  Numerical simulations and finite-size scaling analysis have been carried out
to study standard and inverse percolation of straight rigid rods on triangular
lattices. In the case of standard (inverse) percolation, the lattice is
initially empty(occupied) and linear $k$-mers ($k$ linear consecutive sites)
are randomly and sequentially deposited on(removed from) the lattice,
considering an isotropic and nematic scheme. The study is conducted by
following the behavior of four critical concentrations with the size $k$,
determined for a wide range of $k$ : $(i)$[$(ii)$] standard isotropic[nematic]
percolation threshold $\theta_{c,k}$[$\vartheta_{c,k}$], and $(iii)$[$(iv)$]
inverse isotropic[nematic] percolation threshold
$\theta^i_{c,k}$[$\vartheta^i_{c,k}$]. The obtained results indicate that:
$(1)$ $\theta_{c,k}$[$\theta^i_{c,k}$] exhibits a non-monotonous dependence
with $k$. It decreases[increases], goes through a minimum[maximum] around $k =
11$, then increases and asymptotically converges towards a definite value for
large $k$ $\theta_{c,k \rightarrow \infty}=0.500(2)$[$\theta^i_{c,k \rightarrow
\infty}=0.500(1)$]; $(2)$ $\vartheta_{c,k}$[$\vartheta^i_{c,k}$] rapidly
increases[decreases] and asymptotically converges towards a definite value for
infinitely long $k$-mers $\vartheta_{c,k \rightarrow
\infty}=0.5334(6)$[$\vartheta^i_{c,k \rightarrow \infty}=0.4666(6)$]; $(3)$ for
both models, the curves of standard and inverse percolation thresholds are
symmetric with respect to $\theta = 0.5$. Thus, a complementary property is
found $\theta_{c,k} + \theta^i_{c,k} = 1$ (and $\vartheta_{c,k} +
\vartheta^i_{c,k} = 1$), which has not been observed in other regular lattices.
This condition is analytically validated by using exact enumeration of
configurations for small systems; and $(4)$ in all cases, the model presents
percolation transition for the whole range of $k$.
