Converting between different ways of representing integer partitions

Question

I'm aware of two ways of representing integer partitions, and I'm trying to understand how to write the mapping between then in terms of mathematical notation.

Firstly, we can say that an integer parition of $n$ is a sequence $\lambda = (\lambda_1,\ldots,\lambda_k)$ such that $\lambda_i \in \mathbb{N}_{+}$, $\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_k$ and $\sum_{i = 1}^k \lambda_i = n$.

Or equivalently, we can say that an integer partition of $n$ is an $n$-tuple $\eta \in \{0,1,...,n\}^n$ such that $\sum_{i = 1}^n i \,\eta_i = n.$

I understand both definitions fine, and I've found it straightforward to write an algorithm in Mathematica to convert from one to the other. Starting with $\lambda$, I count the number of each element of size $i$ from $1$ up to $n$ and assign the value to $\eta_i$. Starting from $\eta$, I count $i$ down from $n$ to $1$ and put $\eta_i$ of the number $i$ into the sequence $\lambda.$

For the application I'm currently working with, it's more natural to use the second representation. I've recently been directed to findstat.org for a list of possible properties of integer paritions, however they use the first representation. I don't understand how to write the operations that I use programatically to map between the two representations in terms of mathematical notation, can someone please help to show me how to write down this mapping?

Answer 1

We can provide a mapping by introducing an intermediate step taking the ring of formal power series $\mathbb{Z}[[X]]$ with integer coefficients.

We consider the general case and as small example a partition of $11$: \begin{align*} 4+3+3+1=11 \end{align*}

We have a sequence $\lambda\in\mathbb{N}^k$: \begin{align*} &\lambda=\left(\lambda_i\right)_{1\leq i\leq k}&\lambda_1\geq\lambda_2\geq\cdots\geq\lambda_k&\qquad\qquad\\ &&\sum_{i=1}^k\lambda_i=n\qquad\qquad&\qquad\qquad\\ \\ &\lambda=(4,3,3,1)&4\geq 3\geq 3\geq 1\qquad&\qquad\qquad\\ &&4+3+3+1=11&\qquad\qquad \end{align*} and a sequence $\eta\in\{0,1,\ldots,n\}^n$: \begin{align*} &\eta=\left(\eta_j\right)_{1\leq j\leq n}&\sum_{j=1}^nj\eta_j=n\quad\qquad\qquad&\qquad\qquad\\ \\ &\eta=(1,0,2,1,0,0,0,0,0,0,0)&\qquad1\cdot1+2\cdot 3+1\cdot 4 = 11&\qquad\qquad\\ \end{align*}

We define a mapping $\phi$: \begin{align*} &\phi: \mathbb{N}^k\rightarrow \mathbb{Z}[[X]]\\ &\phi(\lambda)=\phi\left(\left(\lambda_i\right)_{1\leq i\leq k}\right)=\sum_{i=1}^kx^{\lambda_i}\\ \\ &\phi: \mathbb{N}^4\rightarrow \mathbb{Z}[[X]]\\ &\phi(\lambda)=\phi\left(\color{blue}{\left(4,3,3,1\right)}\right)=x^4+x^3+x^3+x^1\tag{1}\\ &\quad\qquad\qquad\qquad\qquad=x+2x^3+x^4\tag{2}\\ \end{align*}

We observe the power series representation shows the structure of $\lambda$ in the representation (1). But we also have the structure of $\eta$ when looking at the coefficients of representation (2). In the following it is convenient to use the coefficient of operator $[x^k]$ to denote the coefficient of $x^k$ in a series.

We define a mapping $\psi$: \begin{align*} &\psi:\mathbb{Z}[[X]]\rightarrow \mathbb{N}^n\\ &\psi\left(L(x)\right)=\left([x^i]L(x)\right)_{1\leq i\leq n}\\ \\ &\psi: \mathbb{Z}[[X]]\rightarrow \mathbb{N}^{11}\\ &\psi\left(x+2x^3+x^4\right)=\left([x^i]\left(x+2x^3+x^4\right)\right)_{1\leq i\leq 11}\\ &\qquad\qquad\qquad\qquad=\color{blue}{\left(1,0,2,1,0,0,0,0,0,0,0\right)}\tag{3} \end{align*}

We conclude from (1) - (3) the mapping $\phi\circ\psi$: \begin{align*} &\color{blue}{\phi\circ\psi: \mathbb{N}^k\rightarrow \mathbb{N}^n}\\ &\color{blue}{\left(\phi\circ\psi\right)\left(\left(\lambda_i\right)_{1\leq i\leq k}\right) \rightarrow \left([x^j]\sum_{i=1}^kx^{\lambda_i}\right)_{1\leq j\leq n}} \end{align*} does the job.