The Multinomial coefficients in this layer represent the total number of possibilities to choose a,then b,then c things from a total of 6 objects.
As long as a+b+c=6, "6 choose a, b, c" exists.
As you can see, if we start with the entry 6 choose 0,0,6... we can move 2 spots to the left, which is equivalent to gaining 2 x's and losing two z's (x^2*y^0*z^4) then we can move up-left one spot, which is equivalent to gaining a y and losing a z (x^2*y^1*z^3), the corresponding coefficient is 60 (6 choose 2, choose 1, choose 3)