\[2\frac{1}{13} \cdot 1\frac{4}{9} - 5\frac{1}{6}\ :2\frac{7}{12} = \frac{27}{13} \cdot \frac{13}{9} - \frac{31}{6}\ :\frac{31}{12} =\]
\[= 3 - \frac{31}{6} \cdot \frac{12}{31} = 3 - 2 = 1.\]