By Wilf H.
This fabric is meant for a path that may mix a examine of combinatorial buildings with introductory recursive programming in Maple.
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Extra resources for East side, West side. Lectures on combinatorial objects with Maple
Example text
Thus, the list of all k-subsets of n elements corresponds to a list of nk codewords of n 0’s and 1’s. The 10 3-subsets of {1, 2, 3, 4, 5} correspond to the list 00000, 01000, 01100, 01110, 10000, 10100, 10110, 11000, 11010, 11100 of codewords. Note that these are arranged in lexicographic order, considered as 5 letter words over an alphabet of two letters {0, 1}. It is easy to design a recursive program that will list, in lexicographic order, all of the codewords that correspond to walks that originate at a fixed vertex (n, k).
For this example partition Π, 8 is not in a singleton class, so it is preceded in the list by all of the 73 partitions in which 8 does live in a singleton class. Further, 8 does not live in the first class of Π, so Π is also preceded by all 74 partitions that are obtained by inserting 8 into the first class of some partition of 7 letters into 4 classes. Thus, if Π denotes the partition obtained from Π by deleting 8, then rank(Π) = 7 7 + + rank(Π ), 3 4 which is the idea of the recursive algorithm.
Imagine that all of the partitions of n letters into exactly k classes have been strewn on the sidewalks and streets, all around the town. Now take a walking tour and inspect them as they lie on the ground. If, in a certain set partition, the letter n lives in a class all by itself, then move that set partition to the east side of town, and if, on the contrary, n lives in a class with at least one other letter, then move it to the west side. Now instead of one large collection of set partitions all over the place, we have two nice neat piles, one on the east side, and the other on the west side of town.