a formal definition of self-conjugacy

Question

C(n) := number of integer partitions of number

For instance: 43 = 7 + 5 + 5 + 4 + 4 + 4 + 3 + 2 + 2 + 2 + 2 + 2 + 1

 -------------------------------------------------
 Integer Partition of 43 with 13 parts

we can use this to make a picture( Ferrers diagrams).

Symmetry about the down-to-the-right diagonal makes the partition self-conjugate.

Question: Write a formal definition of self-conjugacy that refers only to the parts $c_1, c_2, c_3$, ... and without referring to a picture( Ferrers diagrams).

Answer 1

Let $n=c_1+c_2+\ldots+c_m$ be a partition written in non-increasing order. For $k=1,\ldots,c_1$ let

$$d_k=|\{\ell\in[m]:c_\ell\ge k\}|\;;$$

then

$$\begin{align*} \sum_{k=1}^{c_1}d_k&=\sum_{k=1}^{c_1}|\{\ell\in[m]:c_\ell\ge k\}|\\ &=\sum_{k=1}^{c_1}\sum_{\ell=1}^m[c_\ell\ge k]\\ &=\sum_{\ell=1}^m\sum_{k=1}^{c_1}[c_\ell\ge k]\\ &=\sum_{\ell=1}^mc_\ell\\ &=n\;, \end{align*}$$

where $[c_\ell\ge k]$ is an Iverson bracket.

Moreover it’s clear that $d_1\ge d_2\ge\ldots\ge d_{c_1}$, so $n=d_1+d_2+\ldots+d_{c_1}$ is a partition of $n$ written in non-increasing order, and in fact it is the partition conjugate to the original partition. Thus, the original partition is self conjugate if and only if $c_1=m$, and

$$c_k=d_k=|\{\ell\in[m]:c_\ell\ge k\}|$$

for $k=1,\ldots,m$.