Bsrgvty

I have the following code to align several equations:

beginaligned
s_1 & :=sum_1 leq j leq m X_j \
s_2 & :=sum_1 leq j<k leq m X_j cdot X_k \
 & vdots \
s_k & :=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 \
 & vdots \
s_m & :=X_1 X_2 cdots X_m
endaligned

I would like to ask why the colon does not align with the vdots and a solution for it.

Welcome to TeX-SE! It does not align because a colon is a relation, i.e. of the mathrel type. If you want to align it "by hand" you could just make the vdots of that type, too.

documentclassarticle
usepackageamsmath
begindocument
[
beginaligned
s_1 & :=sum_1 leq j leq m X_j \
s_2 & :=sum_1 leq j<k leq m X_j cdot X_k \
 & mathrelvdots \
s_k & :=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 \
 & mathrelvdots\
s_m & :=X_1 X_2 cdots X_m
endaligned]
enddocument

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 :=:

documentclassarticle

usepackagemathtools

begindocument

beginalign*
 s_1 &coloneqqsum_1 leq j leq m X_j \
 s_2 &coloneqqsum_1 leq j<k leq m X_j cdot X_k \
 vdotswithins_k& \
 s_k &coloneqqsum_1 leq j_1<j_2<cdots<j_k leq m
 X_j_1 cdot X_j_2 cdot cdots cdot X_j_k \
 vdotswithins_k& \
 s_m &coloneqq X_1 X_2 cdots X_m
endalign*

beginalign*
 s_1 &coloneqqsum_1 leq j leq m X_j \
 s_2 &coloneqqsum_1 leq j<k leq m X_j cdot X_k \
 &vdotswithincoloneqq \
 s_k &coloneqqsum_1 leq j_1<j_2<cdots<j_k leq m
 X_j_1 cdot X_j_2 cdot cdots cdot X_j_k \
 &vdotswithincoloneqq \
 s_m &coloneqq X_1 X_2 cdots X_m
endalign*

enddocument

My interpretation of changing of the answer given by the user @marmot using:

• The macro coegual instead of := simply because it provides the perfect alignment between : and =;


• Using the negative space with the command mkern I tried to bring the products of the elements close to the summation because I do not like them distant;


• I have used substack command because the subscripts on the summation did not bring me the products far away.

documentclass[a4paper,12pt]article
usepackageamsmath,amssymb
newcommandcoegualmathrelmathop:=
begindocument
[
beginaligned
s_1 & coegual sum_1 leq j leq m X_j \
s_2 & coegual sum_1 leq j < k leq m mkern-5mu X_j cdot X_k \
 & mathrelvdots \
s_k & coegual 
sum_substack1 leq j_1< j_2 <cdots\ cdots< j_k leq m mkern-15mu X_j_1 cdot X_j_2 cdot cdots cdot X_j_k \
 & mathrelvdots\
s_m & coegual X_1 X_2 cdots X_m
endaligned
]
enddocument