| On Vince Vatter’s Brilliant Extension of Doron Zeilberger’s Enumeration Schemes for Herb Wilf’s Classes Retail vs. Wholesale Enumeration (2008) | |||||||||||||
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| An enumerative retailer studies one problem at a time. On the other hand, an enumerative wholesaler studies a family of problems, and tries to design algorithms that can be implemented to solve lots and lots of problems from that family. Doron Zeilberger’s Original Enumeration Schemes In [Z1] I tackled the problem of enumerating Wilf classes, i.e. finding “schemes ” for enumerating sets of permutations avoiding a given set of patterns. Alas, the success rate was rather disappointing. Vince Vatter’s Brilliant Extension In [V], my brilliant disciple, Vince Vatter, introduced a far-reaching extension: gap vectors! This made the success rate much higher! Zeilberger’s original approach My original approach was to teach the computer some elementary ‘logical reasoning’, and find enumeration schemes that way. This was done in a Maple package (still available from my website) called WILF. To check the ‘logical ’ program, I also wrote an empirical program, called HERB. The Maple package HERB found enumeration schemes by testing them for permutations of size n ≤ N0, where N0 was some positive integer chosen by the user. But to rigorously prove the validity of | |||||||||||||
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