BUG: Empty string multilevel column names

[prhbrt]

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Reproducible Example

Empty string multilevel column names that follow an indexed column are removed from results.

    import pandas as pd
    
    data = pd.DataFrame({('b', '', '', '', 'a', ''): []})
    
    print(data.columns)
    print(data['b'].columns)

results in this:

    MultiIndex([('b', '', '', '', 'a', '')], )
    MultiIndex([('a', '')], )

Issue Description

Missing column levels after indexing. This seems to be explicitly programmed behavior, that recursively eats empty top column names, but I can't find anything about this in the user guide on indexing, nor why empty column names should be treated differently.

Expected Behavior

MultiIndex([('', '', '', 'a', '')], ) was expected for the second output.

Installed Versions

This issue reproduces on:

Google colab:

------------------ commit : 66e3805 python : 3.7.15.final.0 python-bits : 64 OS : Linux OS-release : 5.10.133+ Version : #1 SMP Fri Aug 26 08:44:51 UTC 2022 machine : x86_64 processor : x86_64 byteorder : little LC_ALL : None LANG : en_US.UTF-8 LOCALE : en_US.UTF-8

pandas : 1.3.5
numpy : 1.21.6
pytz : 2022.6
dateutil : 2.8.2
pip : 21.1.3
setuptools : 57.4.0
Cython : 0.29.32
pytest : 3.6.4
hypothesis : None
sphinx : 1.8.6
blosc : None
feather : 0.4.1
xlsxwriter : None
lxml.etree : 4.9.1
html5lib : 1.0.1
pymysql : None
psycopg2 : 2.9.5 (dt dec pq3 ext lo64)
jinja2 : 2.11.3
IPython : 7.9.0
pandas_datareader: 0.9.0
bs4 : 4.6.3
bottleneck : None
fsspec : 2022.10.0
fastparquet : None
gcsfs : None
matplotlib : 3.2.2
numexpr : 2.8.4
odfpy : None
openpyxl : 3.0.10
pandas_gbq : 0.17.9
pyarrow : 6.0.1
pyxlsb : None
s3fs : None
scipy : 1.7.3
sqlalchemy : 1.4.43
tables : 3.7.0
tabulate : 0.8.10
xarray : 0.20.2
xlrd : 1.1.0
xlwt : 1.3.0
numba : 0.56.4

Main-branch (2.0.0.dev0+697.g8020bf1b2)

INSTALLED VERSIONS ------------------ commit : 91111fd python : 3.10.6.final.0 python-bits : 64 OS : Linux OS-release : 5.15.0-53-generic Version : #59-Ubuntu SMP Mon Oct 17 18:53:30 UTC 2022 machine : x86_64 processor : x86_64 byteorder : little LC_ALL : None LANG : en_US.UTF-8 LOCALE : en_US.UTF-8

pandas : 1.5.1
numpy : 1.23.0
pytz : 2022.1
dateutil : 2.8.2
setuptools : 59.6.0
pip : 22.0.2
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : 3.0.3
lxml.etree : 4.9.1
html5lib : None
pymysql : None
psycopg2 : 2.9.1
jinja2 : 3.1.2
IPython : 8.4.0
pandas_datareader: None
bs4 : 4.11.1
bottleneck : 1.3.5
brotli : None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : 3.5.2
numba : None
numexpr : None
odfpy : None
openpyxl : 3.0.10
pandas_gbq : None
pyarrow : None
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : 1.8.1
snappy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : 2.0.1
xlwt : None
zstandard : None
tzdata : None

1.5.1:

INSTALLED VERSIONS ------------------ commit : 91111fd python : 3.10.6.final.0 python-bits : 64 OS : Linux OS-release : 5.15.0-53-generic Version : #59-Ubuntu SMP Mon Oct 17 18:53:30 UTC 2022 machine : x86_64 processor : x86_64 byteorder : little LC_ALL : None LANG : en_US.UTF-8 LOCALE : en_US.UTF-8

pandas : 1.5.1
numpy : 1.23.0
pytz : 2022.1
dateutil : 2.8.2
setuptools : 59.6.0
pip : 22.0.2
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : 3.0.3
lxml.etree : 4.9.1
html5lib : None
pymysql : None
psycopg2 : 2.9.1
jinja2 : 3.1.2
IPython : 8.4.0
pandas_datareader: None
bs4 : 4.11.1
bottleneck : 1.3.5
brotli : None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : 3.5.2
numba : None
numexpr : None
odfpy : None
openpyxl : 3.0.10
pandas_gbq : None
pyarrow : None
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : 1.8.1
snappy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : 2.0.1
xlwt : None
zstandard : None
tzdata : None

[phofl]

We have a couple of tests that are testing exactly this behavior. So don't think this should be considered a bug. (test_frame_mixed_depth_get)

This is from way back and has been around since forever (#2596) one of the tests was added there

[phofl]

We should maybe document this?

[prhbrt]

We should maybe document this?

I think that's important, this behavior causes obscure KeyErrors when not considered.

I'm also interested in the reasoning behind this explicitly implemented and tested behavior.