  In arXiv:0907.3784, we introduced a variant of non-commutative
Donaldson-Thomas theory in a combinatorial way, which is related with
topological vertex by a wall-crossing phenomenon. In this paper, we (1) provide
an alternative definition in a geometric way, (2) show that the two definitions
agree with each other and (3) compute the invariants using the vertex operator
method, following Okounkov-Reshetikhin-Vafa and Young. The stability parameter
in the geometric definition determines the order of the vertex operators and
hence we can understand the wall-crossing formula in non-commutative
Donaldson-Thomas theory as the commutator relation of the vertex operators.
