### Citations

114 | Permutations with restricted patterns and Dyck paths
- Krattenthaler
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Citation Context ...steps to obtain a one-size-smaller Dyck path). Proposition 1. [4] The number of indecomposable 321-avoiding permutations on [n] is Cn−1. Proof. One method is to observe that Krattenthaler’s bijection =-=[5]-=- from 321-avoiding permutations on [n] to Dyck paths of size n preserves components in the obvious sense 2 and so sends indecomposable permutations to indecomposable paths. Given a nonnegative path, s... |

1 | The number of {1243, 2134}-avoiding permutations
- Callan
- 2013
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Citation Context ...We show that permutations avoiding both of the (classical) patterns 4321 and 3241 have the algebraic generating function conjectured by Vladimir Kruchinin. 1 Introduction This paper is a companion to =-=[1]-=-, which established the algebraic generating function for {1243, 2134}-avoiding permutations conjectured by Vaclav Kotesovec [2]. In similar vein, Vladimir Kruchinin [3] has conjectured the generating... |

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comment in Formula section of sequence A164651 in The On-Line Encyclopedia of Integer Sequences
- Kotesovec
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Citation Context ...ured by Vladimir Kruchinin. 1 Introduction This paper is a companion to [1], which established the algebraic generating function for {1243, 2134}-avoiding permutations conjectured by Vaclav Kotesovec =-=[2]-=-. In similar vein, Vladimir Kruchinin [3] has conjectured the generating function 1 1− xC(xC(x)) for {4321, 3241}-avoiding permutations, where C(x) := 1− √ 1−4x 2x denotes the generating function for ... |

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comment in Formula section of sequence A165543 in The OnLine Encyclopedia of Integer Sequences
- Kruchinin
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Citation Context ...n This paper is a companion to [1], which established the algebraic generating function for {1243, 2134}-avoiding permutations conjectured by Vaclav Kotesovec [2]. In similar vein, Vladimir Kruchinin =-=[3]-=- has conjectured the generating function 1 1− xC(xC(x)) for {4321, 3241}-avoiding permutations, where C(x) := 1− √ 1−4x 2x denotes the generating function for the Catalan numbers. We will show that {4... |

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Catalan addendum (version of 25 May 2013), item (t6). The current version is available online at http://www-math.mit.edu/˜rstan/ec
- Stanley
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Citation Context ...it a Dyck path into its (indecomposable) components. The number of indecomposable Dyck paths of size n is Cn−1 (delete the first and last steps to obtain a one-size-smaller Dyck path). Proposition 1. =-=[4]-=- The number of indecomposable 321-avoiding permutations on [n] is Cn−1. Proof. One method is to observe that Krattenthaler’s bijection [5] from 321-avoiding permutations on [n] to Dyck paths of size n... |

1 | A combinatorial interpretation of the Catalan transform of the Catalan numbers, preprint
- Callan
- 1111
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Citation Context ...ts nonnegative paths of n + m upsteps U = (1, 1) and n downsteps D = (1,−1), where nonnegative means the path never dips below ground level, the horizontal line through its initial vertex (see, e.g., =-=[6]-=-). A nonnegative path of n upsteps and n downsteps is a Dyck path and its size is n. A nonempty Dyck path is indecomposable if its only return to ground level is at the end. The returns to ground leve... |