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BUG: DataFrame.dtypes doesn't include backend for string columns #50099

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@jrbourbeau

Description

@jrbourbeau
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Pandas version checks

  • I have checked that this issue has not already been reported.

  • I have confirmed this bug exists on the latest version of pandas.

  • I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd

df = pd.DataFrame({"x": ["foo", "bar", "baz"], "y": ["a", "b", "c"], "z": [1, 2, 3]})
df = df.astype({"x": "string[pyarrow]", "y": "string[python]", "z": "int64[pyarrow]"})
print(f"df.dtypes: \n{df.dtypes}")
print(f"{df.x.dtypes = }")
print(f"{df.y.dtypes = }")

Issue Description

The output of DataFrame.dtypes doesn't include [pyarrow] or [python] when using string extension types. This makes it unclear what dtypes are in a DataFrame. It's also not consistent with Series.dtypes, which includes [pyarrow] / [python] information, and other pyarrow-backed dtypes like int64[pyarrow]

The above code snippet outputs

df.dtypes:
x            string
y            string
z    int64[pyarrow]
dtype: object
df.x.dtypes = string[pyarrow]
df.y.dtypes = string[python]

Expected Behavior

I'd expect string[pyarrow] and string[python] to be in output of DataFrame.dtypes

Installed Versions

INSTALLED VERSIONS
------------------
commit           : 8dab54d6573f7186ff0c3b6364d5e4dd635ff3e7
python           : 3.9.15.final.0
python-bits      : 64
OS               : Darwin
OS-release       : 21.6.0
Version          : Darwin Kernel Version 21.6.0: Thu Sep 29 20:12:57 PDT 2022; root:xnu-8020.240.7~1/RELEASE_X86_64
machine          : x86_64
processor        : i386
byteorder        : little
LC_ALL           : None
LANG             : en_US.UTF-8
LOCALE           : en_US.UTF-8

pandas           : 1.5.2
numpy            : 1.21.6
pytz             : 2022.6
dateutil         : 2.8.2
setuptools       : 59.8.0
pip              : 22.3.1
Cython           : None
pytest           : 7.2.0
hypothesis       : None
sphinx           : 4.5.0
blosc            : None
feather          : None
xlsxwriter       : None
lxml.etree       : None
html5lib         : 1.1
pymysql          : None
psycopg2         : None
jinja2           : 3.1.2
IPython          : 8.7.0
pandas_datareader: None
bs4              : 4.11.1
bottleneck       : None
brotli           :
fastparquet      : 2022.11.0
fsspec           : 2022.11.0
gcsfs            : None
matplotlib       : None
numba            : 0.56.4
numexpr          : 2.8.3
odfpy            : None
openpyxl         : None
pandas_gbq       : None
pyarrow          : 10.0.1
pyreadstat       : None
pyxlsb           : None
s3fs             : 2022.11.0
scipy            : 1.9.3
snappy           :
sqlalchemy       : 1.4.44
tables           : 3.7.0
tabulate         : None
xarray           : 2022.11.0
xlrd             : None
xlwt             : None
zstandard        : None
tzdata           : None

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    BugDataFrameDataFrame data structureDataFrame data structureOutput-Formatting__repr__ of pandas objects, to_string__repr__ of pandas objects, to_stringStringsString extension data type and string dataString extension data type and string data

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