BUG: In master, math with periodindex + index of offset produces wrong result

[Dr-Irv]

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
p = pd.Period("2012-1-1", freq="D")
as_period_index = pd.period_range("2012-1-1", periods=10, freq="D")
offset_index = p - as_period_index
print(offset_index)
result = p + offset_index
print(result)

Issue Description

In pandas 1.5.2, here is the result:

Index([ <0 * Days>,  <-1 * Day>, <-2 * Days>, <-3 * Days>, <-4 * Days>,
       <-5 * Days>, <-6 * Days>, <-7 * Days>, <-8 * Days>, <-9 * Days>],
      dtype='object')
PeriodIndex(['2012-01-01', '2011-12-31', '2011-12-30', '2011-12-29',
             '2011-12-28', '2011-12-27', '2011-12-26', '2011-12-25',
             '2011-12-24', '2011-12-23'],
            dtype='period[D]')

In master version 2.0.0.dev0+881.gcab77e3d6c

Index([ <0 * Days>,  <-1 * Day>, <-2 * Days>, <-3 * Days>, <-4 * Days>,
       <-5 * Days>, <-6 * Days>, <-7 * Days>, <-8 * Days>, <-9 * Days>],
      dtype='object')
Index([2012-01-01, 2011-12-31, 2011-12-30, 2011-12-29, 2011-12-28, 2011-12-27,
       2011-12-26, 2011-12-25, 2011-12-24, 2011-12-23],
      dtype='object')

So the result used to be PeriodIndex, now it is Index. This was picked up in our tests for pandas-stubs

Expected Behavior

Should return PeriodIndex when you add offsets

Installed Versions

INSTALLED VERSIONS

commit : cab77e3
python : 3.8.15.final.0
python-bits : 64
OS : Windows
OS-release : 10
Version : 10.0.19045
machine : AMD64
processor : Intel64 Family 6 Model 158 Stepping 13, GenuineIntel
byteorder : little
LC_ALL : None
LANG : None
LOCALE : English_United States.1252

pandas : 2.0.0.dev0+881.gcab77e3d6c
numpy : 1.23.5
pytz : 2022.6
dateutil : 2.8.2
setuptools : 65.4.1
pip : 22.3.1
Cython : 0.29.32
pytest : 7.2.0
hypothesis : 6.60.0
sphinx : 4.5.0
blosc : None
feather : None
xlsxwriter : 3.0.3
lxml.etree : 4.9.1
html5lib : 1.1
pymysql : 1.0.2
psycopg2 : 2.9.3
jinja2 : 3.0.3
IPython : 8.7.0
pandas_datareader: 0.10.0
bs4 : 4.11.1
bottleneck : 1.3.5
brotli :
fastparquet : 2022.12.0
fsspec : 2021.11.0
gcsfs : 2021.11.0
matplotlib : 3.6.2
numba : 0.56.4
numexpr : 2.8.0
odfpy : None
openpyxl : 3.0.10
pandas_gbq : 0.17.9
pyarrow : 9.0.0
pyreadstat : 1.2.0
pyxlsb : 1.0.10
s3fs : 2021.11.0
scipy : 1.9.3
snappy :
sqlalchemy : 1.4.44
tables : 3.7.0
tabulate : 0.9.0
xarray : 2022.12.0
xlrd : 2.0.1
zstandard : 0.19.0
tzdata : None
qtpy : None
pyqt5 : None

[ramvikrams]

If we change the pandas/_libs/_tslibs/period.pyi and add the def __add__(self, other: Index) -> PeriodIndex: ... the result does not changes, so if not here where should the change ne done

[Dr-Irv]

If we change the pandas/_libs/_tslibs/period.pyi and add the def __add__(self, other: Index) -> PeriodIndex: ... the result does not changes, so if not here where should the change ne done

The error is in the pyx or py files, not the pyi files.

[jbrockmendel]

This is probably the result of #49999, specifically moving away from doing type inference on object-dtype results. Use result.infer_objects(copy=False) to get the old behavior

(the reversed op offset_index + p raises a TypeError that looks very wrong to me)

[Dr-Irv]

I find this change to be inconsistent. Aside from the TypeError that should be investigated, this behavior seems inconsistent:

p = pd.Period("2012-1-1", freq="D")
as_period_index = pd.period_range("2012-1-1", periods=10, freq="D")
offset_index = p - as_period_index
print(offset_index)
result = p + offset_index
print(result)
iresult = pd.Index([p]*10) + offset_index
print(iresult)

This produces output:

Index([ <0 * Days>,  <-1 * Day>, <-2 * Days>, <-3 * Days>, <-4 * Days>,
       <-5 * Days>, <-6 * Days>, <-7 * Days>, <-8 * Days>, <-9 * Days>],
      dtype='object')
Index([2012-01-01, 2011-12-31, 2011-12-30, 2011-12-29, 2011-12-28, 2011-12-27,
       2011-12-26, 2011-12-25, 2011-12-24, 2011-12-23],
      dtype='object')
PeriodIndex(['2012-01-01', '2011-12-31', '2011-12-30', '2011-12-29',
             '2011-12-28', '2011-12-27', '2011-12-26', '2011-12-25',
             '2011-12-24', '2011-12-23'],
            dtype='period[D]')

In the first addition, you are adding a Period plus an index of offsets, the latter has type object, and you get an Index.
In the second addition, you are adding a PeriodIndex plus an index of offsets, the latter has type object and you get a PeriodIndex.

So that PR documents the change as:

Changed behavior of :class:Index, :class:Series, and :class:DataFrame arithmetic methods when working with object-dtypes, the results no longer do type inference on the result of the array operation

That's not really correct, because in the second addition I did, you added one Index that had a dtype of Period and a second index that had a dtype of object, and it kept the dtype of Period (as I would expect). So it seems that if you do arithmetic on a single object that has a type, then it doesn't do the inference. This could get really confusing.

Why did you make that change?