  We define special Hamiltonian-paths and special permutations of the up-facing
dark tiles on a checked triangular grid related to the generalized
Sierpi\'{n}ski Gasket. Our definitions and observations make possible the
generalization of the Sierpi\'{n}ski Arrowhead Curve for all orders. We produce
these symmetric recursive curves in many ways by two kinds of asymmetric paths
which are in a bijective relation and unambiguously transformable into each
other in any order. These node-rewriting and edge-rewriting recursive curves
keep their self-avoiding and simple properties after the transformation and
their cardinality specifies a new integer sequence. We show a transformation
table to change the curves into each other and we give another table to change
them into Lindenmayer-system strings both by the absolute direction codes of
their edges.
