@{ $genes{$r[0]}{$r[1]}{$r[2]} }
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is short for

@{ $genes{$r[0]}->{$r[1]}->{$r[2]} }
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which is equivalent to

@{ ${ ${ $genes{$r[0]} }{$r[1]} }{$r[2]} }
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There are two notations for dereferencing.

• The embedded* notation (${ $array }[4]) can be used to dereference anything that can be deferenced, but it's a bit clunky.

• The arrow notation ($array->[4]) is cleaner, but it can only be used for some forms of dereferencing.

There's no arrow equivalent for @{ $array }, so using push @{ ... }, $r[3] was necessary.

* — I just made up that name. I don't know of any existing name for it.