How can I define a function to index a ragged matrix?

Consider the $$m \times n$$ matrix $$\mathbf{A} = \begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1n} \\ A_{21} & A_{22} & \cdots & A_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ A_{m1} & A_{m2} & \cdots & A_{mn} \end{bmatrix}$$ such that $$\mathbf{A} \in \mathbb{R}^{m\times n}$$. I can define a function $$f : \{1,2,\dots,m\} \times \{1,2,\dots,n\} \mapsto \mathbb{R}$$ that returns the element in the $$i^{\text{th}}$$ row and $$j^{\text{th}}$$ column of $$\mathbf{A}$$ as $$f(i,j) = A_{ij}$$ Now suppose that I define the ragged matrix $$\mathbf{A} = \begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1n_{1}} \\ A_{21} & A_{22} & \cdots & A_{2n_{2}} \\ \vdots & \vdots & \ddots & \vdots \\ A_{m1} & A_{m2} & \cdots & A_{mn_{K}} \end{bmatrix}$$ where $$n_1,n_2,\dots,n_K \in \mathbb{N}$$ and $$n_i \neq n_j \ \forall i \neq j$$. Is there a similar way to define a function $$f$$ to index this ragged matrix?

My initial attempt at this is as follows: first, define a function $$M : \{1,2,\dots,m\} \mapsto \mathbb N$$ that returns the number of elements in the $$i^{\text{th}}$$ row. Next, define $$g : \{1,2,\dots,m\} \mapsto \mathbb{R}^{M(i)}$$ that returns the elements in the $$i^{\text{th}}$$ row. I am stuck here because I am aware that the notation $$g : \{1,2,\dots,m\} \mapsto \mathbb{R}^{M(i)}$$ doesn't really make sense. I think the function $$f$$ will consist of some composition of $$g$$ with $$M$$, but I am not sure.

$$I = \{ (i,j) : i \in \{1,2,\ldots,m\} \land j \in \{1,2,\ldots,n_i\} \}.$$
Then define $$f : I \to \mathbb R$$ by $$f(i,j) = A_{ij}.$$
Actually making sense of the definition of the "ragged matrix" may be even harder, and representing the matrix harder still. Note how in the question the elements $$A_{1n_1},$$ $$A_{1n_2},$$ and $$A_{1n_m}$$ all ended up vertically aligned despite the assertion that $$n_1 \neq n_2 \neq n_m \neq n_1.$$ Can you write out these rows of the "matrix" in such a way that it makes sense both for the case $$n_1 < n_2$$ and the case $$n_1 > n_2$$?