\mathrm{\varnothing }$, the empty sequence with $k=0$; for $n\ge 1$ one has ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_k\ge 1$ and $k\le n$. We write $\lambda ⊢n$. Trailing zeros are not parts: a finite sequence of nonnegative integers ending in $0$ is not a partition, so the number of parts of $\lambda$ is determined by $\lambda$ and $({\lambda }_1,\dots ,{\lambda }_k)$ is a different data type from $({\lambda }_1,\dots ,{\lambda }_k,0)$.

English Young diagrams. The (English) Young diagram of a nonempty $\lambda$ is $[\lambda ]:=\{(i,j):1\le i\le k,1\le j\le {\lambda }_i\}\subseteq \mathbb{N}\times \mathbb{N},$ where $i$ numbers the rows downward and $j$ numbers the columns rightward. An element of $[\lambda ]$ is a node, or box, of the diagram. Row $i$ of $[\lambda ]$ carries ${\lambda }_i$ nodes, and column $j$ carries ${\lambda }_j^{\mathrm{\prime }}:=\mathrm{\#}\{i:{\lambda }_i\ge j\}$ nodes, a quantity that is $0$ beyond the last column (and for every $j\ge 1$ when $\lambda =\mathrm{\varnothing }$). Since a partition is weakly decreasing, $[\lambda ]$ determines $\lambda$: if $[\mu ]=[\lambda ]$, then row $i$ of $[\lambda ]$ is row $i$ of $[\mu ]$, so ${\mu }_i={\lambda }_i$ for every $i$ and $\mu =\lambda$. In particular $[\mathrm{\varnothing }]=\mathrm{\varnothing }$, the empty diagram.

Conjugation. The conjugate partition of a nonempty $\lambda$ is ${\lambda }^{\mathrm{\prime }}:=({\lambda }_1^{\mathrm{\prime }},{\lambda }_2^{\mathrm{\prime }},\dots ,{\lambda }_{{\lambda }_1}^{\mathrm{\prime }}),$ the sequence of column heights of $[\lambda ]$, with ${\mathrm{\varnothing }}^{\mathrm{\prime }}:=\mathrm{\varnothing }$ for the empty partition. This is again a partition of $n$, and $[{\lambda }^{\mathrm{\prime }}]$ is the transpose of $[\lambda ]$: for $1\le i\le {\lambda }_1$ and $1\le j\le k$, $(i,j)\in [{\lambda }^{\mathrm{\prime }}]⟺j\le {\lambda }_i^{\mathrm{\prime }}⟺{\lambda }_j\ge i⟺(j,i)\in [\lambda ]\mathrm{.}$ Outside these bounds neither diagram contains the corresponding node; for the empty partition both diagrams are empty. Conjugation is an involution, ${\lambda }^{\mathrm{\prime }\mathrm{\prime }}=\lambda$, because ${\lambda }_i^{\mathrm{\prime }\mathrm{\prime }}=\mathrm{\#}\{j:{\lambda }_j^{\mathrm{\prime }}\ge i\}$ counts the columns of $[\lambda ]$ of height at least $i$, that is, the columns $j$ with at least $i$ rows of length $\ge j$, which by weak decrease is exactly the set of $j\le {\lambda }_i$. Thus $\lambda \mapsto {\lambda }^{\mathrm{\prime }}$ is a bijection on the partitions of $n$, exchanging the number of parts with the largest part for nonempty partitions.

Size zero. The symmetric group of $\{1,\dots ,n\}$ is $S_n:=\mathrm{Sym}(\{1,\dots ,n\})$ (The symmetric group $\mathrm{Sym}(X)$: the bijections of a set $X$ under composition) for every $n\ge 0$, and we fix $S_0:=\mathrm{Sym}(\mathrm{\varnothing })=\{1\},$ the trivial group; for $n=0$ the set $\{1,\dots ,n\}$ is empty and the group acts trivially on every size-zero object below. This is the convention used whenever the constructions of this page are read at $n=0$.

• Indexing conventions. Rows are numbered from top to bottom and columns from left to right, and the row lengths are weakly decreasing, so the diagram is left-aligned and top-aligned inside its bounding rectangle of ${\lambda }_1$ columns and $k$ rows. This is the English convention; the French convention (rows weakly increasing downward) is not used here.

• Conjugation transposes the diagram. The identity $[{\lambda }^{\mathrm{\prime }}]=[\lambda {]}^T$ says that summing over the parts of ${\lambda }^{\mathrm{\prime }}$ is summing over the columns of $[\lambda ]$: double counting the nodes of $[\lambda ]$ by rows gives $n$ and by columns gives ${\lambda }_1^{\mathrm{\prime }}+\cdots +{\lambda }_{{\lambda }_1}^{\mathrm{\prime }}$, so ${\lambda }^{\mathrm{\prime }}$ is a partition of $n$. A partition equal to its conjugate is self-conjugate; the diagonal nodes of $[\lambda ]$ are the fixed points of the transpose.