REF: EA value_counts -> _value_counts
Instead of returning a Series, return a tuple with the index and values to be passed to Series.

Where possible I've changed the methods to use `_values_for_factorized` in the hopes of converging on a base class implementation.  This is proving elusive, suggestions welcome.  cc @TomAugspurger @jorisvandenbossche 

xref #22843, #23074.
@simonjayhawkins suggestions for making mypy happy here? does this make possible EA.value_counts? > does this make possible EA.value_counts?

Still trying to figure out what the base class implementation looks like.  I think it involves _values_for_factorized, but that seems to have inconsistent copy/view semantics. > @simonjayhawkins suggestions for making mypy happy here?

adding
```
    def __init__(self, values, freq=None, dtype=None, copy=False):
        pass
```

to DatetimeLikeArrayMixin works. may want to make DatetimeLikeArrayMixin an abstract base class instead of mixin. The docstring states "Shared Base/Mixin class for DatetimeArray, TimedeltaArray, PeriodArray". > I think it involves _values_for_factorized, but that seems to have inconsistent copy/view semantics.

Can you clarify what you mean with this?

---

Personally, I would even go for removing `(_)value_counts` entirely from the ExtensionArray (and move the logic to `algorithms.value_counts`. But that is of course only possible if it can be implemented generically based on `_values_for_factorize` (which right now is not yet possible?)
 >> I think it involves _values_for_factorized, but that seems to have inconsistent copy/view semantics.

> Can you clarify what you mean with this?

Suppose we just want to go through the existing implementations and substitute in _values_for_factorize wherever feasible:

- DTA/TDA/PA use i8values, which matches _values_for_factorize()[0]
- BooleanArray/IntegerArray calls goes through the Index constructor, so not immediately obvious
- Categorical uses np.arange
- IntervallArray is equivalent to using _values_for_factorize.
- StringArray uses self._ndarray, which is similar to _values_for_factorize, but _values_for_factorize makes a copy and then masks

_values_for_factorize copy/view behavior:

- BooleanArray/IntegerArray does an "astype" followed by masking, so we get a copy
- Categorical does an "astype", so we get a copy
- DTA/TDA/PA -> view
- PandasArray -> view
- StringArray -> copy+mask
- Sparse -> calls np.asarray, copy vs view varies.
 And how is the copy/view semantic important for a value_counts implementation? I'd also be happy to see ExtensionArray.value_counts go away if possible, in favor of `pd.value_counts(values)`. > I'd also be happy to see ExtensionArray.value_counts go away if possible, in favor of `pd.value_counts(values)`.

really? that is a giant step backwards for the api
virtually everything is a method, so now you want top level functions? where did this come from > And how is the copy/view semantic important for a value_counts implementation?

If the existing implementations dont make a copy and the _values_for_factorize-based ones do, thats a performance hit that im not eager to take.  Since I don't fully understand the distinction between _values_for_factorize vs _values_for_argsort, I find it worth holding off on implementing a general version until I find the right attribute. > really? that is a giant step backwards for the api

The *method* that users can use is `pd.Series.value_counts`. For that to work, the EA does not need to have a `value_counts` method. >  Since I don't fully understand the distinction between _values_for_factorize vs _values_for_argsort, I find it worth holding off on implementing a general version until I find the right attribute.

Since `value_counts` is doing a factorization (with counting included), it's certainly `_values_for_factorize` we should be using for this, and not `_values_for_argsort`, I would think. 

Anyway, regardless of the "possible general implementation" discussion, I think this PR is a step forward, so fine with first focusing on the content right now in this PR. Though in its current state, the PR breaks API right? Won't`Categorical.value_counts` raise an AttributeError right now? what is the reason we are so against an EA.value_counts() method? We have one now, and IIRC you didn't want this on the base class, though this PR actually makes that a very simpl impl  now. > what is the reason we are so against an EA.value_counts() method? 

For me its about dependency structure.  I don't want our EAs depending on Series/DataFrame/Index (and want to change the handful of places that they currently do) > Though in its current state, the PR breaks API right? Won'tCategorical.value_counts raise an AttributeError right now?

Yes.  Easy to reinstate i guess > > what is the reason we are so against an EA.value_counts() method?
>
> For me its about dependency structure. I don't want our EAs depending on Series/DataFrame/Index (and want to change the handful of places that they currently do)

Yes, for me the same. And next to the actual code dependency structure, there is also the mental model: for me, Array is something independent of Series/Index, while Series/Index holds arrays. It's biizarre to me for an array method to return a Series. 

Also, I doubt anyone is actually using this. Users have `Series.value_counts` available for this. @jbrockmendel can you provide a depretion for ``.value_counts()``? closing to clear the queue, will revisit after indexing fixes are done @jorisvandenbossche I'm now trying to implement more ExtensionIndex methods in terms of the backing EA, am having trouble similar to the trouble with `value_counts`.  Did you ever try to implement this?