By Ainouche A., Schiermeyer I.
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Additional resources for 0-Dual Closures for Several Classes of Graphs
D1 Find all of the partitions of ͕A, B, C, D, E, F͖ into three two-element parts (type (2, 2, 2)). You should find 15 partitions. Suppose we compare the partitions in problem D1 with distributions of the set ͕A, B, C, D, E, F͖ into three boxes with two letters going into each box. In other words, we are comparing partitions of type (2, 2, 2) with distributions of type (2, 2, 2). The number of these distributions is 6 2, 2, 2 ϭ 6! ϭ 90 8 D2* Explain why the number of distributions counted in the preceding calculation is six times the number of partitions found in problem D1.
The answer to both is the sum of all of the distribution numbers in which m ϭ number of balls ϭ length of each word n ϭ number of boxes ϭ number of letters in the set 38 COMBINATORICS and the numbers m1 , . . ,mn Ն1 m1 ϩиииϩmn ϭm m m1 , . . , mn Notation The sum above is denoted by T (m, n). ) C30 Calculate T (6, 3) and T (6, 4). C31 Find T (m, 1) and T (m, m) for any m. Interpret the results in terms of Standard Problems #13 and #14. C32* Find the number of five-letter words that use letters from the set ͕A, B, C, D, E͖ and contain exactly three different letters.
Notation m m1 , m2 , . . , m n ϭ m! m2 ! и и и mn ! We now explain why this value gives the correct result for distributions of type (m1 , . . , mn ). The number of ways to select m1 balls to go into box 1 is mm1 . Of the remaining m Ϫ m1 balls, m2 of them must go into box 2. There 1 ways to select them. Continue in this way. The resulting number of are mϪm m2 distributions of all m balls into the n boxes is the product m m1 m Ϫ m1 m2 m Ϫ m1 Ϫ m2 m Ϫ m1 Ϫ и и и Ϫ mnϪ1 иии m3 mn C21 Use factorials to show that the product above is equal to the distribution number m m1 , m2 , .