2014
DOI: 10.1016/j.aam.2014.01.006
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Using functional equations to enumerate 1324-avoiding permutations

Abstract: We consider the problem of enumerating permutations with exactly r occurrences of the pattern 1324 and derive functional equations for this general case as well as for the pattern avoidance (r = 0) case. The functional equations lead to a new algorithm for enumerating length n permutations that avoid 1324. This approach is used to enumerate the 1324-avoiders up to n = 31. We also extend those functional equations to account for the number of inversions and derive analogous algorithms.

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Cited by 16 publications
(27 citation statements)
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“…At present, the best estimate seems to be due to Conway and Guttmann [78]. They extended the approach of Johansson and Nakamura [106] (who had computed the first 31 terms of the enumeration) to compute the first 36 terms of the enumeration of this class and then applied the methodology laid out in Guttmann [98] to approximate that Av n (1324) ∼ Cµ n ν √ n n g , where C ≈ 9.5, ν ≈ 0.04, g ≈ −1.1. and µ-the growth rate-is approximately 11.60.…”
Section: Avoiding a Longer Permutationmentioning
confidence: 95%
“…At present, the best estimate seems to be due to Conway and Guttmann [78]. They extended the approach of Johansson and Nakamura [106] (who had computed the first 31 terms of the enumeration) to compute the first 36 terms of the enumeration of this class and then applied the methodology laid out in Guttmann [98] to approximate that Av n (1324) ∼ Cµ n ν √ n n g , where C ≈ 9.5, ν ≈ 0.04, g ≈ −1.1. and µ-the growth rate-is approximately 11.60.…”
Section: Avoiding a Longer Permutationmentioning
confidence: 95%
“…Despite the abundance of data we have for this example, we are not able to fit its generating function to any algebraic differential equation. Interestingly this means that in the chain of classes Av(4231, 4123, 4312) ⊂ Av(4231, 4312) ⊂ Av(4231), the first class is easy to enumerate (we can compute terms in polynomial time) seems to lack a D-finite generating function, the second has an algebraic generating function (see Section 2.2 where we analyze it as a C-machine), and the third seems very difficult to enumerate (the current record is 50 terms, computed by Conway, Guttmann, and Zinn-Justin [13] building on the work of Johansson and Nakamura [21] and Conway and Guttmann [12]).…”
Section: Structure Of the Papermentioning
confidence: 99%
“…At this point there is an isomorphism to the JN algorithm [20]. An example of all the computations done (in the order that they are finished) for permutations of length 6 is given in table 2…”
Section: Basic Algorithmmentioning
confidence: 99%
“…In this paper we give details of an improved algorithm for the enumeration of such PAPs, with which we obtained five further terms in the OGF beyond the existing longest known sequence, due to Johansson and Nakamura [20], using comparable computing resources. We will refer to their algorithm as the JN algorithm.…”
Section: Introductionmentioning
confidence: 99%