2007
DOI: 10.1007/s00236-007-0053-0
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A general exhaustive generation algorithm for Gray structures

Abstract: Starting from a succession rule for Catalan numbers, we define a procedure encoding and listing the objects enumerated by these numbers such that two consecutive codes of the list differ only for one digit. Gray code we obtain can be generalized to all the succession rules with the stability property: each label (k) has in its production two labels c 1 and c 2 , always in the same position, regardless of k. Because of this link, we define Gray structures the sets of those combinatorial objects whose constructi… Show more

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Cited by 11 publications
(13 citation statements)
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“…The main tool is an infinite tree with integer node labels, and a set of production rules for creating the children of a node based on its label. Bacchelli, Barcucci, Grazzini, and Pergola [BBGP04] also used ECO for exhaustive generation, deriving an efficient algorithm for generating the corresponding root-tonode label sequences in the ECO tree in lexicographic order, which was later turned into a Gray code [BGPP07]. Dukes, Flanagan, Mansour, and Vajnovszki [DFMV08], Baril [Bar09], and Do, Tran and Vajnovszki [DTV19] used ECO for deriving Gray codes for different classes of patternavoiding permutations, which works under certain regularity assumptions on the production rules.…”
Section: Introductionmentioning
confidence: 99%
“…The main tool is an infinite tree with integer node labels, and a set of production rules for creating the children of a node based on its label. Bacchelli, Barcucci, Grazzini, and Pergola [BBGP04] also used ECO for exhaustive generation, deriving an efficient algorithm for generating the corresponding root-tonode label sequences in the ECO tree in lexicographic order, which was later turned into a Gray code [BGPP07]. Dukes, Flanagan, Mansour, and Vajnovszki [DFMV08], Baril [Bar09], and Do, Tran and Vajnovszki [DTV19] used ECO for deriving Gray codes for different classes of patternavoiding permutations, which works under certain regularity assumptions on the production rules.…”
Section: Introductionmentioning
confidence: 99%
“…Similarly to what has been done for classical rules [4], one can try to develop general exhaustive generation algorithms based on mixed succession rules, maybe finding a new way of defining general Gray codes depending only on the form of the mixed succession rule under consideration.…”
Section: Final Remarksmentioning
confidence: 99%
“…The variety of problems in which the ECO method has shown its soundness ranges from enumerative and bijective combinatorics to random [2] and exhaustive [4] generation.…”
Section: Introductionmentioning
confidence: 99%
“…Our work is different from similar work for combinatorial classes having the same counting sequence, see for instance [6,22]. Indeed, as Savage [21,Section 7] points out: 'Since bijections are known between most members of the Catalan family, a Gray code for one member of the family gives implicitly a listing scheme for every other member of the family.…”
Section: Introductionmentioning
confidence: 95%