k! is k factorial... from 1 to 4 is what's given..
Ek... that is just [k,0,0,0....]T [0,k,0,0...]T to whatever your dimension is.
so i did that [1,0,0,0]T [0,2!,0,0]T [0,0,3!,0]T [0,0,0,4!]T ... which turned out to be [1,0,0,0]T [0,2,0,0]T [0,0,6,0]T [0,0,0,24]T
state vectors are...