Permutations of multisets Lecture 3 (Brualdi Ch. 2.4)
Wednesday, September 9th Lecture 3 page 2 ,
De nition: A multiset M is a set S together with a function
R : S → {0, 1, 2, . . .} ∪ { }
where R(x) is the repetition number of x. By the cardinality (or size) of M we ∞ mean |M| = R(x). x∈S X Example: How many ways can we order letters in
MISSISSIPPI ? Lecture 3 page 3 ,
Theorem: Given a multiset
M = {n1 · a1, n2 · a2,..., nk · ak}.
The number of permutations of M is
where n = |M| = n1 + n2 + ... + nk.
Example: Given nite set S, |S| = 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 ,
De nition: Given multiset M. For an integer k > 0a k-permutation of M is a sequence
a1, a2,..., ak
where multiset {a1, a2,..., ak} is contained in M.
Example: For the multiset
M = { · a1, · a2,..., · am}. Find the number of k-permutations for any k 0. ∞ ∞ > ∞
Example: For the multiset
M = {1 · a1, 1 · a2, . . . , 1 · am}.
Find the number of k-permutations for any k > 0. Lecture 3 page 5 ,
Example: For the multiset
M = {5 · x, 7 · y, 8 · z}.
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?