We investigate a generalization of stacks that we call C-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that C-machines generate, and how these systems of functional equations can be iterated and sometimes solved. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by C-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their counting sequences, seem to not have D-finite generating functions.
The set of 123-avoiding permutations (alias words in {1, ..., n} 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 {1, . . . , n}. 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 {1, . . . , n} 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.
Preface: How many permutations are there of length googol+30 avoiding an increasing subsequence of length googol?This number is way too big for our physical universe, but the number of permutations of length googol+30 that contain at least one increasing subsequence of length googol is a certain integer that may be viewed in http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/gessel64.pdf. Hence the number of permutations of length googol+30 avoiding an increasing subsequence of length googol is (googol + 30)! minus the above small number.
Counting the "Bad Guys"Recall that thanks to Robinson-Schensted ([Rob][Sc]), the number of permutations of length n that do not contain an increasing subsequence of length d is given by G d (n) := λ⊢n #rows(λ)
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