Associative words in the symmetric group of degree three

Let G be a group. An element $w(x,y)$ of the absolutely free group on free generators $x,y$ is called an associative word in $G$ if the equality $w(w(g_1,g_2),g_3)=w(g_1,w(g_2,g_3))$ holds for all $g_1,g_2 \in G$. In this paper we determine all associative words in the symmetric group on three letters.