Mathtools: using substack to align limits

You could just use the aligned environment to do the alignment:

Notes:

• The difference in the first two is where the k: is located.
• There must be a better way to enforce \scriptstyle.

Code:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
Ok?
\[
S_1 = \sum_{\mathclap{k:~\substack{\setlength{\jot}{-0.8\baselineskip}\everymath{\scriptstyle}\begin{aligned}
   \scriptstyle X(S_k) &\scriptstyle= x_i,\\
   \scriptstyle Y(S_k) &\scriptstyle= y_j,
  \end{aligned}
  }}}
  x_i + y_i
  \quad\text{or }
S_1 = \sum_{\mathclap{\substack{\setlength{\jot}{-0.8\baselineskip}\everymath{\scriptstyle}\begin{aligned}
   \scriptstyle k:~ X(S_k) &\scriptstyle= x_i,\\
   \scriptstyle Y(S_k) &\scriptstyle= y_j,
  \end{aligned}
  }}}
  x_i + y_i
\]
Okay
\[
S_2 = \sum_{\mathclap{\substack{\setlength{\jot}{-0.8\baselineskip}\begin{aligned}
    \scriptstyle X(S_k) &\scriptstyle= x_i,\\
    \scriptstyle  Y(S_k) &\scriptstyle= y_j,
  \end{aligned}
  }}}
  x_i + y_i
\]
\end{document}

A variant of my answer to Align in Substack

\documentclass{article}
\usepackage{mathtools}

\makeatletter
\newcommand{\subalign}[2][c]{%
  \if#1c\vcenter\else\vtop\fi{%
    \Let@ \restore@math@cr \default@tag
    \baselineskip\fontdimen10 \scriptfont\tw@
    \advance\baselineskip\fontdimen12 \scriptfont\tw@
    \lineskip\thr@@\fontdimen8 \scriptfont\thr@@
    \lineskiplimit\lineskip
    \ialign{\hfil$\m@th\scriptstyle##$&$\m@th\scriptstyle{}##$\hfil\crcr
      #2\crcr
    }%
  }%
}
\makeatother


\begin{document}

\[
\sum_{\mathclap{
  k:\;\subalign[t]{
  X(S_k) &= x_i,\\
  Y(S_k) &= y_j}
}}f(k)+g(k)+h(k)
\qquad
\sum_{\mathclap{
  k:\;\subalign{
  X(S_k) &= x_i,\\
  Y(S_k) &= y_j}
}}f(k)+g(k)+h(k)
\]
\end{document}

The \subalign command has an optional argument, which can be t or c (default c) for setting the vertical alignment.

As far as I understand the following correction will be sufficient:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
Distortion
\[
\sum_{k:\substack{
  \mathllap{X(S_k)} = \mathrlap{x_i,}\\
  \mathllap{Y(S_k)} = \mathrlap{y_j,}}
} % closing - PS
\]

Distortion corrected
\[
\sum_{k:\ \substack{
  {X(S_k)} = \mathrlap{x_i,}\\
  {Y(S_k)} = \mathrlap{y_j,}}
} % closing - PS
\]


\end{document}