Permutations of multisets Lecture 3 (Brualdi Ch. 2.4) Wednesday, September 9th Lecture 3 page 2 , Denition: A multiset M is a set S together with a function R : S ! f0, 1, 2, . .g [ f g where R(x) is the repetition number of x. By the cardinality (or size) of M we 1 mean jMj = R(x). x2S X Example: How many ways can we order letters in MISSISSIPPI ? Lecture 3 page 3 , Theorem: Given a multiset M = fn1 · a1, n2 · a2,..., nk · akg. The number of permutations of M is where n = jMj = n1 + n2 + ... + nk. Example: Given nite set S, jSj = n and labelled boxes Box 1, Box 2, . , Box k. Distribute all the elements of S into boxes as follows: Box1 gets exactly n1 elements. Box2 gets exactly n2 elements. .............................. Box k gets exactly nk elements. How many ways to distribute elements are there? Lecture 3 page 4 , Denition: Given multiset M. For an integer k > 0a k-permutation of M is a sequence a1, a2,..., ak where multiset fa1, a2,..., akg is contained in M. Example: For the multiset M = f · a1, · a2,..., · amg. Find the number of k-permutations for any k 0. 1 1 > 1 Example: For the multiset M = f1 · a1, 1 · a2, . , 1 · amg. Find the number of k-permutations for any k > 0. Lecture 3 page 5 , Example: For the multiset M = f5 · x, 7 · y, 8 · zg. Find the number of 18-permutations of M. Lecture 3 page 6 , Consider n × n chessboard. Recall in game of chess rooks move horizontally or vertically. 2 rooks are in non-attacking position if they are in distinct rows and distinct columns. Example: How many ways we can place 3 mutually non-attacking rooks on a3 × 3 chessboard. Example: How many ways we can place n mutually non-attacking rooks on a n × n chessboard. Lecture 3 page 7 , Example: Now me color the rocks with k distinct colors, require: n1 rooks get color 1 n2 rooks get color 2 .................. nk rooks get color k. How many colored rook arrangements are possible? Lecture 3 page 8 , Example: A committee of 7 people is chosen from a club that has a membership of 10 men and 12 women. The committee must have at least 2 women and men. How many ways to form a committee?.