Let $$x^0 = 1, x^1 = x, x^2 = x(x-1), x^3 = x(x-1)(x-2), x^4 = x(x-1)(x-2)(x-3), etc $$.
Let $$V = P_d$$ then $$B = [x^0, x^1, ..., x^n]$$ and $$E = [x, .., x_d]$$are both basis of V. Let S1 be the change of basis matrix $P_e ->_b $and let S2 be the change of basis matrix $P_b->_e$. Both of these d+1 by d-1 matrices are upper triangular. Number the rows and columns of the matrices: 0, ..., d instead of 0, ..., d + 1.
Find then the matrices for d = 5, thereby computing the Stirling matrices.