Question

Complete number 29 ALL parts and SHOW ALL WORK!

gic and those where it does not.) 2。Suppose B is an n × n board and r.(B) is the coefficient of xin the rook polynomial R(x, B). Use recurrence relations to compute r(B) if (a) B has all squares darkened; (b) B has only the main diagonal lightened.

Answer 1

Let B be alpha board of and decomposes into dis - joined sub - boards B_1 and B_2 then rk (B) = e k (B_1) r (B_2) + rk - 1 (B_1) r (B_2) + + ro (B_1) (B_2) then rk (B) = rk (B_1) ro (B_2) + rk - 1 (B_1) r1 (B_2) + + ro (B_1) (B_2). we defined the roots polynomial R (x, B) of the Borad B as follows B as follows. R(x, B) = ro (B) + r_1 (B) + R_2 (B) r_2 + r_3 (B) x_3 + + n (B) xnt let B be a board of sources that into dis - joint sub - boards B_2 then R (x, B) = R (x, B_1) R (x_1 B_2) proof: R(x_1 B) = ro (B) + r_1 (B) x + r_2 (B) x 2 + r_3 (B) x_3 1 + r_1 (B_1) ro (B_2) + r_0 (B_1) x + r_1 (B_2) x + r_2 (B_1) ro (B_2) + r_1 (B_1)