BUG: all-na corner case for str.cat
State: closed

- [x] closes #24044, split off from #23167
- [x] tests added / passed
- [x] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [x] whatsnew entry

Labels:
name: Bug
name: Reshaping
description: Concat, Merge/Join, Stack/Unstack, Explode
name: Strings
description: String extension data type and string data

Comments:
Hello @h-vetinari! Thanks for submitting the PR.

 - There are no PEP8 issues in the file [`pandas/core/strings.py`](https://github.com/h-vetinari/pandas/blob/f2c4943034ae271f6d5fbc435244189460c915c1/pandas/core/strings.py) !

 - There are no PEP8 issues in the file [`pandas/tests/test_strings.py`](https://github.com/h-vetinari/pandas/blob/f2c4943034ae271f6d5fbc435244189460c915c1/pandas/tests/test_strings.py) !


# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=h1) Report
> Merging [#24045](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/b45eb265dd88648fb02ebb0fdcfb168364ab664e?src=pr&el=desc) will **increase** coverage by `49.88%`.
> The diff coverage is `100%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/24045/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=tree)

```diff
@@             Coverage Diff             @@
##           master   #24045       +/-   ##
===========================================
+ Coverage   42.46%   92.34%   +49.88%     
===========================================
  Files         161      161               
  Lines       51557    51557               
===========================================
+ Hits        21892    47612    +25720     
+ Misses      29665     3945    -25720
```

| Flag | Coverage Δ | |
|---|---|---|
| #multiple | `90.74% <100%> (?)` | |
| #single | `42.46% <0%> (ø)` | :arrow_up: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/strings.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvc3RyaW5ncy5weQ==) | `98.58% <100%> (+65.58%)` | :arrow_up: |
| [pandas/core/computation/pytables.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvY29tcHV0YXRpb24vcHl0YWJsZXMucHk=) | `92.37% <0%> (+0.3%)` | :arrow_up: |
| [pandas/io/pytables.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2lvL3B5dGFibGVzLnB5) | `92.3% <0%> (+0.92%)` | :arrow_up: |
| [pandas/util/\_test\_decorators.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL3V0aWwvX3Rlc3RfZGVjb3JhdG9ycy5weQ==) | `93.24% <0%> (+4.05%)` | :arrow_up: |
| [pandas/compat/\_\_init\_\_.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvbXBhdC9fX2luaXRfXy5weQ==) | `58.36% <0%> (+8.17%)` | :arrow_up: |
| [pandas/core/config\_init.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvY29uZmlnX2luaXQucHk=) | `99.24% <0%> (+9.84%)` | :arrow_up: |
| [pandas/core/reshape/util.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvcmVzaGFwZS91dGlsLnB5) | `100% <0%> (+11.53%)` | :arrow_up: |
| [pandas/compat/numpy/\_\_init\_\_.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvbXBhdC9udW1weS9fX2luaXRfXy5weQ==) | `92.85% <0%> (+14.28%)` | :arrow_up: |
| [pandas/core/computation/common.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvY29tcHV0YXRpb24vY29tbW9uLnB5) | `85.71% <0%> (+14.28%)` | :arrow_up: |
| [pandas/core/api.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvYXBpLnB5) | `100% <0%> (+14.81%)` | :arrow_up: |
| ... and [120 more](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree-more) | |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=footer). Last update [b45eb26...f2c4943](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

# [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=h1) Report
> Merging [#24045](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=desc) into [master](https://codecov.io/gh/pandas-dev/pandas/commit/6dd130a08189ce9cfebf1a49f4ba98a9721b49a8?src=pr&el=desc) will **not change** coverage.
> The diff coverage is `0%`.

[![Impacted file tree graph](https://codecov.io/gh/pandas-dev/pandas/pull/24045/graphs/tree.svg?width=650&token=eZ4WkYLtcO&height=150&src=pr)](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=tree)

```diff
@@           Coverage Diff           @@
##           master   #24045   +/-   ##
=======================================
  Coverage   42.45%   42.45%           
=======================================
  Files         161      161           
  Lines       51561    51561           
=======================================
  Hits        21888    21888           
  Misses      29673    29673
```

| Flag | Coverage Δ | |
|---|---|---|
| #single | `42.45% <0%> (ø)` | :arrow_up: |

| [Impacted Files](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=tree) | Coverage Δ | |
|---|---|---|
| [pandas/core/strings.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvc3RyaW5ncy5weQ==) | `33% <0%> (ø)` | :arrow_up: |
| [pandas/core/frame.py](https://codecov.io/gh/pandas-dev/pandas/pull/24045/diff?src=pr&el=tree#diff-cGFuZGFzL2NvcmUvZnJhbWUucHk=) | `38.7% <0%> (ø)` | :arrow_up: |

------

[Continue to review full report at Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=continue).
> **Legend** - [Click here to learn more](https://docs.codecov.io/docs/codecov-delta)
> `Δ = absolute <relative> (impact)`, `ø = not affected`, `? = missing data`
> Powered by [Codecov](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=footer). Last update [6dd130a...474ddb2](https://codecov.io/gh/pandas-dev/pandas/pull/24045?src=pr&el=lastupdated). Read the [comment docs](https://docs.codecov.io/docs/pull-request-comments).

@jreback Ping on green.
thanks @h-vetinari