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East side, West side. Lectures on combinatorial objects with by Wilf H.

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By Wilf H.

This fabric is meant for a path that might mix a research of combinatorial buildings with introductory recursive programming in Maple.

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N−1} into k −1 classes and then insert the letter n as a new singleton class. / nk , we will choose uar a partition of {1, . . , n − 1} into With probability 1 − n−1 k−1 k classes and then insert the letter n into a randomly chosen one of those k classes. The Maple program9 follows. k); RETURN([op(RandSetPtns(n-1,k)),class()]) fi; fi; end: 9 See the program notes on page 65. 7 Ranking set partitions What is the rank of the partition Π := {1, 4}{2, 7, 8}{3}{5, 6} of the set {1, 2, . . , 8} into 4 classes in the list of all partitions of that set into 4 classes?

It begins as shown in Fig. 3. Look at the sums of the rows in this Stirling triangle. These are the numbers 1, 2, 5, 15, 52, . , and they are called the Bell numbers. The Bell number b(n) is the number of partitions of a set of n elements. So b(4) = 15 means that there are 15 ways to partition a set of 4 elements. Can you write them all out? Let’s take some particular number in this array and try to visualize what it counts. Like the 6 that appears in the (n, k) = (4, 3) position in the array. It says that a set of 4 things can be partitioned into 3 classes in exactly 6 ways.

For this we enter > List(3,5,2); which gets the answer [[1, 1, 1, 1, 1, 1], [1, 2, 2, 2, 2, 1], 2], [1, 2, 2, 2], [1, 1, 1, 2, 1], [1, 1, 2, 1, 2, 1, 1], [1, 1, 2, 2, 1], [1, 2, [1, 1, 1, 2, 2], [1, 1, 2, 1, 2], 1, 2], [1, 1, 2, 2, 2], [1, 2, 1, 2, 2, 2]] 1], [1, 2, 1, 1, 2, 1], [1, [1, 2, 1, 1, 2, 2], [1, 2, • In the list of all partitions of the set {1, 2, 3, 4, 5} into 2 classes, what is the successor of the partition [1,1,2,2,1]? For this we enter > Successor(3,5,2,[1,1,2,2,1]); and Maple says [1,2,1,2,1] My thanks to Joanna Nordlicht for her assistance in writing this package.

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