Counting segmented permutations using bicoloured Dyck paths
Claesson, Anders (2005) Counting segmented permutations using bicoloured Dyck paths. The Electronic Journal of Combinatorics, 12. R39. ISSN 10778926
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A bicoloured Dyck path is a Dyck path in which each upstep is assigned one of two colours, say, red and green. We say that a permutation π is σsegmented if every occurrence o of σ in π is a segmentoccurrence (i.e., o is a contiguous subword in π). We show combinatorially the following two results: The 132segmented permutations of length n with k occurrences of 132 are in onetoone correspondence with bicoloured Dyck paths of length 2n−4k with k red upsteps. Similarly, the 123segmented permutations of length n with k occurrences of 123 are in onetoone correspondence with bicoloured Dyck paths of length 2n−4k with k red upsteps, 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 upsteps, each of height less than h. This generating function is expressed in terms of Chebyshev polynomials of the second kind.


Item type: Article ID code: 49794 Dates: DateEvent17 August 2005PublishedSubjects: Science > Mathematics > Electronic computers. Computer science Department: Faculty of Science > Computer and Information Sciences Depositing user: Pure Administrator Date deposited: 14 Oct 2014 13:49 Last modified: 08 Apr 2024 21:44 Related URLs: URI: https://strathprints.strath.ac.uk/id/eprint/49794