@article{Shar:2016aa,
    abstract = {The set of 123-avoiding permutations (alias words in {\$}{\$}{\{}{$\backslash$}{\{}1, {$\backslash$}ldots, n{$\backslash$}{\}}{\}}{\$}{\$}with exactly 1 occurrence of each letter) is famously enumerated by the ubiquitous Catalan numbers, whose generating function C(x) famously satisfies the algebraic equation {\$}{\$}{\{}C(x) = 1 + xC(x)\^{}{\{}2{\}}{\}}{\$}{\$}. Recently, Bill Chen, Alvin Dai, and Robin Zhou found (and very elegantly proved) an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of {\$}{\$}{\{}{$\backslash$}{\{}1, {$\backslash$}ldots, n{$\backslash$}{\}}{\}}{\$}{\$}. Inspired by the Chen-Dai-Zhou result, we present an algorithm for finding such an algebraic equation for the ordinary generating function enumerating 123- avoiding words with exactly r occurrences of each of {\$}{\$}{\{}{$\backslash$}{\{}1, {$\backslash$}ldots, n{$\backslash$}{\}}{\}}{\$}{\$}for any positive integer r, thereby proving that they are algebraic, and not merely D-finite (a fact that is promised by WZ theory). Our algorithm consists of presenting an algebraic enumeration scheme, combined with the Buchberger algorithm.},
    author = {Shar, Nathaniel and Zeilberger, Doron},
    date = {2016/06/01},
    date-added = {2023-11-6 16:15:21 +0100},
    date-modified = {2023-11-06 16:14:51 +0100},
    doi = {10.1007/s00026-016-0308-y},
    id = {Shar2016},
    isbn = {0219-3094},
    journal = {Annals of Combinatorics},
    number = {2},
    pages = {387--396},
    title = {The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2, . . . , n Are Always Algebraic},
    url = {https://doi.org/10.1007/s00026-016-0308-y},
    volume = {20},
    year = {2016},
    bdsk-url-1 = {https://doi.org/10.1007/s00026-016-0308-y}
}

@article{Shar:2016aa, abstract = {The set of 123-avoiding permutations (alias words in {\$}{\$}{{}{$\backslash$}{{}1, {$\backslash$}ldots, n{$\backslash$}{}}{}}{\$}{\$}with exactly 1 occurrence of each letter) is famously enumerated by the ubiquitous Catalan numbers, whose generating function C(x) famously satisfies the algebraic equation {\$}{\$}{{}C(x) = 1 + xC(x)\^{}{{}2{}}{}}{\$}{\$}. Recently, Bill Chen, Alvin Dai, and Robin Zhou found (and very elegantly proved) an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of {\$}{\$}{{}{$\backslash$}{{}1, {$\backslash$}ldots, n{$\backslash$}{}}{}}{\$}{\$}. Inspired by the Chen-Dai-Zhou result, we present an algorithm for finding such an algebraic equation for the ordinary generating function enumerating 123- avoiding words with exactly r occurrences of each of {\$}{\$}{{}{$\backslash$}{{}1, {$\backslash$}ldots, n{$\backslash$}{}}{}}{\$}{\$}for any positive integer r, thereby proving that they are algebraic, and not merely D-finite (a fact that is promised by WZ theory). Our algorithm consists of presenting an algebraic enumeration scheme, combined with the Buchberger algorithm.}, author = {Shar, Nathaniel and Zeilberger, Doron}, date = {2016/06/01}, date-added = {2023-11-6 16:15:21 +0100}, date-modified = {2023-11-06 16:14:51 +0100}, doi = {10.1007/s00026-016-0308-y}, id = {Shar2016}, isbn = {0219-3094}, journal = {Annals of Combinatorics}, number = {2}, pages = {387--396}, title = {The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2, . . . , n Are Always Algebraic}, url = {https://doi.org/10.1007/s00026-016-0308-y}, volume = {20}, year = {2016}, bdsk-url-1 = {https://doi.org/10.1007/s00026-016-0308-y} }