Fix DataFrame.to_string() justification
- [ ] closes #16839, #13032
- [ ] tests added / passed
- [ ] passes `git diff upstream/master -u -- "*.py" | flake8 --diff`
- [ ] whatsnew entry

Fixes justification with `DataFrame.to_string(index=False)`, but breaks other tests :-(

There are two side-effects of this PR, which will likely make it unacceptable, but wanted to get suggestions if possible.

1. It changes the expected behavior with trailing spaces as shown below.

```
import pandas as pd
def wrap_to_string(df, **kwargs):
    s = df.to_string(**kwargs)
    print(str(kwargs).center(40, '='))
    for i, line in enumerate(s.split('\n')):
        print(f'^{line}$-{i}')
    print('~' * 40)
df = pd.DataFrame({'x': [11, -22], 'y': ['aaa', ' ']})
wrap_to_string(df)
wrap_to_string(df, line_width=1)
wrap_to_string(df, index=False)
wrap_to_string(df, index=False, line_width=1)
```

Current output with `master`
```
==================={}===================  # Looks good
^    x    y$-0
^0  11  aaa$-1
^1 -22     $-2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
==========={'line_width': 1}============  # Looks good (to me anyways) but has trailing spaces
^    x  \$-0
^0  11   $-1
^1 -22   $-2
^$-3
^     y  $-4
^0  aaa  $-5
^1       $-6
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
============{'index': False}============  # broken justification (this is what I want to fix)
^x    y$-0
^11  aaa$-1
^-22$-2                                   # the last value (space) disappeared completely
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
==={'index': False, 'line_width': 1}====  # still broken
^x  \$-0
^11   $-1
^-22   $-2
^$-3
^  y  $-4
^aaa$-5                                   # the last value (space) disappeared completely
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
```

Output with this PR

```
==================={}===================  # No change from master
^    x    y$-0
^0  11  aaa$-1
^1 -22     $-2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
==========={'line_width': 1}============  # No change from master
^    x  \$-0
^0  11   $-1
^1 -22   $-2
^$-3
^     y  $-4
^0  aaa  $-5
^1       $-6
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
============{'index': False}============  # Justification is fixed
^  x    y$-0
^ 11  aaa$-1
^-22     $-2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
==={'index': False, 'line_width': 1}====  # Justification is fixed
^  x  \$-0
^ 11   $-1
^-22   $-2
^$-3
^  y  $-4
^aaa  $-5
^     $-6
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
```




2. It breaks a bunch of other tests that use 
`IntArrayFormatter._format_strings` because the formatting changed from `'{x: d}'` which adds a leading space for positive values, to `'{x:d}'`. My impression is that this PR eliminates an arguably unnecessary extra space for positive integers, but I haven't looked at those tests in too much detail (eg I've never used to_latex).

The tests that break and an example is shown below.

```
pandas/tests/arrays/categorical/test_repr.py::TestCategoricalRepr::test_print_none_width FAILED                                                                                                                  [  4%]
pandas/tests/io/formats/test_format.py::TestDataFrameFormatting::test_east_asian_unicode_frame FAILED                                                                                                            [  8%] 
pandas/tests/io/formats/test_format.py::TestDataFrameFormatting::test_datetimelike_frame FAILED                                                                                                                  [ 12%]
pandas/tests/io/formats/test_format.py::TestDataFrameFormatting::test_to_string_no_header FAILED                                                                                                                 [ 16%]
pandas/tests/io/formats/test_format.py::TestDataFrameFormatting::test_to_string_specified_header FAILED                                                                                                          [ 20%]
pandas/tests/io/formats/test_format.py::TestDataFrameFormatting::test_to_string_index_formatter FAILED                                                                                                           [ 25%]
pandas/tests/io/formats/test_format.py::TestDataFrameFormatting::test_to_string_line_width FAILED                                                                                                                [ 29%] 
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_east_asian_unicode_series FAILED                                                                                                              [ 33%] 
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_period FAILED                                                                                                                                 [ 37%]
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_truncate_ndots FAILED                                                                                                                         [ 41%] 
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_to_string_name FAILED                                                                                                                         [ 45%] 
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_to_string_dtype FAILED                                                                                                                        [ 50%]
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_to_string_length FAILED                                                                                                                       [ 54%]
pandas/tests/io/formats/test_format.py::TestSeriesFormatting::test_to_string_header FAILED                                                                                                                       [ 58%]
pandas/tests/io/formats/test_to_latex.py::TestToLatex::test_to_latex FAILED                                                                                                                                      [ 62%] 
pandas/tests/io/formats/test_to_latex.py::TestToLatex::test_to_latex_multiindex FAILED                                                                                                                           [ 66%] 
pandas/tests/io/formats/test_to_latex.py::TestToLatex::test_to_latex_multicolumnrow FAILED                                                                                                                       [ 70%] 
pandas/tests/io/formats/test_to_latex.py::TestToLatex::test_to_latex_longtable FAILED                                                                                                                            [ 75%] 
pandas/tests/io/formats/test_to_latex.py::TestToLatex::test_to_latex_no_header FAILED                                                                                                                            [ 79%] 
pandas/tests/series/test_repr.py::TestSeriesRepr::test_multilevel_name_print FAILED                                                                                                                              [ 83%]
pandas/tests/series/test_repr.py::TestCategoricalRepr::test_categorical_repr FAILED                                                                                                                              [ 87%]
pandas/tests/series/test_repr.py::TestCategoricalRepr::test_categorical_series_repr FAILED                                                                                                                       [ 91%] 
pandas/tests/series/test_repr.py::TestCategoricalRepr::test_categorical_series_repr_ordered FAILED                                                                                                               [ 95%]
pandas/tests/sparse/test_format.py::TestSparseSeriesFormatting::test_sparse_int FAILED                                                                                                                           [100%] 
```

```
______________________________________________________________________________________ TestCategoricalRepr.test_print_none_width _______________________________________________________________________________________
                                                                                                                                                                                                                        
self = <pandas.tests.arrays.categorical.test_repr.TestCategoricalRepr object at 0x12892def0>                                                                                                                            
                                                                                                                                                                                                                        
    def test_print_none_width(self):                                                                                                                                                                                    
        # GH10087                                                                                                                                                                                                       
        a = Series(Categorical([1, 2, 3, 4]))                                                                                                                                                                           
        exp = u("0    1\n1    2\n2    3\n3    4\n" +                                                                                                                                                                    
                "dtype: category\nCategories (4, int64): [1, 2, 3, 4]")                                                                                                                                                 
                                                                                                                                                                                                                        
        with option_context("display.width", None):                                                                                                                                                                    
>           assert exp == repr(a)                                                                                                                                                                                      
E           AssertionError: assert '0    1\n1   ... [1, 2, 3, 4]' == '0   1\n1   2\... [1, 2, 3, 4]'                                                                                                                   
E             - 0    1                                                                                                                                                                                                 
E             ?  -                                                                                                                                                                                                      
E             + 0   1                                                                                                                                                                                                   
E             - 1    2                                                                                                                                                                                                 
E             ?  -                                                                                                                                                                                                      
E             + 1   2                                                                                                                                                                                                   
E             - 2    3                                                                                                                                                                                                 
E             ?  -                                                                                                                                                                                                     
E             + 2   3                                                                                                                                                                                                  
E             - 3    4                                                                                                                                                                                                  
E             ?  -                                                                                                                                                                                                      
E             + 3   4                                                                                                                                                                                                   
E               dtype: category                                                                                                                                                                                         
E               Categories (4, int64): [1, 2, 3, 4]                                                                                                                                                                     
                                                                                                                                                                                                                       
pandas/tests/arrays/categorical/test_repr.py:59: AssertionError
```

The root issue, I think, is that we have `'{x: d}'.format(x=x)`, but the current tests don't want the leading space for a simple `pd.DataFrame({'x': [0, 1], 'y': [2, 3]}).to_string(index=False)`.

I have [another attempt](https://github.com/gshiba/pandas/commit/05592c74c56f997fc19d1a332f16914ebb22b6e9) at this issue, which explicitly expects the leading space, but may not be palatable to some...

Thanks.
I updated the original description to include current and proposed behavior. I just updated one of the test modules that was failing (`pandas/tests/io/formats/test_format.py`) and there are 4 more modules to go.

It seems a lot of things assume/depend on the `% d` formatting, which makes me think that my [other approach](https://github.com/gshiba/pandas/tree/fix-to-string2) is more sensible... Please let me know what you think.

---

As an aside, while fixing `test_format.py` I think I found some (unrelated?) formatting issues...

```
import pandas as pd
df = pd.DataFrame(123, [0, 10], [0, 11])
print(df.to_string())
print('---')
s = pd.Series([0, 100, 200])
print(s.to_string(max_rows=2))
```
outputs
```
     0    11  # '0' is misaligned
0   123  123
10  123  123
---
0      0
    ...   # dots are misaligned
2    200
``` #22505 is probably a better fix Closing in favor of #22505. Will reopen if needed.