BUG: Fix Index.equals between object and string

pandas/core/indexes/base.py

@@ -5481,11 +5481,7 @@ def equals(self, other: Any) -> bool:
             # quickly return if the lengths are different
             return False
 
-        if (
-            isinstance(self.dtype, StringDtype)
-            and self.dtype.na_value is np.nan
-            and other.dtype != self.dtype
-        ):
+        if isinstance(self.dtype, StringDtype) and other.dtype != self.dtype:
             # TODO(infer_string) can we avoid this special case?
             # special case for object behavior
             return other.equals(self.astype(object))

pandas/tests/frame/test_arithmetic.py

@@ -11,7 +11,7 @@
 import numpy as np
 import pytest
 
-from pandas.compat import HAS_PYARROW
+import pandas.util._test_decorators as td
 
 import pandas as pd
 from pandas import (
@@ -2183,19 +2183,28 @@ def test_enum_column_equality():
     tm.assert_series_equal(result, expected)
 
 
-def test_mixed_col_index_dtype(using_infer_string):
+@pytest.mark.parametrize(
+    "dtype",
+    [
+        "string[python]",
+        pytest.param(
+            "string[pyarrow]",
+            marks=td.skip_if_no("pyarrow"),
+        ),
+        pytest.param(
+            "str",
+            marks=td.skip_if_no("pyarrow"),
+        ),
+    ],
+)
+def test_mixed_col_index_dtype(dtype):
     # GH 47382
     df1 = DataFrame(columns=list("abc"), data=1.0, index=[0])
     df2 = DataFrame(columns=list("abc"), data=0.0, index=[0])
-    df1.columns = df2.columns.astype("string")
+    df1.columns = df2.columns.astype(dtype)
     result = df1 + df2
     expected = DataFrame(columns=list("abc"), data=1.0, index=[0])
-    if using_infer_string:
-        # df2.columns.dtype will be "str" instead of object,
-        #  so the aligned result will be "string", not object
-        if HAS_PYARROW:
-            dtype = "string[pyarrow]"
-        else:
-            dtype = "string"
-        expected.columns = expected.columns.astype(dtype)
+
+    expected.columns = expected.columns.astype(dtype)
+
     tm.assert_frame_equal(result, expected)

pandas/tests/indexes/test_base.py

@@ -40,6 +40,7 @@
     ensure_index,
     ensure_index_from_sequences,
 )
+from pandas.testing import assert_series_equal
 
 
 class TestIndex:
@@ -1717,3 +1718,51 @@ def test_is_monotonic_pyarrow_list_type():
     idx = Index([[1], [2, 3]], dtype=pd.ArrowDtype(pa.list_(pa.int64())))
     assert not idx.is_monotonic_increasing
     assert not idx.is_monotonic_decreasing
+
+
+@pytest.mark.parametrize(
+    "dtype",
+    [
+        "string[python]",
+        pytest.param(
+            "string[pyarrow]",
+            marks=td.skip_if_no("pyarrow"),
+        ),
+        pytest.param(
+            "str",
+            marks=td.skip_if_no("pyarrow"),
+        ),
+    ],
+)
+def test_index_equals_different_string_dtype(dtype):
+    # GH 61099
+    idx_obj = Index(["a", "b", "c"])
+    idx_str = Index(["a", "b", "c"], dtype=dtype)
+
+    assert idx_obj.equals(idx_str)
+    assert idx_str.equals(idx_obj)
+
+
+@pytest.mark.parametrize(
+    "dtype",
+    [
+        "string[python]",
+        pytest.param(
+            "string[pyarrow]",
+            marks=td.skip_if_no("pyarrow"),
+        ),
+        pytest.param(
+            "str",
+            marks=td.skip_if_no("pyarrow"),
+        ),
+    ],
+)
+def test_index_comparison_different_string_dtype(dtype):
+    # GH 61099
+    idx = Index(["a", "b", "c"])
+    s_obj = Series([1, 2, 3], index=idx)
+    s_str = Series([4, 5, 6], index=idx.astype(dtype))
+
+    expected = Series([True, True, True], index=["a", "b", "c"])
+    result = s_obj < s_str
+    assert_series_equal(result, expected)