To calculate (2), it helps to visualize the $S_n$ as the successive positions in a random walk.
The knowledge of the invariant subspaces of an operator helps us to visualize its action.
Go to the list of words starting with: a
b
c
d
e
f
g
h
i
j
k
l
m
n
o
p
q
r
s
t
u
v
w
y
z