Conjugate integer partition identity

Question

Let $\lambda:n = n_1 + \dots + n_k, \ \mu:m = m_1 + \dots + m_l$ be integer partitions of $n, m$ respectively. Now, define two operations $\circ, \bullet$ as follows:$\lambda \circ \mu: (n + m) = (n_1 + m_1) + (n_2 + m_2) + \dots \ $ and $\lambda \bullet \mu$ are the parts of $\lambda$ and $\mu$ together. I am asked to prove that $$(\lambda \circ \mu)^* = \lambda^* \bullet \mu^*$$ where $\gamma^*$ denotes the conjugate partition of a partition $\gamma$. I am not totally sure how to prove this; I thought to use $D(\lambda)$, the diagram of a partition lambda, but have not gotten anywhere.

Answer 1

This is part of Exercise 2.9.17 (a) in Darij Grinberg and Victor Reiner, Hopf Algebras in Combinatorics, version of 19 April 2020 (also available as arXiv:1409.8356v6). (Should the numbering of exercises change, you can find this exercise by searching for the words "we define two new partitions". Once you found it, you can then find its solution using the table of contents.) Note that your $\lambda \circ \mu$ is called $\lambda + \mu$ in these notes, and your $\lambda \bullet \mu$ is called $\lambda \sqcup \mu$, whereas your $\lambda^*$ is called $\lambda^t$.

There is only one trick to the whole proof: First show the "dual" identity $\left(\lambda \bullet \mu\right)^\ast = \lambda^\ast \circ \mu^\ast$ (or, in my notations, $\left(\lambda \sqcup \mu\right)^t = \lambda^t + \mu^t$), which easily follows from the definition of conjugate partitions ($\left(\nu^t\right)_i = \left(\text{the number of } j \text{ such that } \nu_j \geq i \right)$). Now, apply it to $\lambda^\ast$ and $\mu^\ast$ instead of $\lambda$ and $\mu$, and recall that conjugating partitions is an involution.