# Partition Analysis [F050-06]

### Project Lead

### Project Duration

01/03/2013 - 28/02/2021### Project URL

Go to Website## Partners

### The Austrian Science Fund (FWF)

## Publications

### 2018

### 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, Xin Li, Zuhair Nashed (ed.), pp. -. 2018. World Scientific Publishing, 978-981-3228-87-0. [url]@

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}

}

**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}

}

### 2017

### 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]@

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}

}

**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}

}

### 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]@

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}

}

**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}

}

### Overpartitions and ternary quadratic forms

#### xinhua xiong

The Ramanujan Journal 42(2), pp. 429-442. 2017. issn:1382-4090.@

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}

}

**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}

}

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

#### Liangjie Ye

Journal of Symbolic Computation, to appear, pp. 1-25. 2017. -. [pdf]@

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}

}

**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}

}

### 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]@

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}

}

**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}

}

### 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]@

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}

}

**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}

}

### 2016

### 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.@

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}

}

**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}

}

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

#### Peter Paule, Cristian-Silviu Radu

In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series, George E. Andrews, Frank Garvan (ed.), pp. 625-640. 2016. Springer, 2194-1009. [pdf]@

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}

}

**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}

}

### 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.@

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}

}

**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}

}

### 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]@

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}

}

**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}

}

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

#### Xinhua Xiong

arXiv:1605.09508, submitted to journal., pp. 1-10. 2016. ISSN.@

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}

}

**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}

}

### 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.@

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}

}

**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}

}

### 2015

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

#### F. Breuer

In: Computer Algebra and Polynomials, J. Gutierrez, J. Schicho, M. Weimann (ed.), Lecture Notes in Computer Science 8942, pp. 1-29. 2015. 978-3-319-15080-2. [url]@

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},

url = {http://link.springer.com/chapter/10.1007/978-3-319-15081-9_1}

}

**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},

url = {http://link.springer.com/chapter/10.1007/978-3-319-15081-9_1}

}

### 2014

### Enumerating Colorings, Tensions and Flows in Cell Complexes

#### M. Beck, F. Breuer, L. Godkin, J. L. Martin

Journal of Combinatorial Theory, Series A 122(4), pp. 82-106. 2014. 0097-3165. [url]@

author = {M. Beck and F. Breuer and L. Godkin and J. L. Martin},

title = {{Enumerating Colorings, Tensions and Flows in Cell Complexes}},

language = {english},

journal = {Journal of Combinatorial Theory, Series A},

volume = {122},

number = {4},

pages = {82--106},

isbn_issn = {0097-3165},

year = {2014},

refereed = {yes},

length = {25},

url = {http://dx.doi.org/10.1016/j.jcta.2013.10.002}

}

**article**{RISC5108,author = {M. Beck and F. Breuer and L. Godkin and J. L. Martin},

title = {{Enumerating Colorings, Tensions and Flows in Cell Complexes}},

language = {english},

journal = {Journal of Combinatorial Theory, Series A},

volume = {122},

number = {4},

pages = {82--106},

isbn_issn = {0097-3165},

year = {2014},

refereed = {yes},

length = {25},

url = {http://dx.doi.org/10.1016/j.jcta.2013.10.002}

}

### Symmetry and Prime Divisibility Properties of Partitions of $n$ into Exactly $m$ Parts

#### B. Kronholm, A. Larsen

Annals of Combinatorics, pp. -. March 2014. Springer Basel, Basel, Switzerland, ISSN 0218-0006. [url] [pdf]@

author = {B. Kronholm and A. Larsen},

title = {{Symmetry and Prime Divisibility Properties of Partitions of $n$ into Exactly $m$ Parts}},

language = {english},

abstract = {Let p(n,m) denote the number of partitions of n into exactly m parts. In this paper we uncover new congruences for the function p(n,m) and give an alternate proof to a known theorem in addition to extending it. The methods of proof rely on identifying generating functions to polynomials and then using the symmetric properties of those polynomials. The theorems proved here provide further motivation and description for a full characterisation of Ramanujan-like divisibility statements about the partition numbers p(n,m).},

journal = {Annals of Combinatorics},

pages = {--},

publisher = {Springer Basel},

address = {Basel, Switzerland},

isbn_issn = {ISSN 0218-0006},

year = {2014},

month = {March},

refereed = {yes},

length = {0},

url = {http://www.springer.com/new+%26+forthcoming+titles+%28default%29/journal/26}

}

**article**{RISC4976,author = {B. Kronholm and A. Larsen},

title = {{Symmetry and Prime Divisibility Properties of Partitions of $n$ into Exactly $m$ Parts}},

language = {english},

abstract = {Let p(n,m) denote the number of partitions of n into exactly m parts. In this paper we uncover new congruences for the function p(n,m) and give an alternate proof to a known theorem in addition to extending it. The methods of proof rely on identifying generating functions to polynomials and then using the symmetric properties of those polynomials. The theorems proved here provide further motivation and description for a full characterisation of Ramanujan-like divisibility statements about the partition numbers p(n,m).},

journal = {Annals of Combinatorics},

pages = {--},

publisher = {Springer Basel},

address = {Basel, Switzerland},

isbn_issn = {ISSN 0218-0006},

year = {2014},

month = {March},

refereed = {yes},

length = {0},

url = {http://www.springer.com/new+%26+forthcoming+titles+%28default%29/journal/26}

}

### 2013

### Generalized Congruence Properties of the Restricted Partition Function p(n, m)

#### B. Kronholm

The Ramanujan Journal 30(3), pp. 425-436. 2013. 1572-9303. [url]@

author = {B. Kronholm},

title = {{Generalized Congruence Properties of the Restricted Partition Function p(n,m)}},

language = {english},

journal = {The Ramanujan Journal},

volume = {30},

number = {3},

pages = {425--436},

isbn_issn = {1572-9303},

year = {2013},

refereed = {yes},

length = {12},

url = {http://link.springer.com/article/10.1007/s11139-012-9382-x}

}

**article**{RISC4833,author = {B. Kronholm},

title = {{Generalized Congruence Properties of the Restricted Partition Function p(n,m)}},

language = {english},

journal = {The Ramanujan Journal},

volume = {30},

number = {3},

pages = {425--436},

isbn_issn = {1572-9303},

year = {2013},

refereed = {yes},

length = {12},

url = {http://link.springer.com/article/10.1007/s11139-012-9382-x}

}

### 2012

### A Result on Ramanujan-Like Congruence Properties of the Restricted partition Function p(n, m) Across Both Variables

#### B. Kronholm

INTEGERS 12(A63), pp. 1-6. 2012. 1553-1732. [url] [pdf]@

author = {B. Kronholm},

title = {{A Result on Ramanujan-Like Congruence Properties of the Restricted partition Function p(n,m) Across Both Variables}},

language = {english},

journal = {INTEGERS},

volume = {12},

number = {A63},

pages = {1--6},

isbn_issn = {1553-1732},

year = {2012},

refereed = {yes},

length = {6},

url = {http://www.integers-ejcnt.org/vol12.html}

}

**article**{RISC4834,author = {B. Kronholm},

title = {{A Result on Ramanujan-Like Congruence Properties of the Restricted partition Function p(n,m) Across Both Variables}},

language = {english},

journal = {INTEGERS},

volume = {12},

number = {A63},

pages = {1--6},

isbn_issn = {1553-1732},

year = {2012},

refereed = {yes},

length = {6},

url = {http://www.integers-ejcnt.org/vol12.html}

}

### 2007

### On Congruence Properties of Consecutive Values of p(n, m)

#### B. Kronholm

INTEGERS 7(A16), pp. 1-6. 2007. 1553-1732. [url] [pdf]@

author = {B. Kronholm},

title = {{On Congruence Properties of Consecutive Values of p(n,m)}},

language = {english},

journal = {INTEGERS},

volume = {7},

number = {A16},

pages = {1--6},

isbn_issn = {1553-1732},

year = {2007},

refereed = {yes},

length = {6},

url = {http://www.integers-ejcnt.org/vol7.html}

}

**article**{RISC4832,author = {B. Kronholm},

title = {{On Congruence Properties of Consecutive Values of p(n,m)}},

language = {english},

journal = {INTEGERS},

volume = {7},

number = {A16},

pages = {1--6},

isbn_issn = {1553-1732},

year = {2007},

refereed = {yes},

length = {6},

url = {http://www.integers-ejcnt.org/vol7.html}

}

### 2005

### On Congruence Properties of p(n, m)

#### B. Kronholm

Proceedings of the American Mathematical Society 133, pp. 2891-2895. 2005. 1088-6826. [url] [pdf]@

author = {B. Kronholm},

title = {{On Congruence Properties of p(n,m)}},

language = {english},

journal = {Proceedings of the American Mathematical Society},

volume = {133},

pages = {2891--2895},

isbn_issn = {1088-6826},

year = {2005},

refereed = {yes},

length = {5},

url = {http://www.ams.org/journals/proc/2005-133-10/S0002-9939-05-07972-4/S0002-9939-05-07972-4.pdf}

}

**article**{RISC4831,author = {B. Kronholm},

title = {{On Congruence Properties of p(n,m)}},

language = {english},

journal = {Proceedings of the American Mathematical Society},

volume = {133},

pages = {2891--2895},

isbn_issn = {1088-6826},

year = {2005},

refereed = {yes},

length = {5},

url = {http://www.ams.org/journals/proc/2005-133-10/S0002-9939-05-07972-4/S0002-9939-05-07972-4.pdf}

}