Combinatorics of Compositions and Words

Heubach, Silvia; Mansour, Toufik

doi:10.1201/9781420072686

Cited by 117 publications

(102 citation statements)

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“…It has been shown using generating functions that the number of n-color compositions of ν is F 2ν , the 2ν-th Fibonacci number [1]. For example: The following relationships between restricted compositions and Fibonacci numbers are also well-known (see [3], for example): The number of compositions of ν having only parts of size 1 or 2, referred to here as 1-2 compositions, is F ν+1 . The number of compositions of ν having only odd parts, referred to here as odd compositions, is F ν .…”

Section: Introductionmentioning

confidence: 99%

New bijections from $n$-color compositions

Shapcott

2013

Journal of Combinatorics

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Combinatorial bijections are given from the set of n-color compositions of ν, for which a part of size n can take on n colors, to the set of compositions of 2ν − 1 having only parts of size 1 or 2, the set of compositions of 2ν having only odd parts, and the set of compositions of 2ν + 1 having no parts of size 1. A generalized bijection based on similar ideas is then given between the set of compositions of ν into parts congruent to a(mod b) and the set of compositions of ν + b − a into parts congruent to b(mod a) with each part greater than b − a.

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Section: Introductionmentioning

confidence: 99%

New bijections from $n$-color compositions

Shapcott

2013

Journal of Combinatorics

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“…We will call the random partition model described in the preceding sentence a uniform model and denote it by Π n,k because of its similarity to the uniform random graph model. Some related results on the issues concerning the largest parts and multiplicities of parts in random compositions can be found in Knopfmacher and Robbins [24], Hitczenko and Louchard [11], Hitczenko and Savage [12] and the book by Heubach and Mansour [10]. In these results, however, the number of parts is not a parameter so it seems they are not useful in our considerations.…”

Section: A Packing Number Asymptotic Formula For K = O(n)mentioning

confidence: 84%

Universal Cycle Packings and Coverings for k-Subsets of an n-Set

DźBski

Lonc

2016

Graphs and Combinatorics

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A cyclic sequence of elements of [n] is an (n, k)-Ucycle packing (respectively, (n, k)-Ucycle covering) if every k-subset of [n] appears in this sequence at most once (resp. at least once) as a subsequence of consecutive terms. Let p n,k be the length of a longest (n, k)-Ucycle packing and c n,k the length of a shortest (n, k)-Ucycle covering. We show that, for a fixed k, p n,and c n,, for some β < 1. Finally, we show that if k = o(n), then p n,k = n k (1 − o(1)).

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“…Jelínek and Mansour [11] and Brändén and Mansour [1] give bijective proofs of the same fact. For more information on pattern avoidance in words see [10]. It turns out that the bijection above can be used to give another bijective proof of this fact.…”

Section: A Bijection and Pattern Avoidance In Wordsmentioning

confidence: 99%

Pattern Avoidance in Ordered Set Partitions

et al. 2014

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In this paper we consider the enumeration of ordered set partitions avoiding a permutation pattern of length 2 or 3. We provide an exact enumeration for avoiding the permutation 12. We also give exact enumeration for ordered partitions with 3 blocks and ordered partitions with n-1 blocks avoiding a permutation of length 3. We use enumeration schemes to recursively enumerate 123-avoiding ordered partitions with any block sizes. Finally, we give some asymptotic results for the growth rates of the number of ordered set partitions avoiding a single pattern; including a Stanley-Wilf type result that exhibits existence of such growth rates.

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Combinatorics of Compositions and Words

Cited by 117 publications

References 0 publications

New bijections from $n$-color compositions

New bijections from $n$-color compositions

Universal Cycle Packings and Coverings for k-Subsets of an n-Set

Pattern Avoidance in Ordered Set Partitions

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