I don't think these should align on the dots of :=
, but rather should be centered with respect to either the s_k
or perhaps the relation :=
. The dots in :=
serve a different function to those in \vdots
. The package mathtools
provides the command \vdotswithin{...}
for centering such \vdots
with respect to the material ...
; it also provides \coloneqq
for a better version of :=
:

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{align*}
s_{1} &\coloneqq\sum_{1 \leq j \leq m} X_{j} \\
s_{2} &\coloneqq\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
\vdotswithin{s_{k}}& \\
s_{k} &\coloneqq\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m}
X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
\vdotswithin{s_{k}}& \\
s_{m} &\coloneqq X_{1} X_{2} \cdots X_{m}
\end{align*}
\begin{align*}
s_{1} &\coloneqq\sum_{1 \leq j \leq m} X_{j} \\
s_{2} &\coloneqq\sum_{1 \leq j<k \leq m} X_{j} \cdot X_{k} \\
&\vdotswithin{\coloneqq} \\
s_{k} &\coloneqq\sum_{1 \leq j_{1}<j_{2}<\cdots<j_{k} \leq m}
X_{j_{1}} \cdot X_{j_{2}} \cdot \cdots \cdot X_{j_{k}} \\
&\vdotswithin{\coloneqq} \\
s_{m} &\coloneqq X_{1} X_{2} \cdots X_{m}
\end{align*}
\end{document}