Abstract
In this paper, the methods and results in enumeration and generation of Rota–Baxter words in Guo and Sit (Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Math. Comp. Sci., vol. 4, Sp. Issue (2,3), 2011) are generalized and applied to a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra with one generator. A differential Rota–Baxter algebra is an associative algebra with two operators modeled after the differential and integral operators, which are related by the First Fundamental Theorem of Calculus. Differential Rota–Baxter words are words formed by concatenating differential monomials in the generator with images of words under the Rota–Baxter operator. Their totality is a canonical basis of a free, non-commutative, non-unitary, ordinary differential Rota–Baxter algebra. A free differential Rota–Baxter algebra can be constructed from a free Rota–Baxter algebra on a countably infinite set of generators. The order of the derivation gives another dimension of grading on differential Rota–Baxter words, enabling us to generalize and refine results from Guo and Sit to enumerate the set of differential Rota–Baxter words and outline an algorithm for their generation according to a multi-graded structure. Enumeration of a basis is often a first step to choosing a data representation in implementation of algorithms involving free algebras, and in particular, free differential Rota–Baxter algebras and several related algebraic structures on forests and trees. The generating functions obtained can be used to provide links to other combinatorial structures.
Similar content being viewed by others
References
Aguiar, M., Moreira, W.: Combinatorics of the free Baxter algebra. (Electronic) J. Combin. 13(1) (2006), Research Paper 17, 38 pp
Ebrahimi-Fard K., Guo L.: Rota–Baxter algebras and dendriform algebras. J. Pure Appl. Algebra 212, 320–339 (2008)
Grossman R.L., Larson R.G.: Differential algebra structures on families of trees. Adv. Appl. Math. 35(1), 97–119 (2005)
Guo L.: Operated semigroups, Motzkin paths and rooted trees. J. Algebr. Combin. 29(1), 35–62 (2009)
Guo L., Keigher W.: On differential Rota–Baxter algebras. J. Pure Appl. Algebra 212(3), 522–540 (2008)
Guo, L., Sit, W.: Enumeration and generating functions of Rota–Baxter words. In: Regensburger, G., Rosenkranz, M., Sit, W. (eds.) Algebraic and Algorithmic Aspects of Differential and Integral Operators (AADIOS), Math. Comp. Sci., vol. 4, Sp. Issue (2,3) (2011). doi:10.1007/s11786-010-0061-2
Nijenhuis A., Wilf H.: Combinatorial Algorithms, 2nd edn. Academic Press, New York (1978)
Rosenkranz M., Regensburger G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebra. J. Symbolic Comput. 43(8), 515–544 (2008)
Sloane, N., et al.: On-Line Encyclopedia of Integer Seqences. http://www.research.att.com/~njas/sequences/index.html
Author information
Authors and Affiliations
Corresponding author
Additional information
Li Guo acknowledges support from NSF grants DMS 0505643 and DMS 1001855. William Sit acknowledges support from NSF grant CCF-0430722.
Rights and permissions
About this article
Cite this article
Guo, L., Sit, W.Y. Enumeration and Generating Functions of Differential Rota–Baxter Words. Math.Comput.Sci. 4, 339–358 (2010). https://doi.org/10.1007/s11786-010-0062-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-010-0062-1
Keywords
- Differential Rota–Baxter words
- Differential Rota–Baxter algebras
- Generating functions
- Compositions
- Colorings
- Enumerative combinatorics