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## JavaScript (Node.js), 51 bytes

```
n=[];f=(a)=>a&&a.map?(n.push(a.length),f(a[0]),n):n
```

Run f on your input, e.g. `f([[],[]])`.

---

Explanation:

```
n=[]; // make a global array to hold the results
f=(a)=> // make the function
	a&&a.map? // is a defined, and an array?
		(n.push(a.length), // add the length to the results
		f(a[0]),           // do it again on the first item
		n)                 // ...and return n
	// ...if not, return n
	:n
```