An exception to the preceding example takes place when the indexes contain the same exact elements in the same order. When this occurs, a Cartesian product does not take place, and the indexes instead align by their position. Notice here that each element aligned exactly by position and that the data type remained an integer:

>>> s1 = pd.Series(index=list('aaabb'), data=np.arange(5))
>>> s2 = pd.Series(index=list('aaabb'), data=np.arange(5))
>>> s1 + s2
a    0
a    2
a    4
b    6
b    8
dtype: int64

If the elements of the index are identical, but the order is different between the Series, a Cartesian product occurs. Let's change the order of the index in s2 and rerun the same operation:

>>> s1 = pd.Series(index=list('aaabb'), data=np.arange(5))
>>> s2 = pd.Series(index=list('bbaaa'), data=np.arange(5))
>>> s1 + s2
a    2
a    3
a    4
a    3
a    4
a    5
a    4
a    5
a    6
b    3
b    4
b    4
b    5
dtype: int64

It is quite interesting that pandas has two drastically different outcomes for this same operation. If a Cartesian product was the only choice for pandas, then something as simple as adding DataFrame columns together would explode the number of elements returned.

In this recipe, each Series had a different number of elements. Typically, array-like data structures in Python and other languages do not allow operations to take place when the operating dimensions do not contain the same number of elements. Pandas allows this to happen by aligning the indexes first before completing the operation.