@article{NAKAMURA2013356,
    Abstract = {One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a ``formula'', or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate functional equations, alas, with an unbounded number of ``catalytic variables'', but then described a clever way, using multivariable calculus, on how to get enumeration schemes. Alas, their method becomes very complicated for r larger than 1. In the present article we describe a far simpler way to squeeze the necessary information, in polynomial time, for increasing patterns of any length and for any number of occurrences r.},
    Author = {Nakamura, Brian and Zeilberger, Doron},
    File = {Using Noonan–Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes - 1-s2.0-S0196885812001054-main - a.pdf},
    ISSN = {0196-8858},
    Journal = {Advances in Applied Mathematics},
    Keywords = {Permutation patterns, Functional equations, Exact enumeration, Algorithm},
    Number = {3},
    Pages = {356-366},
    Title = {Using Noonan--Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes},
    URL = {https://www.sciencedirect.com/science/article/pii/S0196885812001054},
    Volume = {50},
    Year = {2013},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0196885812001054},
    bdsk-url-2 = {https://doi.org/10.1016/j.aam.2012.10.003},
    date-added = {2023-02-26 07:58:30 +0100},
    date-modified = {2023-02-26 07:58:30 +0100},
    doi = {10.1016/j.aam.2012.10.003}
}

@article{NAKAMURA2013356, Abstract = {One of the most challenging problems in enumerative combinatorics is to count Wilf classes, where you are given a pattern, or set of patterns, and you are asked to find a formula'', or at least an efficient algorithm, that inputs a positive integer n and outputs the number of permutations avoiding that pattern. In 1996, John Noonan and Doron Zeilberger initiated the counting of permutations that have a prescribed, r, say, occurrences of a given pattern. They gave an ingenious method to generate functional equations, alas, with an unbounded number ofcatalytic variables'', but then described a clever way, using multivariable calculus, on how to get enumeration schemes. Alas, their method becomes very complicated for r larger than 1. In the present article we describe a far simpler way to squeeze the necessary information, in polynomial time, for increasing patterns of any length and for any number of occurrences r.}, Author = {Nakamura, Brian and Zeilberger, Doron}, File = {Using Noonan–Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes - 1-s2.0-S0196885812001054-main - a.pdf}, ISSN = {0196-8858}, Journal = {Advances in Applied Mathematics}, Keywords = {Permutation patterns, Functional equations, Exact enumeration, Algorithm}, Number = {3}, Pages = {356-366}, Title = {Using Noonan--Zeilberger Functional Equations to enumerate (in polynomial time!) generalized Wilf classes}, URL = {https://www.sciencedirect.com/science/article/pii/S0196885812001054}, Volume = {50}, Year = {2013}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0196885812001054}, bdsk-url-2 = {https://doi.org/10.1016/j.aam.2012.10.003}, date-added = {2023-02-26 07:58:30 +0100}, date-modified = {2023-02-26 07:58:30 +0100}, doi = {10.1016/j.aam.2012.10.003} }