In the class ${\mathcal T}$ of triangular maps of
the square we consider the strongest notion of distributional chaos, DC1, originally introduced by Schweizer and Smítal [Trans. Amer. Math.
Soc. 344 (1994), 737 - 854] for continuous maps of the interval. We
show that a map $F\in{\mathcal T}$ is DC1 if $F$ has a periodic orbit
with period $\ne 2^n$, for any $n\ge 0$. Consequently, a map in $\mathcal
T$ is DC1 if it has a homoclinic trajectory. This result is
interesting since, in $\mathcal T$, positive topological entropy does not
imply DC1.