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Fully commutative elements in types $B$ and $D$ are completely characterizedand counted by Stembridge. Recently, Feinberg-Kim-Lee-Oh have extended thestudy of fully commutative elements from Coxeter groups to the complex setting,giving an enumeration of such elements in $G(m,1,n)$. In this note, we prove aconnection between fully commutative elements in $B_n$ and in $G(m,1,n)$, whichallows us to characterize fully commutative elements in $G(m,1,n )$ by patternavoidance. Further, we present a counting formula for such elements in$G(m,1,n)$.

A note on fully commutative elements in complex reflection groups