  In this paper we introduce a new combinatorial approach to analyze the trace
of large powers of Wigner matrices. Our approach is motivated from the paper by
\citet{sosh}. However the counting approach is different. We start with
classical word sentence approach similar to \citet{AZ05} and take the
motivation from \citet{sinaisosh}, \citet{sosh} and
\citet{peche2009universality} to encode the words to objects similar to Dyck
paths. To be precise the map takes a word to a Dyck path with some edges
removed from it. Using this new counting we prove edge universality for large
Wigner matrices with sub-Gaussian entries. One novelty of this approach is
unlike \citet{sinaisosh}, \citet{sosh} and \citet{peche2009universality} we do
not need to assume the entries of the matrices are symmetrically distributed
around $0$. The main technical contribution of this paper is two folded.
Firstly we produce an encoding of the ``contributing words" (for definition one
might look at Section \ref{sec:word}) of the Wigner matrix which retrieves the
edge universality. Hence this is the best one can do. We hope this method will
be applicable to many other scenarios in random matrices. Secondly in course of
the paper we give a combinatorial description of the GOE Tracy Widom law. The
explanation for GUE is very similar. This explanation might be important for
the models where exact calculations are not available but some combinatorial
structures are present.
