DISCRETE STRUCTURES SOLUTION 4 1. Call a partition P of An special, if no subset in P contains two consecutive integers. Denote by S '(n, r ) the number of special partitions of An containing exactly r non-empty subsets. First I prove by induction on n, that S '(n, r ) S (n 1, r 1) for all n 1 and for all r in the range 1 r n . For n = 1 we have S '(1,1) 1 S (0, 0) and the induction basis is proved. So take n 2 and 1 r n , and assume that the claim holds for S '( n 1, r ') for all legitimate indices r ' . Consider the special partition of An1 into r-1 non empty subsets. Adding {n} as singleton gives a special partition of An . Then consider a special partition of An1 into r non-empty subsets. Adding n to any one the r-1 subsets not containing n-1 yields a special partition of An . All special partitions of An into r non-empty subsets can be generated in this way. Subsequently: S '(n, r ) S '(n 1, r 1) (r 1) S '(n 1, r ) S (n 2, r 2) (r 1) S (n 2, r 1) [by the Induction hypothesis] S (n 1, r 1) [by the recurrence for Stirling numbers] And the inductive step is established. For n 1 it then follows that n n Bn' S '(n, 0) S '(n, r ) S '(n, r ) [Since S '(n, 0) 0 ] r 1 r 1 n n 1 r 1 r 1 0 S (n 1, r 1) S (n 1, r 1) Bn 1 2. The characteristic equation is ( X ) X 3 5 X 2 3X 9 X 3 3X 2 2 X 2 6 X 3X 9 X 3 X 2 2 X 3 X 3 X 1 X 3 X 1 X 3 0 2 Therefore, a general solution of the given recurrence is: Tn c(1) n c '3n c "3n Plugging in the initial values gives: c c ' 1, c 3c ' 3c " 2 , c 9c '18c " 28 The solution of this system is c Tn 25 9 7 , c ' , c " . Then we have 16 16 4 25 9 7 (1) n n 3n . 16 16 4 3. We prove all the three properties as follows: [Reflexive] For any a A we have min(a) = min(a). [Symmetric] For any a, b A , if min(a) = min(b), then min(b) = min(a) [Transitive] For any a, b, c A , if min(a) = min(b), min(b) = min(c), then min(a) = min(c). 4. The proof is as follows: n Given L(1) = 3 and L(n) L 1 2 n n So L L 1 2 4 n n i.e. L(n) L 2 4 n Similarly, L(n) L 3 and so on. 8 n In general, L(n) L a a for any positive integer a 2 a Let 2 n , or a log 2 n i.e. L(n) L 1 1 4 n L(n) L log 2 n n i.e. L(n) L 1 log 2 n i.e. L(n) log 2 n 3 . 1 5 . So a particular 2 solution is of the form 1n . Substitution gives d , i.e. d . 5. The characteristic equation x 2 x 1 has roots x Therefore, a general solution is of the form n n 1 5 1 5 Tn u v d 2 2 where u, v are constants to be determined from the initial conditions 1 5 1 5 T1 a u v d , 2 2 3 5 3 5 T2 b u v d 2 2 Solving this system gives, 1 3 5 a 1 5 b 2d 2 5 1 v 3 5 a 1 5 b 2d 2 5 u Notice that the exact values of the constants u, v, w are not very important here. It suffices to know the form of the solution. In particular, since a, b, d are positive, it follows that u > 0. Also 1 5 1 . Consequently, 2 1 5 n Tn . 2 S 6. Let n ,3 denote the total number of bit strings of length n that have three consecutive 0s. Now we have two possibilities, if 1 occurs on the first place or 0 occurs in the first place. If 1 occurs, then the problem reduces to finding the number of bit strings of length n-1 that have three consecutive 0s i.e. S n 1,3 and if 0 occurs in the first place then we look for the 2nd digit. Similarly if 1 occurs in the 2nd digit as well, then we have to solve the relation S n 2,3 , and if again 0 occurs we see the 3rd digit, if 1 occurs here then we solve S n 3,3 otherwise the remaining n-3 places can be either filled with 1 or 0, giving a total number of ways as 2n3 Hence the recurrence relation becomes Sn,3 Sn1,3 Sn2,3 Sn3,3 2n3 S 2,3 0 , S1,3 0 , S 0,3 0 From the above recurrence relation, we get S3,3 1 S 4,3 3 S5,3 8 S 6,3 20 S7,3 47 7. If n is odd, the f(n+2) = 2f(n+1) + 1 and f(n+1) = 2f(n); if n is even, f(n+2) = 2f(n+1) and f(n+1) = 2f(n) + 1. So the result holds in either case. This is not a “pure” recurrence relation because of added 1. But if we set g(n) = f(n) + ½, the g(n+1) + 2g(n) = f(n+1) + 2f(n) + 3/2 = g(n+2). The characteristic equation is x 2 x 2 , with solutions 2 and -1, so g (n) a2n b(1)n . The initial values g(1) = 3/2 and g(2) = 5/2 yield a = 2/3 2 3 1 6 1 2 and b = -1/6. So f (n) 2n (1) n .