week3 - S Tanny c MAT 344F Fall 1997 r distinct balls into...

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S. Tanny MAT 344F Fall, 1997 1 c) r distinct balls into n nondistinct boxes: k = 1 n S (r, k) where S(r,k) = number of ways to partition an r-set into k (non-empty) parts. S(r,k) is called a Stirling number of the Second Kind. Try to compute some S(r,k): eg: S(r,0) = 1 2200 r 0. S(1,1) = 1 S(r,2) = 2 r - 1 - 1 S(r,1) = 1 S(r,3) (ii) 1 (iii) S(r,n) d) r non-distinct balls into n nondistinct boxes: (i) k = 1 n k (r) P , where P k (r) = number of ways to partition the numbers r into k parts, each part > 0 Think of the number r as a sum of r ones: r = 1 + 1 + 1 + + 1 _____________ r times We can group the ones as desired to form different parts. For example, for 3 we have: 3 1 + 2 - (2 parts, one each of size 1,2) 1 + 1 + 1 - (3 parts, all of size 1) 3 - (1 part of size 3) For 4 we have: 4 1 + 3 2 + 2 - (2 parts, each of size 2) 1 + 1 + 2 - (3 parts)
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S. Tanny MAT 344F Fall, 1997 2 (ii) 1 (iii) n (r) k=1 r-n k (r - n) P = P Ex. (i) How many non-negative integer solutions to x 1 + x 2 + x 3 + x 4 = 12 ? (ii) x i > 0 (iii) x 1 2 , x 2 2 , x 3 4 , x 4 0 Solution (i) 15 12 = 15 3 (ii) 11 8
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