ARC: Automated Reasoning in The Class [Erasmus+ Project]
Project Lead
Project Duration
01/10/2019 - 31/08/2022Partners
EU

Publications
2023
[Buchberger]
Is ChatGPT Smarter Than Master’s Applicants?
Bruno Buchberger
Technical report no. 23-04 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). January 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6684,
author = {Bruno Buchberger},
title = {{Is ChatGPT Smarter Than Master’s Applicants?}},
language = {English},
abstract = {During the selection procedure for a particular informatics fellowship program sponsored by Upper Austrian companies, I ask the applicants a couple of simple technical questions about programming, etc., in a Zoom meeting. I put the same questions to the dialogue system ChatGPT, [ChatGPT]. The result surprised me: Nearly all answers of ChatGPT were totally correct and nicely explained. Also, in the dialogues to clarify some critical points in the answers, the explanations by ChatGPT were amazingly clear and goal-oriented.In comparison: I tried out the same questions in the personal Zoom interviews with approximately 30 applicants from five countries. Only the top three candidates (with a GPA of 1.0, i.e., the highest possible GPA in their bachelor’s study) performed approximately equally well in the interview. All the others performed (far) worse than ChatGPT. And, of course, all answers from ChatGPT came within 1 to 10 seconds, whereas most of the human applicants' answers needed lengthy and arduous dialogues.I am particularly impressed by the ability of ChatGPT to extract meaningful and well-structured programs from problem specifications in natural language. In this experiment, I also added some questions that ask for proofs for simple statements in natural language, which I do not ask in the student's interviews. The performance of ChatGPT was quite impressive as far as formalization and propositional logic are concerned. In examples where predicate logic reasoning is necessary, the ChatGPT answers are not (yet?) perfect. I am pleased to see that ChatGPT tries to present the proofs in a “natural style” This is something that I had as one of my main goals when I initiated the Theorema project in 1995. I think we already achieved this in the early stage of Theorema, and we performed this slightly better and more systematically than ChatGPT does.I also tried to develop a natural language input facility for Theorema in 2017, i.e., a tool to formalize natural language statements in predicate logic. However, I could not continue this research for a couple of reasons. Now I see that ChatGPT achieved this goal. Thus, I think that the following combination of methods could result in a significant leap forward:- the “natural style” proving methods that we developed within Theorema (for the automated generation of programs from specifications, the automated verification of programs in the frame of knowledge, and the automated proof of theorems in theories), in particular, my “Lazy Thinking Method” for algorithm synthesis from specifications- and the natural language formalization techniques of ChatGPT.I propose this as a research project topic and invite colleagues and students to contact me and join me in this effort: Buchberger.bruno@gmail.com.},
number = {23-04},
year = {2023},
month = {January},
keywords = {ChatGPT, automated programming, program synthesis, automated proving, formalization of natural language, master's screening},
length = {30},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {Bruno Buchberger},
title = {{Is ChatGPT Smarter Than Master’s Applicants?}},
language = {English},
abstract = {During the selection procedure for a particular informatics fellowship program sponsored by Upper Austrian companies, I ask the applicants a couple of simple technical questions about programming, etc., in a Zoom meeting. I put the same questions to the dialogue system ChatGPT, [ChatGPT]. The result surprised me: Nearly all answers of ChatGPT were totally correct and nicely explained. Also, in the dialogues to clarify some critical points in the answers, the explanations by ChatGPT were amazingly clear and goal-oriented.In comparison: I tried out the same questions in the personal Zoom interviews with approximately 30 applicants from five countries. Only the top three candidates (with a GPA of 1.0, i.e., the highest possible GPA in their bachelor’s study) performed approximately equally well in the interview. All the others performed (far) worse than ChatGPT. And, of course, all answers from ChatGPT came within 1 to 10 seconds, whereas most of the human applicants' answers needed lengthy and arduous dialogues.I am particularly impressed by the ability of ChatGPT to extract meaningful and well-structured programs from problem specifications in natural language. In this experiment, I also added some questions that ask for proofs for simple statements in natural language, which I do not ask in the student's interviews. The performance of ChatGPT was quite impressive as far as formalization and propositional logic are concerned. In examples where predicate logic reasoning is necessary, the ChatGPT answers are not (yet?) perfect. I am pleased to see that ChatGPT tries to present the proofs in a “natural style” This is something that I had as one of my main goals when I initiated the Theorema project in 1995. I think we already achieved this in the early stage of Theorema, and we performed this slightly better and more systematically than ChatGPT does.I also tried to develop a natural language input facility for Theorema in 2017, i.e., a tool to formalize natural language statements in predicate logic. However, I could not continue this research for a couple of reasons. Now I see that ChatGPT achieved this goal. Thus, I think that the following combination of methods could result in a significant leap forward:- the “natural style” proving methods that we developed within Theorema (for the automated generation of programs from specifications, the automated verification of programs in the frame of knowledge, and the automated proof of theorems in theories), in particular, my “Lazy Thinking Method” for algorithm synthesis from specifications- and the natural language formalization techniques of ChatGPT.I propose this as a research project topic and invite colleagues and students to contact me and join me in this effort: Buchberger.bruno@gmail.com.},
number = {23-04},
year = {2023},
month = {January},
keywords = {ChatGPT, automated programming, program synthesis, automated proving, formalization of natural language, master's screening},
length = {30},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Dominici]
Linear functionals and $Delta$- coherent pairs of the second kind
Diego Dominici and Francisco Marcellan
Technical report no. 23-02 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6677,
author = {Diego Dominici and Francisco Marcellan},
title = {{Linear functionals and $Delta$- coherent pairs of the second kind}},
language = {english},
abstract = {We classify all the emph{$Delta$-}coherent pairs of measures of the secondkind on the real line. We obtain $5$ cases, corresponding to all the familiesof discrete semiclassical orthogonal polynomials of class $sleq1.$},
number = {23-02},
year = {2023},
month = {February},
keywords = { Discrete orthogonal polynomials, discrete semiclassical functionals, discrete Sobolev inner products, coherent pairs of discrete measures, coherent pairs of second kind for discrete measures.},
length = {24},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {Diego Dominici and Francisco Marcellan},
title = {{Linear functionals and $Delta$- coherent pairs of the second kind}},
language = {english},
abstract = {We classify all the emph{$Delta$-}coherent pairs of measures of the secondkind on the real line. We obtain $5$ cases, corresponding to all the familiesof discrete semiclassical orthogonal polynomials of class $sleq1.$},
number = {23-02},
year = {2023},
month = {February},
keywords = { Discrete orthogonal polynomials, discrete semiclassical functionals, discrete Sobolev inner products, coherent pairs of discrete measures, coherent pairs of second kind for discrete measures.},
length = {24},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Dominici]
Recurrence relations for the moments of discrete semiclassical functionals of class $sleq2.$
Diego Dominici
Technical report no. 23-05 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). March 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6687,
author = {Diego Dominici },
title = {{Recurrence relations for the moments of discrete semiclassical functionals of class $sleq2.$}},
language = {english},
abstract = {We study recurrence relations satisfied by the moments $lambda_{n}left(zright) $ of discrete linear functionals whose first moment satisfies aholonomic differential equation. We consider all cases when the order of theODE is less or equal than $3$.},
number = {23-05},
year = {2023},
month = {March},
keywords = {Discrete orthogonal polynomials, discrete semiclassical functionals, moments.},
length = {81},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {Diego Dominici },
title = {{Recurrence relations for the moments of discrete semiclassical functionals of class $sleq2.$}},
language = {english},
abstract = {We study recurrence relations satisfied by the moments $lambda_{n}left(zright) $ of discrete linear functionals whose first moment satisfies aholonomic differential equation. We consider all cases when the order of theODE is less or equal than $3$.},
number = {23-05},
year = {2023},
month = {March},
keywords = {Discrete orthogonal polynomials, discrete semiclassical functionals, moments.},
length = {81},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Jimenez Pastor]
An extension of holonomic sequences: $C^2$-finite sequences
A. Jimenez-Pastor, P. Nuspl, V. Pillwein
Journal of Symbolic Computation 116, pp. 400-424. 2023. ISSN: 0747-7171.@article{RISC6636,
author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},
title = {{An extension of holonomic sequences: $C^2$-finite sequences}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {116},
pages = {400--424},
isbn_issn = {ISSN: 0747-7171},
year = {2023},
refereed = {yes},
length = {25}
}
author = {A. Jimenez-Pastor and P. Nuspl and V. Pillwein},
title = {{An extension of holonomic sequences: $C^2$-finite sequences}},
language = {english},
journal = {Journal of Symbolic Computation},
volume = {116},
pages = {400--424},
isbn_issn = {ISSN: 0747-7171},
year = {2023},
refereed = {yes},
length = {25}
}
[Kauers]
Order bounds for $C^2$-finite sequences
M. Kauers, P. Nuspl, V. Pillwein
Technical report no. 23-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6683,
author = {M. Kauers and P. Nuspl and V. Pillwein},
title = {{Order bounds for $C^2$-finite sequences}},
language = {english},
abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},
number = {23-03},
year = {2023},
month = {February},
keywords = {Difference equations, holonomic sequences, closure properties, algorithms},
length = {16},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {M. Kauers and P. Nuspl and V. Pillwein},
title = {{Order bounds for $C^2$-finite sequences}},
language = {english},
abstract = {A sequence is called $C$-finite if it satisfies a linear recurrence with constant coefficients. We study sequences which satisfy a linear recurrence with $C$-finite coefficients. Recently, it was shown that such $C^2$-finite sequences satisfy similar closure properties as $C$-finite sequences. In particular, they form a difference ring. In this paper we present new techniques for performing these closure properties of $C^2$-finite sequences. These methods also allow us to derive order bounds which were not known before. Additionally, they provide more insight in the effectiveness of these computations. The results are based on the exponent lattice of algebraic numbers. We present an iterative algorithm which can be used to compute bases of such lattices.},
number = {23-03},
year = {2023},
month = {February},
keywords = {Difference equations, holonomic sequences, closure properties, algorithms},
length = {16},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Mitteramskogler]
The algebro-geometric method: Solving algebraic differential equations by parametrizations
S. Falkensteiner, J.J. Mitteramskogler, R. Sendra, F. Winkler
Bulletin of the American Mathematical Society, pp. 1-41. 2023. ISSN 0273-0979.@article{RISC6507,
author = {S. Falkensteiner and J.J. Mitteramskogler and R. Sendra and F. Winkler},
title = {{The algebro-geometric method: Solving algebraic differential equations by parametrizations}},
language = {english},
journal = {Bulletin of the American Mathematical Society},
pages = {1--41},
isbn_issn = {ISSN 0273-0979},
year = {2023},
refereed = {yes},
length = {41}
}
author = {S. Falkensteiner and J.J. Mitteramskogler and R. Sendra and F. Winkler},
title = {{The algebro-geometric method: Solving algebraic differential equations by parametrizations}},
language = {english},
journal = {Bulletin of the American Mathematical Society},
pages = {1--41},
isbn_issn = {ISSN 0273-0979},
year = {2023},
refereed = {yes},
length = {41}
}
[Mitteramskogler]
General solutions of first-order algebraic ODEs in simple constant extensions
J. J. Mitteramskogler, F. Winkler
Journal of Systems Science and Complexity (JSSC), pp. 0-0. 2023. 1009-6124.@article{RISC6674,
author = {J. J. Mitteramskogler and F. Winkler},
title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},
language = {english},
journal = {Journal of Systems Science and Complexity (JSSC)},
pages = {0--0},
isbn_issn = {1009-6124},
year = {2023},
refereed = {yes},
length = {0}
}
author = {J. J. Mitteramskogler and F. Winkler},
title = {{General solutions of first-order algebraic ODEs in simple constant extensions}},
language = {english},
journal = {Journal of Systems Science and Complexity (JSSC)},
pages = {0--0},
isbn_issn = {1009-6124},
year = {2023},
refereed = {yes},
length = {0}
}
[Paule]
Ramanujan and Computer Algebra
Peter Paule
In: Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence, K. Alladi, G.E. Andrews, B. Berndt, F. Garvan, K. Ono, P. Paule, S. Ole Warnaar, Ae Ja Yee (ed.), pp. -. 2023. Springer, ISBN x. [pdf]@incollection{RISC6678,
author = {Peter Paule},
title = {{Ramanujan and Computer Algebra}},
booktitle = {{Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence}},
language = {english},
pages = {--},
publisher = {Springer},
isbn_issn = {ISBN x},
year = {2023},
editor = {K. Alladi and G.E. Andrews and B. Berndt and F. Garvan and K. Ono and P. Paule and S. Ole Warnaar and Ae Ja Yee },
refereed = {yes},
length = {0}
}
author = {Peter Paule},
title = {{Ramanujan and Computer Algebra}},
booktitle = {{Srinivasa Ramanujan: His Life, Legacy, and Mathematical Influence}},
language = {english},
pages = {--},
publisher = {Springer},
isbn_issn = {ISBN x},
year = {2023},
editor = {K. Alladi and G.E. Andrews and B. Berndt and F. Garvan and K. Ono and P. Paule and S. Ole Warnaar and Ae Ja Yee },
refereed = {yes},
length = {0}
}
[Paule]
Interview with Peter Paule
Toufik Mansour and Peter Paule
Enumerative Combinatorics and Applications ECA 3:1(#S3I1), pp. -. 2023. ISSN 2710-2335. [doi]@article{RISC6679,
author = {Toufik Mansour and Peter Paule},
title = {{Interview with Peter Paule}},
language = {english},
journal = {Enumerative Combinatorics and Applications },
volume = {ECA 3:1},
number = {#S3I1},
pages = {--},
isbn_issn = {ISSN 2710-2335},
year = {2023},
refereed = {yes},
length = {0},
url = {http://doi.org/10.54550/ECA2023V3S1I1}
}
author = {Toufik Mansour and Peter Paule},
title = {{Interview with Peter Paule}},
language = {english},
journal = {Enumerative Combinatorics and Applications },
volume = {ECA 3:1},
number = {#S3I1},
pages = {--},
isbn_issn = {ISSN 2710-2335},
year = {2023},
refereed = {yes},
length = {0},
url = {http://doi.org/10.54550/ECA2023V3S1I1}
}
[Schneider]
Hypergeometric Structures in Feynman Integrals
J. Blümlein, C. Schneider, M. Saragnese
Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science" to appear, pp. ?-?. 2023. ISSN 1573-7470. arXiv:2111.15501 [math-ph]. [doi]@article{RISC6643,
author = {J. Blümlein and C. Schneider and M. Saragnese},
title = {{Hypergeometric Structures in Feynman Integrals}},
language = {english},
abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},
journal = {Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science"},
volume = {to appear},
pages = {?--?},
isbn_issn = {ISSN 1573-7470},
year = {2023},
note = {arXiv:2111.15501 [math-ph]},
refereed = {yes},
keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},
length = {55},
url = {https://doi.org/10.48550/arXiv.2111.15501}
}
author = {J. Blümlein and C. Schneider and M. Saragnese},
title = {{Hypergeometric Structures in Feynman Integrals}},
language = {english},
abstract = {Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {tt Sigma} in the case of linear difference equations or by applying heuristic methods in the case of partial linear difference equations. In the present context a new type of sums occurs, the Hurwitz harmonic sums, and generalized versions of them. The code {tt HypSeries} transforming classes of differential equations into analytic series expansions is described. Also partial difference equations having rational solutions and rational function solutions of Pochhammer symbols are considered, for which the code {tt solvePartialLDE} is designed. Generalized hypergeometric functions, Appell-,~Kamp'e de F'eriet-, Horn-, Lauricella-Saran-, Srivasta-, and Exton--type functions are considered. We illustrate the algorithms by examples.},
journal = {Annals of Mathematics and Artificial Intelligence, Special issue on " Symbolic Computation in Software Science"},
volume = {to appear},
pages = {?--?},
isbn_issn = {ISSN 1573-7470},
year = {2023},
note = {arXiv:2111.15501 [math-ph]},
refereed = {yes},
keywords = {hypergeometric functions, symbolic summation, expansion, partial linear difference equations, partial linear differential equations},
length = {55},
url = {https://doi.org/10.48550/arXiv.2111.15501}
}
[Schneider]
Refined telescoping algorithms in $R\Pi\Sigma$-extensions to reduce the degrees of the denominators
C. Schneider
Technical report no. 23-01 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Austria. ISSN 2791-4267 (online). February 2023. arXiv:2302.03563 [cs.SC]. Licensed under CC BY 4.0 International. [doi] [pdf]@techreport{RISC6682,
author = {C. Schneider},
title = {{Refined telescoping algorithms in $R\Pi\Sigma$-extensions to reduce the degrees of the denominators}},
language = {english},
abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $R\Pi\Sigma$-ring extensions that are built over general $\Pi\Sigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},
number = {23-01},
year = {2023},
month = {February},
note = {arXiv:2302.03563 [cs.SC]},
keywords = {telescoping, difference rings, reduced denominators, nested sums},
length = {18},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
author = {C. Schneider},
title = {{Refined telescoping algorithms in $R\Pi\Sigma$-extensions to reduce the degrees of the denominators}},
language = {english},
abstract = {We present a general framework in the setting of difference ring extensions that enables one to find improved representations of indefinite nested sums such that the arising denominators within the summands have reduced degrees. The underlying (parameterized) telescoping algorithms can be executed in $R\Pi\Sigma$-ring extensions that are built over general $\Pi\Sigma$-fields. An important application of this toolbox is the simplification of d'Alembertian and Liouvillian solutions coming from recurrence relations where the denominators of the arising sums do not factor nicely.},
number = {23-01},
year = {2023},
month = {February},
note = {arXiv:2302.03563 [cs.SC]},
keywords = {telescoping, difference rings, reduced denominators, nested sums},
length = {18},
license = {CC BY 4.0 International},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}
[Schneider]
Representation of hypergeometric products of higher nesting depths in difference rings
E.D. Ocansey, C. Schneider
J. Symb. Comput. to appear, pp. ?-?. 2023. ISSN: 0747-7171. arXiv:2011.08775 [cs.SC]. [doi]@article{RISC6688,
author = {E.D. Ocansey and C. Schneider},
title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},
language = {english},
journal = {J. Symb. Comput.},
volume = {to appear},
number = {?},
pages = {?--?},
isbn_issn = {ISSN: 0747-7171},
year = {2023},
note = {arXiv:2011.08775 [cs.SC]},
refereed = {yes},
length = {48},
url = {https://doi.org/10.1016/j.jsc.2023.03.002}
}
author = {E.D. Ocansey and C. Schneider},
title = {{Representation of hypergeometric products of higher nesting depths in difference rings}},
language = {english},
journal = {J. Symb. Comput.},
volume = {to appear},
number = {?},
pages = {?--?},
isbn_issn = {ISSN: 0747-7171},
year = {2023},
note = {arXiv:2011.08775 [cs.SC]},
refereed = {yes},
length = {48},
url = {https://doi.org/10.1016/j.jsc.2023.03.002}
}
[Smoot]
A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method
N. Smoot
Journal of Number Theory 242, pp. 112-153. January 2023. ISSN 1096-1658. [doi]@article{RISC6661,
author = {N. Smoot},
title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},
language = {english},
abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},
journal = {Journal of Number Theory},
volume = {242},
pages = {112--153},
isbn_issn = {ISSN 1096-1658},
year = {2023},
month = {January},
refereed = {yes},
length = {42},
url = {https://doi.org/10.1016/j.jnt.2022.07.014}
}
author = {N. Smoot},
title = {{A Congruence Family For 2-Elongated Plane Partitions: An Application of the Localization Method}},
language = {english},
abstract = {George Andrews and Peter Paule have recently conjectured an infinite family of congruences modulo powers of 3 for the 2-elongated plane partition function $d_2(n)$. This congruence family appears difficult to prove by classical methods. We prove a refined form of this conjecture by expressing the associated generating functions as elements of a ring of modular functions isomorphic to a localization of $mathbb{Z}[X]$.},
journal = {Journal of Number Theory},
volume = {242},
pages = {112--153},
isbn_issn = {ISSN 1096-1658},
year = {2023},
month = {January},
refereed = {yes},
length = {42},
url = {https://doi.org/10.1016/j.jnt.2022.07.014}
}
2022
[Banerjee]
Hook Type enumeration and parity of parts in partitions
K. Banerjee, M. G. Dastidar
Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6596, 2022. [pdf]@techreport{RISC6596,
author = {K. Banerjee and M. G. Dastidar},
title = {{Hook Type enumeration and parity of parts in partitions}},
language = {english},
abstract = {This paper is devoted to study an association between hook type enumeration and counting integer partitions subject to parity of its parts. We shall primarily focus on a result of Andrews in two possible direction. First, we confirm a conjecture of Rubey and secondly, we extend the theorem of Andrews in a more general set up. },
number = {RISC6596},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {8}
}
author = {K. Banerjee and M. G. Dastidar},
title = {{Hook Type enumeration and parity of parts in partitions}},
language = {english},
abstract = {This paper is devoted to study an association between hook type enumeration and counting integer partitions subject to parity of its parts. We shall primarily focus on a result of Andrews in two possible direction. First, we confirm a conjecture of Rubey and secondly, we extend the theorem of Andrews in a more general set up. },
number = {RISC6596},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {8}
}
[Banerjee]
Hook type tableaux and partition identities
K. Banerjee, M. G. Dastidar
Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6597, 2022. [pdf]@techreport{RISC6597,
author = {K. Banerjee and M. G. Dastidar},
title = {{Hook type tableaux and partition identities}},
language = {english},
abstract = {In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of Stanley's and Elder's theorems without even the use of partition statistics in general. We primarily focus on to study Stanley's theorem in color partition context.},
number = {RISC6597},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {13}
}
author = {K. Banerjee and M. G. Dastidar},
title = {{Hook type tableaux and partition identities}},
language = {english},
abstract = {In this paper we exhibit the box-stacking principle (BSP) in conjunction with Young diagrams to prove generalizations of Stanley's and Elder's theorems without even the use of partition statistics in general. We primarily focus on to study Stanley's theorem in color partition context.},
number = {RISC6597},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {13}
}
[Banerjee]
Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach
Koustav Banerjee
Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6592, 2022. [pdf]@techreport{RISC6592,
author = {Koustav Banerjee},
title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},
language = {english},
number = {RISC6592},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {5}
}
author = {Koustav Banerjee},
title = {{Positivity of the second shifted difference of partitions and overpartitions: a combinatorial approach}},
language = {english},
number = {RISC6592},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {5}
}
[Banerjee]
Ramanujan's theta functions and parity of parts and cranks of partitions
K. Banerjee, M. G. Dastidar
Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6595, 2022. [pdf]@techreport{RISC6595,
author = {K. Banerjee and M. G. Dastidar},
title = {{Ramanujan's theta functions and parity of parts and cranks of partitions}},
language = {english},
abstract = {In this paper we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation to the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.},
number = {RISC6595},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {13}
}
author = {K. Banerjee and M. G. Dastidar},
title = {{Ramanujan's theta functions and parity of parts and cranks of partitions}},
language = {english},
abstract = {In this paper we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation to the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.},
number = {RISC6595},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {13}
}
[Banerjee]
Inequalities for the partition function arising from truncated theta series
K. Banerjee, M. G. Dastidar
Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6622, 2022. [pdf]@techreport{RISC6622,
author = {K. Banerjee and M. G. Dastidar},
title = {{Inequalities for the partition function arising from truncated theta series}},
language = {english},
number = {RISC6622},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {12}
}
author = {K. Banerjee and M. G. Dastidar},
title = {{Inequalities for the partition function arising from truncated theta series}},
language = {english},
number = {RISC6622},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {12}
}
[Banerjee]
Parity biases in partitions and restricted partitions
Banerjee Koustav, Bhattacharjee Sreerupa, Dastidar Manosij Ghosh, Mahanta Pankaj Jyoti, Saikia Manjil P
European Journal of Combinatorics 103, pp. 103522-103522. 2022. Elsevier, ISSN 0195-6698. [pdf]@article{RISC6606,
author = {Banerjee Koustav and Bhattacharjee Sreerupa and Dastidar Manosij Ghosh and Mahanta Pankaj Jyoti and Saikia Manjil P},
title = {{Parity biases in partitions and restricted partitions}},
language = {english},
journal = {European Journal of Combinatorics},
volume = {103},
pages = {103522--103522},
publisher = {Elsevier},
isbn_issn = {ISSN 0195-6698},
year = {2022},
refereed = {yes},
length = {19}
}
author = {Banerjee Koustav and Bhattacharjee Sreerupa and Dastidar Manosij Ghosh and Mahanta Pankaj Jyoti and Saikia Manjil P},
title = {{Parity biases in partitions and restricted partitions}},
language = {english},
journal = {European Journal of Combinatorics},
volume = {103},
pages = {103522--103522},
publisher = {Elsevier},
isbn_issn = {ISSN 0195-6698},
year = {2022},
refereed = {yes},
length = {19}
}
[Banerjee]
New inequalities for p(n) and log p(n)
K. Banerjee, P. Paule, C. S. Radu, W. H. Zeng
Research Institute for Symbolic Computation, JKU, Linz. Technical report no. RISC6607, 2022. To appear in the Ramanujan Journal. [pdf]@techreport{RISC6607,
author = {K. Banerjee and P. Paule and C. S. Radu and W. H. Zeng},
title = {{New inequalities for p(n) and log p(n)}},
language = {english},
number = {RISC6607},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {37},
type = {To appear in the Ramanujan Journal}
}
author = {K. Banerjee and P. Paule and C. S. Radu and W. H. Zeng},
title = {{New inequalities for p(n) and log p(n)}},
language = {english},
number = {RISC6607},
year = {2022},
institution = {Research Institute for Symbolic Computation, JKU, Linz},
length = {37},
type = {To appear in the Ramanujan Journal}
}