# Lemmas involving two partitions of integers

Question: Does anyone know a reference to the following lemmas involving two partitions? (The proofs are not hard, and may well be previously recorded, but where?) First some notation. Let $r$ be a positive integer. A partition $\lambda=(\lambda_1,\lambda_2,\dots)$ of $r$, written $\lambda\vdash r$, has $\lambda_1\geqslant\lambda_2\geqslant\cdots\geqslant0$, $\lambda_i\in\mathbb{Z}$, and $\sum_{i\geqslant0}\lambda_i=r$. Interchanging rows and columns of the Young diagram of $\lambda$ gives the dual partition $\lambda'=(\lambda'_1,\lambda'_2,\dots)$. For example, $\lambda=(4,3,1,0,\dots)$ has dual $\lambda'=(3,2,2,1,0,\dots)$ and $(\lambda')'=\lambda$.

Lemma 1. If $\lambda\vdash r$ and $\mu\vdash s$, then $\sum_{i\geqslant1}\sum_{j\geqslant1}\min(\lambda_i,\mu_j)=\sum_{i\geqslant1}\lambda'_i\mu'_i$. Alternatively, $$\sum_{i=1}^{\lambda'_1}\sum_{j=1}^{\mu'_1}\min(\lambda_i,\mu_j)=\sum_{i=1}^{\min(\lambda_1,\mu_1)}\lambda'_i\mu'_i.$$

Let $J_r$ be an $r\times r$ Jordan block, and let $J_\lambda$ be the block diagonal matrix $\bigoplus_{i\geqslant1}J_{\lambda_i}$.

Lemma 2. Suppose $\lambda,\mu\vdash r$. Then $J_\lambda J_\mu=J_\mu J_\lambda$ if and only if $\mu=\lambda^{(i)}$ for a unique $i\in\{0,1,\dots,\lambda'_2\}$ where $\lambda^{(i)}:=(\lambda_1,\dots,\lambda_i,1,\dots,1,0,\dots)$.

Edit: For a proof of Lemma 1 see [1, Lemma 3.2] which should appear in 2018.

[1]: https://arxiv.org/pdf/1711.04104.pdf to appear in Ars Math. Contemp.