Gray codes for noncrossing and nonnesting partitions of classical types

Conflitti, Alessandro; Mamede, Ricardo

doi:10.1142/s0218196717500230

Cited by 3 publications

(2 citation statements)

References 10 publications

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“…In addition to these simple non-cyclic constructions, Conflitti and Mamede also provide constructions of cyclic Gray codes for both problems. In a later paper, Conflitti and Mamede [CM17] present similar Gray codes for non-crossing and non-nesting partitions in Weyl groups of type B and D. P44 For all the classes of restricted set partitions discussed in this section, it is open whether they admit Gray codes that change only few entries in the RGS representation in each step.…”

Section: Non-crossing Non-nesting and Sparse Partitionsmentioning

confidence: 90%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

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Section: Non-crossing Non-nesting and Sparse Partitionsmentioning

confidence: 90%

Combinatorial Gray Codes — an Updated Survey

Mütze

2023

Electron. J. Combin.

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“…In addition to these simple non-cyclic constructions, Conflitti and Mamede also provide constructions of cyclic Gray codes for both problems. In a later paper, Conflitti and Mamede [CM17] present similar Gray codes for non-crossing and non-nesting partitions in Weyl groups of type B and D. P45 For all the classes of restricted set partitions discussed in this section, it is open whether they admit Gray codes that change only few entries in the RGS representation in each step. 6.4.…”

Section: Set Partitionsmentioning

confidence: 90%

Combinatorial Gray codes-an updated survey

Mütze¹

2022

Preprint

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A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a 'small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605-629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

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Listing all delta partitions of a given set: Algorithm design and results

Nofal

2024

Open Computer Science

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Let α \alpha be a set of n n elements and δ \delta be a nonnegative integer. A δ \delta -partition of α \alpha is a set of pairwise disjoint nonempty subsets of α \alpha such that the union of the subsets is equal to α \alpha and every subset has a size greater than δ \delta . We formulate an algorithm for computing all δ \delta -partitions of a given n n -element set and show that the algorithm runs in O ( n ) {\mathcal{O}}\left(n) space and O ( n ) {\mathcal{O}}\left(n) delay time between any two successive outputs of δ \delta -partitions of the given set. An application of the notion of δ \delta -partitions is illustrated in the following scheduling problem. Suppose a factory has n n machines and m ≤ n m\le n jobs to complete daily. Every job can be accomplished by operating at least δ + 1 \delta +1 machines. A machine cannot work on multiple jobs simultaneously. According to a utilization policy of the factory’s management, no machine is allowed to be idle, so all machines should be running on some job. Find a daily schedule of the factory’s machines satisfying all the mentioned constraints. Let α \alpha be the set of the factory’s machines. Then, an α \alpha ’s δ \delta -partition with m m subsets is a legal schedule if every subset (in the δ \delta -partition) includes exclusively δ + 1 \delta +1 or more machines that run on the same job.

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Gray codes for noncrossing and nonnesting partitions of classical types

Cited by 3 publications

References 10 publications

Combinatorial Gray Codes — an Updated Survey

Combinatorial Gray Codes — an Updated Survey

Combinatorial Gray codes-an updated survey

Listing all delta partitions of a given set: Algorithm design and results

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