Index position of $(x_{i})_{i\in I}$ vs $\left(x_{i}\right)_{i\in I}$

Question

It seems that the basic syntax

(x_{i})_{i\in I}

leads to a different index position than

\left(x_{i}\right)_{i\in I}

Specificically, the lower index in the last example has a bigger vertical offset. This seems independent of the content between the brackets, as the same effect shows up with the even simpler example

(x)_{i\in I}
\left(x\right)_{i\in I}

What is the cause or rationale of this behavior? Since I might accidentally mix both syntaxes in my document, what is the best practice to ensure a uniform alignment of indices?

Answer 1

It's not actually due to \left and \right.

Look at the following plain TeX code:

$(x_{i})_{i\in I}$ $\left(x_{i}\right)_{i\in I}$ ${(x_{i})}_{i\in I}$
\bye

As you see, the last two samples typeset the same: the subscript is to the whole subformula, not just to the right parenthesis.

To the contrary with

$x_{x}$ $\left.x\right._{x}$
\bye

we get the same height (only spaces differ, but it's expected):

so you see that the problem is not with \left and \right.

What's the best practice? Not using \left and \right unless really necessary.

Answer 2

You actually ask two separate questions.

What is the cause ... of this behavior?

Consider the following, slightly more elaborate examples that involve mathematical delimiters, such as ) and ], and adjacent subscript and superscript terms.

\documentclass{article}      
\begin{document} 
$()_I$   ${()}_I$ $\left(\right)_I$ ${\left(\right)}_I$
$()^2$   ${()}^2$ $\left(\right)^2$ ${\left(\right)}^2$
$[]_u^v$  ${[]}_u^v$
\end{document}

The two main things to note in this screenshot are:

• In a given row, the round parentheses and square brackets all have the same heights and depths. Same for the subscript and superscript terms in a given row.

• In a given row, the subscript and superscript terms are placed more compactly, i.e., with smaller vertical offsets, in the first case than in the subsequent case[s].

So, in a given row, what distinguishes the first case from the subsequent cases? Take, say, the second row.

• In the first case, the "math atom" that immediately precedes the ^ token (TeX's superscript material initiator) is ), which has TeX status "math-close".

• In the second through fourth cases, the "math atoms" -- or should that be "math molecules"? -- that precede the ^ tokens are {()}^2, \left(\right)^2, and {\left(\right)}^2, respectively. What do these math atoms have in common? It's that their TeX status is "math-ordinary". Aside: encasing any math atom in curly braces changes its status to math-ordinary. And, the entire \left(\right) math atom also has status math-ordinary.

Now, TeX's superscript and subscript placement rules differ importantly in terms of the vertical offsets that get applied, depending on whether the material that immediately precedes the ^ and _ tokens has status math-open/math-close or not. In particular, what you've (re)discovered is that the placement is tighter if the preceding math atom has status math-close than if its status is math-ordinary.

What is the ... rationale [for] this behavior?

My impression is that there are two separate reasons for TeX applying smaller offsets if the math atom that immediately precedes the _ and ^ tokens has status math-open/close.

• Compare the outputs of $(a+b)^2 = a^2+2ab+b^2$ and $\left(a+b\right)^2 = a^2+2ab+b^2$.

  

  In the former case, all three instances of 2 are placed at the same height, which I would argue looks natural and harmonious. In the latter case, the first instance of 2 is placed noticeably higher than the other two; to me, that seems artificial, unnecessary, and visually distracting. (Aside: Having worked with TeX and LaTeX for more than 30 years, my opinion in this matter may well be a tad biased...)

• In an inline-math context, applying smaller vertical offsets to any subscript and superscript terms that may be present significantly raises the odds that that the entire paragraph can be typeset with even spacing between lines. In fine typesetting, having even line spacing in a paragraph is considered to be very desirable.

Finally, you asked,

Since I might accidentally mix both syntaxes in my document, what is the best practice to ensure a uniform alignment of indices?

It's quite simple, really: Don't use the \left and \right auto-sizing directives unless doing so serves a discernible valid purpose. Applying this stricture to some of the preceding material suggests that there is no valid reason for using \left and \right in $\left(a+b\right)^2=a^2+2ab+b^2$...

Answer 3

Mico and egreg convinced me that it is best to avoid \left...\right unless there are reasons to use that.

Mico says "... applying smaller vertical offsets to any subscript and superscript terms that may be present significantly raises the odds that that the entire paragraph can be typeset with even spacing between lines. In fine typesetting, having even line spacing in a paragraph is considered to be very desirable."

So the behaviour of )_ is by design.

How is that done and why it happens that the behaviour changes in cases like \left(...\right)_I and {)}_I ? I think here is the answer:

UPDATE: There is a later answer below better explaining what happens.

Special behaviour is built into the font parameters of character ). In a math font each character has about 22 parameters some of which have something to do with indexes:

s13 sup1 (alternative superscript position)
s14 sup2 (alternative superscript position)
s15 sup3 (alternative superscript position)
s16 sub1 (alternative subscript position)
s17 sub2 (alternative subscript position)
s18 sup-drop (further control of superscript positioning)
s19 sub-drop (further control of subscript positioning)

With {(} or \left...\right this parameters are out of question because in front of _, no more a character of a font is sitting. UPDATE: Or quite the opposite as says the answer that I already mentioned.

Answer 4

While not quite answering the question a note about how the technical implementation for this is done which didn't fit into a comment:

The deciding factor is not so much the mathclass (except mathop which is special) but the question is the object which is accented.

Whenever a single character from a math font is accented, then it's height and depth are completely ignored and instead the scripts are set as if they attach to an empty list.

Take for example the following plain TeX document which attaches scripts to huge parens in different math classes:

$$
)_a
\mathchar"0321_a % This is a math ordinary
\mathchar"5321_a % This is a math close
{}_a % an empty list to compare
$$
\bye

You can see that they all attach the subscript at exactly the same height:

If the script attaches to anything else than a single character then this special case does not trigger and the height and depth of the character are taken into account. Generally this already happens when just surrounding some math atom with {}. But as a further exception adding {} around a single character which already is a math ordinary character is redundant and therefore optimized away in many situations, so the single character rules apply again:

$$
{)}_a % A math close character in braces
{x\mathchar"0321}_a % This are multiple math ordinary in braces
{\mathchar"0321}_a % This is one math ordinary character in braces so the braces are ignored
{\mathchar"5321}_a % This is a math close character in braces
{}_a % an empty list for comparision
$$
\bye

Here the subscripts move down to adjust for the depth of the characters except in the case where the braces are redundant and therefore removed:

The reason why it is done like this is that single character usually all have roughly the same size (unless you intentionally use weird characters like the \mathchar I used for testing). So one height looks reasonable next to all characters and looks much more consistent than to have different script heights for every characters. Everything more complicated than a single character can not be relied upon to have usual sizes, so the more general placement has to be used.