Counting segmented permutations using bicoloured Dyck paths

Anders Claesson

Research output: Contribution to journalArticle

Abstract

A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation π is σ-segmented if every occurrence o of σ in π is a segment-occurrence (i.e., o is a contiguous subword in π).

We show combinatorially the following two results: The 132-segmented permutations of length n with k occurrences of 132 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps. Similarly, the 123-segmented permutations of length n with k occurrences of 123 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps, each of height less than 2.

We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length 2n with k red up-steps, each of height less than h. This generating function is expressed in terms of Chebyshev polynomials of the second kind.
Language English R39 18 The Electronic Journal of Combinatorics 12 Published - 17 Aug 2005

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Dyck Paths
Counting
Permutation
Polynomials
One to one correspondence
Color
Generating Function
Subword
Chebyshev Polynomials

Keywords

• bicoloured Dyck path
• Dyck path
• segmented permutations

Cite this

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title = "Counting segmented permutations using bicoloured Dyck paths",
abstract = "A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation π is σ-segmented if every occurrence o of σ in π is a segment-occurrence (i.e., o is a contiguous subword in π).We show combinatorially the following two results: The 132-segmented permutations of length n with k occurrences of 132 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps. Similarly, the 123-segmented permutations of length n with k occurrences of 123 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps, each of height less than 2.We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length 2n with k red up-steps, each of height less than h. This generating function is expressed in terms of Chebyshev polynomials of the second kind.",
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author = "Anders Claesson",
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Counting segmented permutations using bicoloured Dyck paths. / Claesson, Anders.

In: The Electronic Journal of Combinatorics, Vol. 12, R39, 17.08.2005.

Research output: Contribution to journalArticle

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N2 - A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation π is σ-segmented if every occurrence o of σ in π is a segment-occurrence (i.e., o is a contiguous subword in π).We show combinatorially the following two results: The 132-segmented permutations of length n with k occurrences of 132 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps. Similarly, the 123-segmented permutations of length n with k occurrences of 123 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps, each of height less than 2.We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length 2n with k red up-steps, each of height less than h. This generating function is expressed in terms of Chebyshev polynomials of the second kind.

AB - A bicoloured Dyck path is a Dyck path in which each up-step is assigned one of two colours, say, red and green. We say that a permutation π is σ-segmented if every occurrence o of σ in π is a segment-occurrence (i.e., o is a contiguous subword in π).We show combinatorially the following two results: The 132-segmented permutations of length n with k occurrences of 132 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps. Similarly, the 123-segmented permutations of length n with k occurrences of 123 are in one-to-one correspondence with bicoloured Dyck paths of length 2n−4k with k red up-steps, each of height less than 2.We enumerate the permutations above by enumerating the corresponding bicoloured Dyck paths. More generally, we present a bivariate generating function for the number of bicoloured Dyck paths of length 2n with k red up-steps, each of height less than h. This generating function is expressed in terms of Chebyshev polynomials of the second kind.

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