Generalized Permutations and The Multinomial Theorem

Transcription

1 Generalized Permutations and The Multinomial Theorem 1 / 19

2 Overview The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 2 / 19

3 Outline The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 3 / 19

4 The Binomial Theorem Theorem (x + y) n = n [ C(n, r) x n r y r] r=0 4 / 19

5 Binary Sequences Count the number of binary sequences of length n in two different ways. 5 / 19

6 Outline The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 6 / 19

7 Generalized Permutations Definition Let X be a set of n not necessarily distinct objects belonging to k different nonempty groups such that 1. all the objects in a group are identical 2. an object in a group is not identical to an object in another group. A generalized permutation of X is an arrangement in a row of the n objects of X. Anagrams are generalized permutations. A famous contemporary example: IAMLORDVOLDEMORT TOMMARVOLORIDDLE 7 / 19

8 The Number of Anagrams Theorem If the set X of n objects consists of k different nonempty groups such that group i has n i identical objects for 1 i k, then the number of generalized permutations of X is n! (n 1!)(n 2!) (n k!). [anagram tool] Example Determine the number of generalized permutations of the 5 letters that appear in the word LEMMA. 8 / 19

9 The Number of Anagrams Theorem If the set X of n objects consists of k different nonempty groups such that group i has n i identical objects for 1 i k, then the number of generalized permutations of X is n! (n 1!)(n 2!) (n k!). [anagram tool] Example Determine the number of generalized permutations of the 6 letters that appear in the word TSETSE. 9 / 19

10 Some Identities Definition P(n; n 1, n 2,..., n k ) := P(n,n 1+ +n k ) (n 1!)(n 2!) (n k!) Proposition We have the following combinatorial identities: 1. P(n; r) = P(n; n r) = P(n; r, n r) 2. P(n; r) = P(n,r) r!. 10 / 19

11 The Allocation Interpretation of Generalized Permutations Theorem If there are n i identical objects in group i for 1 i k and if r = n n k is the total number of the objects in these k groups, then these r objects can be placed in n distinct locations so that each location receives at most one object in P(n; n 1, n 2,..., n k ) ways. In particular, if each group has exactly one object, then this number of allocations is P(n, r). 11 / 19

12 Outline The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 12 / 19

13 The Multinomial Theorem Theorem In a typical term of the expansion of (x 1 + x x k ) n the variable x i appears n i times (where n 1 + n n k = n) and the coefficient of this typical term is P(n; n 1, n 2,..., n k ) = n! (n 1!)(n 2!) (n k!). 13 / 19

14 Outline The Binomial Theorem Generalized Permutations The Multinomial Theorem Circular and Ring Permutations 14 / 19

15 Circular Permutations Circular permutations are a variant of the r-permutations of a set X of n distinct elements we have been considering. Suppose that we now assume that two permutations are the same provided that one can be obtained from the other by cycling. For example, the 3-permutations of the set X = {A, B, C} given by ABC, CAB, and BCA are the same when considered as circular permutations. 15 / 19

16 Circular Permutations Circular permutations are a variant of the r-permutations of a set X of n distinct elements we have been considering. Suppose that we now assume that two permutations are the same provided that one can be obtained from the other by cycling. For example, the 3-permutations of the set X = {A, B, C} given by ABC, CAB, and BCA are the same when considered as circular permutations. Proposition The number of circular permutations of a set of n elements is P(n, n) n = (n 1)!. 16 / 19

17 Ring Permutations Supposing that two permutations are the same provided that one can be obtained from the other by cycling or by mirror reversal, we obtain the notion of a ring permutation. For example, the 3-permutations of the set X = {A, B, C} given by ABC, CAB, BCA, CBA, BAC, and ACB are the same when considered as ring permutations. 17 / 19

18 Ring Permutations Supposing that two permutations are the same provided that one can be obtained from the other by cycling or by mirror reversal, we obtain the notion of a ring permutation. For example, the 3-permutations of the set X = {A, B, C} given by ABC, CAB, BCA, CBA, BAC, and ACB are the same when considered as ring permutations. Proposition The number of ring permutations of a set of n elements is 1 P(n, n) 2 n = (n 1)! / 19

19 Acknowledgements Statements of results follow the notation and wording of Balakrishnan s Introductory Discrete Mathematics. Some examples follow Rosen s Discrete Mathematics and Its Applications. 19 / 19