# Partition Analysis [F050-06]

### Project Duration

01/03/2013 - 28/02/2021

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## Publications

[Hemmecke]

### The Generators of all Polynomial Relations among Jacobi Theta Functions

#### Ralf Hemmecke, Silviu Radu, Liangjie Ye

In: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, , Texts & Monographs in Symbolic Computation 18-09, pp. 259-268. 2019. Springer International Publishing, Cham, 978-3-030-04479-4. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf. [url]
@incollection{RISC5913,
author = {Ralf Hemmecke and Silviu Radu and Liangjie Ye},
title = {{The Generators of all Polynomial Relations among Jacobi Theta Functions}},
booktitle = {{Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory}},
language = {english},
abstract = {In this article, we consider the classical Jacobi theta functions$\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomialrelations among them with coefficients in$K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ isgenerated by just two polynomials, that correspond to well knownidentities among Jacobi theta functions.},
series = {Texts & Monographs in Symbolic Computation},
number = {18-09},
pages = {259--268},
publisher = {Springer International Publishing},
isbn_issn = {978-3-030-04479-4},
year = {2019},
editor = {Johannes Blümlein and Carsten Schneider and Peter Paule},
refereed = {yes},
length = {9},
url = {https://doi.org/10.1007/978-3-030-04480-0_11}
}

[Hemmecke]

### Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$

Technical report no. 18-03 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. January 26 2018. Accepted for publication in the Journal of Symbolic Computation. [pdf]
@techreport{RISC5561,
author = {Ralf Hemmecke and Silviu Radu},
title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},
language = {english},
abstract = {We describe an algorithm that, given a positive integer $N$,computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind$\eta$-functions of level $N$, i.e., among the elements of$\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where$1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of$N$.More precisely, we find a finite generating set (which is also aGr\"obner basis of the ideal $\ker\phi$ where\begin{gather*} \phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)], \quad E_k\mapsto \eta(\delta_k\tau), \quad k=1,\ldots,n.\end{gather*}},
number = {18-03},
year = {2018},
month = {January 26},
note = {Accepted for publication in the Journal of Symbolic Computation},
keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},
length = {18},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Jiu]

### The Method of Brackets in Experimental Mathematics

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

In: Frontiers in Orthogonal Polynomials and q-Series, , pp. -. 2018. World Scientific Publishing, 978-981-3228-87-0. [url]
@incollection{RISC5497,
author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},
title = {{The Method of Brackets in Experimental Mathematics}},
booktitle = {{Frontiers in Orthogonal Polynomials and q-Series}},
language = {english},
pages = {--},
publisher = {World Scientific Publishing},
isbn_issn = {978-981-3228-87-0},
year = {2018},
editor = {Xin Li and Zuhair Nashed},
refereed = {no},
length = {0},
url = {http://www.worldscientific.com/worldscibooks/10.1142/10677}
}
[Magnusson]

### The functional equation of Dedekind's $\eta$-function

#### Tobias Magnusson

June 15 2018. [pdf] [tex]
@techreport{RISC5701,
author = {Tobias Magnusson},
title = {{The functional equation of Dedekind's $\eta$-function}},
language = {English},
year = {2018},
month = {June 15},
length = {14},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

[Jiu]

### Integral representations of equally positive integer-indexed harmonic sums at infinity

#### L. Jiu

Research in Number Theory 3(10), pp. 1-4. 2017. 2363-9555. [url]
@article{RISC5385,
author = {L. Jiu},
title = {{Integral representations of equally positive integer-indexed harmonic sums at infinity}},
language = {English},
abstract = {We identify a partition-theoretic generalization of Riemann zeta function and the equally positive integer-indexed harmonic sums at infinity, to obtain the generating function and the integral representations of the latter. The special cases coincide with zeta values at positive integer arguments.},
journal = {Research in Number Theory},
volume = {3},
number = {10},
pages = {1--4},
isbn_issn = {2363-9555},
year = {2017},
refereed = {no},
length = {4},
url = {https://resnumtheor.springeropen.com/articles/10.1007/s40993-017-0074-x}
}
[Jiu]

### An extension of the method of brackets. Part 1

#### Ivan Gonzalez, Karen Kohl, Lin Jiu, and Victor H. Moll

Open Mathematics (formerly Central European Journal of Mathematics) 15, pp. 1181-1211. 2017. 2391-5455. [url]
@article{RISC5483,
author = {Ivan Gonzalez and Karen Kohl and Lin Jiu and and Victor H. Moll},
title = {{An extension of the method of brackets. Part 1}},
language = {english},
abstract = {The method of brackets is an efficient method for the evaluation of a large class of definite integrals on the half-line. It is based on a small collection of rules, some of which are heuristic. The extension discussed here is based on the concepts of null and divergent series. These are formal representations of functions, whose coefficients $a_n$ have meromorphic representations for $n\in\mathbb{C}$, but might vanish or blow up when $n\in\mathbb{N}$. These ideas are illustrated with the evaluation of a variety of entries from the classical table of integrals by Gradshteyn and Ryzhik.},
journal = {Open Mathematics (formerly Central European Journal of Mathematics)},
volume = {15},
pages = {1181--1211},
isbn_issn = {2391-5455},
year = {2017},
refereed = {no},
length = {31},
url = {https://www.degruyter.com/view/j/math.2017.15.issue-1/math-2017-0100/math-2017-0100.xml?format=INT}
}
[Xiong]

### Overpartitions and ternary quadratic forms

#### xinhua xiong

The Ramanujan Journal 42(2), pp. 429-442. 2017. issn:1382-4090.
@article{RISC5417,
author = {xinhua xiong},
title = {{Overpartitions and ternary quadratic forms}},
language = {english},
journal = {The Ramanujan Journal},
volume = {42},
number = {2},
pages = {429--442},
isbn_issn = {issn:1382-4090},
year = {2017},
refereed = {yes},
length = {13}
}
[Ye]

### Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations

#### Liangjie Ye

Journal of Symbolic Computation, to appear, pp. 1-25. 2017. -. [pdf]
@article{RISC5286,
author = {Liangjie Ye},
title = {{Elliptic Function Based Algorithms to Prove Jacobi Theta Function Relations}},
language = {english},
journal = {Journal of Symbolic Computation, to appear},
pages = {1--25},
isbn_issn = {-},
year = {2017},
refereed = {yes},
length = {25}
}
[Ye]

### A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions

#### Liangjie Ye

Journal of Symbolic Computation 82, pp. 134-163. 2017. ISSN: 0747-7171. [pdf]
@article{RISC5455,
author = {Liangjie Ye},
title = {{A Symbolic Decision Procedure for Relations Arising among Taylor Coefficients of Classical Jacobi Theta Functions}},
language = {english},
journal = {Journal of Symbolic Computation },
volume = {82},
pages = {134--163},
isbn_issn = {ISSN: 0747-7171},
year = {2017},
refereed = {yes},
length = {30}
}
[Ye]

### Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities

#### Liangjie Ye

RISC and the DK program Linz. PhD Thesis. 2017. Updated version in June 2017. [pdf]
@phdthesis{RISC5463,
author = {Liangjie Ye},
title = {{Complex Analysis Based Computer Algebra Algorithms for Proving Jacobi Theta Function Identities}},
language = {english},
year = {2017},
note = {Updated version in June 2017},
translation = {0},
school = {RISC and the DK program Linz},
length = {122}
}

[Kronholm]

### A Polyhedral Model of Partitions with Bounded Differences and a Bijective Proof of a Theorem of Andrews, Beck, and Robbins

#### Brandt Kronholm, Felix Breuer

Research in Number Theory, pp. 1-15. March 2016. Springer, 2363-9555.
@article{RISC5268,
author = {Brandt Kronholm and Felix Breuer},
title = {{A Polyhedral Model of Partitions with Bounded Differences and a Bijective Proof of a Theorem of Andrews, Beck, and Robbins}},
language = {english},
journal = {Research in Number Theory},
pages = {1--15},
publisher = {Springer},
isbn_issn = {2363-9555},
year = {2016},
month = {March},
refereed = {yes},
length = {15}
}
[Paule]

### A New Witness Identity for $11|p(11n+6)$

In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series, , pp. 625-640. 2016. Springer, 2194-1009. [pdf]
@inproceedings{RISC5329,
author = {Peter Paule and Cristian-Silviu Radu},
title = {{A New Witness Identity for $11|p(11n+6)$}},
booktitle = {{Analytic Number Theory, Modular Forms and q-Hypergeometric Series}},
language = {english},
pages = {625--640},
publisher = {Springer},
isbn_issn = { 2194-1009},
year = {2016},
editor = { George E. Andrews and Frank Garvan},
refereed = {yes},
length = {16}
}
[Xiong]

### Overpartition function modulo 16 and some binary quadratic forms

#### Xinhua Xiong

Technical report no. 5 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. ISSN: 1793-0421, 8 2016.
@techreport{RISC5309,
author = {Xinhua Xiong},
title = {{Overpartition function modulo 16 and some binary quadratic forms}},
language = {english},
number = {5},
isbn_issn = { ISSN: 1793-0421},
year = {2016},
month = {8},
length = {13},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Xiong]

### Overpartition function modulo 16 and some binary quadratic forms

#### Xinhua Xiong

International Journal of Number Theory 12(5), pp. 1195-1208. 2016. ISSN 1793-0421.
@article{RISC5310,
author = {Xinhua Xiong},
title = {{Overpartition function modulo 16 and some binary quadratic forms}},
language = {english},
journal = {International Journal of Number Theory},
volume = {12},
number = {5},
pages = {1195--1208},
isbn_issn = {ISSN 1793-0421},
year = {2016},
refereed = {yes},
length = {13}
}
[Xiong]

### A positivity conjecture related first positive rank and crank moments for overpartitions

#### Xinhua Xiong

Proc. Japan Acad. Ser. A: Math. Sci., 92 (2016), no. 11,, pp. 117-120. 2016. ISSN . [url]
@article{RISC5311,
author = {Xinhua Xiong},
title = {{A positivity conjecture related first positive rank and crank moments for overpartitions}},
language = {english},
journal = { Proc. Japan Acad. Ser. A: Math. Sci., 92 (2016), no. 11,},
pages = {117--120},
isbn_issn = {ISSN },
year = {2016},
refereed = {yes},
length = {6},
url = {http://arxiv.org/abs/1605.09135}
}
[Xiong]

### Small Values of Coefficients of a Half Lerch Sum

#### Xinhua Xiong

arXiv:1605.09508, submitted to journal., pp. 1-10. 2016. ISSN.
@article{RISC5313,
author = {Xinhua Xiong},
title = {{Small Values of Coefficients of a Half Lerch Sum}},
language = {english},
journal = {arXiv:1605.09508, submitted to journal.},
pages = {1--10},
isbn_issn = {ISSN},
year = {2016},
refereed = {yes},
length = {10}
}
[Xiong]

### Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities

#### Xinhua Xiong

arXiv:1607.07583, submitted to journal., pp. 1-26. 2016.
@article{RISC5339,
author = {Xinhua Xiong},
title = {{Euler's partition theorem for all moduli and new companions to Rogers-Ramanujan-Andrews-Gordon identities}},
language = {english},
journal = {arXiv:1607.07583, submitted to journal.},
pages = {1--26},
isbn_issn = {?},
year = {2016},
refereed = {yes},
length = {26}
}

[Breuer]

### An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics

#### F. Breuer

In: Computer Algebra and Polynomials, , Lecture Notes in Computer Science 8942, pp. 1-29. 2015. 978-3-319-15080-2. [url]
@inproceedings{RISC5110,
author = {F. Breuer},
title = {{An Invitation to Ehrhart Theory: Polyhedral Geometry and its Applications in Enumerative Combinatorics}},
booktitle = {{Computer Algebra and Polynomials}},
language = {english},
series = {Lecture Notes in Computer Science},
volume = {8942},
pages = {1--29},
isbn_issn = {978-3-319-15080-2},
year = {2015},
editor = { J. Gutierrez and J. Schicho and M. Weimann },
refereed = {yes},
length = {29},
}
[Breuer]

### Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems

#### Felix Breuer, Zafeirakis Zafeirakopoulos

Technical report no. 15-09 in RISC Report Series, Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz, Schloss Hagenberg, 4232 Hagenberg, Austria. January 2015. [pdf]
@techreport{RISC5153,
author = {Felix Breuer and Zafeirakis Zafeirakopoulos},
title = {{Polyhedral Omega: A New Algorithm for Solving Linear Diophantine Systems}},
language = {english},
abstract = {Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omega combines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon's iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok's short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omega is significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omega the simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds.},
number = {15-09},
year = {2015},
month = {January},
keywords = {Linear Diophantine system, linear inequality system, integer solutions, partition analysis, partition theory, polyhedral geometry, rational function, symbolic cone, generating function, implementation},
length = {49},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}
[Breuer]

### Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts

#### Felix Breuer, Dennis Eichhorn, Brandt Kronholm

arXiv , August 2015. [url] [pdf] [pdf]
@techreport{RISC5163,
author = {Felix Breuer and Dennis Eichhorn and Brandt Kronholm},
title = {{Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts}},
language = {english},
abstract = {In this paper, we use a branch of polyhedral geometry, Ehrhart theory,to expand our combinatorial understanding of congruences for partitionfunctions.Ehrhart theory allows us to give a new decomposition of partitions,which in turn allows us to define statistics called {\it supercranks}that combinatorially witness every instance of divisibility of$p(n,3)$ by any prime $m \equiv -1 \pmod 6$, where $p(n,3)$ isthe number of partitions of $n$ intothree parts.A rearrangement of lattice points allows us todemonstrate with explicit bijections how to divide these sets of partitions into $m$ equinumerous classes.The behavior for primes $m' \equiv 1 \pmod 6$ is also discussed.},
year = {2015},
month = {August},
howpublished = {arXiv },
keywords = {Integer partitions, Polyhedral Geometry, Combinatorics, Freeman Dyson, Ramanujan, Ehrhart, Crank, Generating Function, },
length = {28},
url = {http://arxiv.org/abs/1508.00397},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}