Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series

Chen, K. T.; Fox, Ralph H.; Lyndon, Roger C.

doi:10.1007/978-1-4612-2096-1_10

Cited by 103 publications

(192 citation statements)

References 2 publications

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“…The regular words were introduced independently by Lyndon in [38]. In [38,36,39], the authors also considered regular nonassociative words, which satisfy the following conditions: the letters are regular nonassociative words; removing the brackets in a regular nonassociative word results…”

Section: §2 Reduced Gröbner-shirshov Bases Of Lie Algebrasmentioning

confidence: 99%

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Gröbner–Shirshov bases of the Lie algebra $D^{+}_{n}$

Koryukin

2011

St. Petersburg Math. J.

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Abstract. Over a field of characteristic 0, the reduced Gröbner-Shirshov bases (RGShB) are computed in the positive part D + n of the simple finite-dimensional Lie algebra D n for the canonical generators corresponding to simple roots, under an arbitrary ordering of these generators (i.e., an aritrary basis among the n! bases is fixed and analyzed). In this setting, the RGShBs were previously computed by the author for the Lie algebras A + n , B + n , and C + n . For one ordering of the generators, the RGShBs of these algebras were calculated by Bokut and Klein (1996).

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Section: §2 Reduced Gröbner-shirshov Bases Of Lie Algebrasmentioning

confidence: 99%

“…In [36,39], it was shown that the mapping [·] is bijective and the set of Lie monomials [u], where u is a regular associative word, is a basis of the algebra Lie(X).…”

Section: §2 Reduced Gröbner-shirshov Bases Of Lie Algebrasmentioning

confidence: 99%

Gröbner–Shirshov bases of the Lie algebra $D^{+}_{n}$

Koryukin

2011

St. Petersburg Math. J.

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“…The next step is to construct two bijections φ fix and φ pix of D * n (r) onto W n (r) enabling us to calculate certain multivariable statistical distributions on words. As mentioned in the introduction, φ fix relates to the algebra of Lyndon words, first introduced by Chen, Fox and Lyndon [Ch58], popularized in Combinatorics by Schützenberger [Sch65] and now set in common usage in Lothaire [Lo83]. The second bijection φ pix relates to the less classical H-factorization, the analog for words of the hook factorization introduced by Gessel [Ge91].…”

Section: Two Multivariable Generating Functions For Wordsmentioning

confidence: 99%

Fix-mahonian calculus III; a quadruple distribution

Foata¹,

Han

2007

Monatsh Math

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A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points "fix", the second the so-called "pix" statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where "des" and "exc" stand for the number of descents and excedances, respectively.

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“…In combinatorics on words, a Lyndon word is defined as a (generally) finite word which is strictly minimal for the lexicographic order of its conjugacy class; the set of Lyndon words permits the unique maximal factorization of any given string [8,27]. Introduced originally by Lyndon in 1954 as standard lexicographic sequences [29], Lyndon words have been studied extensively and are steadily finding an increasing range of applications: string combinatorics and algorithmics [15,36], constructing bases in free Lie algebras [34], succinct suffix-prefix matching of highly periodic strings [32], constructing de Bruijn sequences [16], musicology [7], computing the lexicographically smallest or largest substring in a string [3], string matching [5,11], and in relation to cryptanalysis [33].…”

Section: Introductionmentioning

confidence: 99%

Indeterminate String Factorizations and Degenerate Text Transformations

Daykin

Watson

2017

Math.Comput.Sci.

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The data explosion problem continues to escalate requiring novel and ingenious solutions. Pattern inference focusing on repetitive structures in data is a vigorous field of endeavor aimed at shrinking volumes of data by means of concise descriptions. The Burrows-Wheeler transformation computes a permutation of a string of letters over an alphabet, and is well-suited to compression-related applications due to its invertability and data clustering properties. For space efficiency the input to the transform can be preprocessed into Lyndon factors. Rather than this classic deterministic approach for letter based strings, we consider scenarios with uncertainty regarding the data: a position in an indeterminate or degenerate string is a set of letters. We first define indeterminate Lyndon words and establish their associated unique string factorization; then we introduce the novel degenerate Burrows-Wheeler transformation which may apply the indeterminate Lyndon factorization. A core computation in Burrows-Wheeler type transforms is the linear sorting of all conjugates of the input string-we achieve this in the degenerate case with lex-extension ordering. Like the original forms, indeterminate Lyndon factorization and the degenerate transform and its inverse can all be computed in linear time and space with respect to total input size of degenerate strings. Regular molecular biological strings yield a wealth of applications of big data-an important motivation for generalizing to degenerate strings is their extensive use in expressing polymorphism in DNA sequences.

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Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series

Cited by 103 publications

References 2 publications

Gröbner–Shirshov bases of the Lie algebra $D^{+}_{n}$

Gröbner–Shirshov bases of the Lie algebra $D^{+}_{n}$

Fix-mahonian calculus III; a quadruple distribution

Indeterminate String Factorizations and Degenerate Text Transformations

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