Ferrers and Young diagrams, conjugate partitions, self-conjugacy, and the Durfee square

Let $\lambda$ be a partition written in the page convention of Conventions for integer partitions, Ferrers diagrams, and the twelvefold-way table. For a nonempty partition, write

$\lambda =({\lambda }_1,\dots ,{\lambda }_r),{\lambda }_1\ge \cdots \ge {\lambda }_r>0.$

Also adjoin the empty partition

${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}:=()$

of $0$.

The Ferrers diagram (or Young diagram) of a nonempty partition $\lambda$ is the set of cells

$\{(i,j):1\le i\le r,1\le j\le {\lambda }_i\},$

drawn as left-justified rows, with row $i$ containing ${\lambda }_i$ cells.

The Ferrers diagram of ${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ is the empty set of cells.

The conjugate partition of a nonempty partition $\lambda$ is obtained by transposing this diagram: for $1\le j\le {\lambda }_1$, its $j$-th part is the number of rows of $\lambda$ of length at least $j$. Equivalently,

${\lambda }_j^{\mathrm{\prime }}=\mathrm{\mid }\{i:{\lambda }_i\ge j\}\mathrm{\mid }(1\le j\le {\lambda }_1)\mathrm{.}$

The conjugate of ${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ is again ${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$.

The partition is self-conjugate when ${\lambda }^{\mathrm{\prime }}=\lambda$.

The Durfee length of ${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ is $0$. For a nonempty partition $\lambda$, the Durfee length $d(\lambda )$ is the largest integer $d\ge 1$ such that ${\lambda }_d\ge d$. The Durfee square is the $d(\lambda )\times d(\lambda )$ square of cells in the upper-left corner of the Ferrers diagram, and for ${\mathrm{\varnothing }}_{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}$ it is the empty $0\times 0$ square.