BibTex RIS Kaynak Göster

The part-frequency matrices of a partition

Yıl 2016, , 177 - 186, 09.08.2016
https://doi.org/10.13069/jacodesmath.41075

Öz

A new combinatorial object is introduced, the part-frequency matrix sequence of a partition, which
is elementary to describe and is naturally motivated by Glaisher’s bijection. We prove results that
suggest surprising usefulness for such a simple tool, including the existence of a related statistic that
realizes every possible Ramanujan-type congruence for the partition function. To further exhibit its
research utility, we give an easy generalization of a theorem of Andrews, Dixit and Yee [1] on the mock
theta functions. Throughout, we state a number of observations and questions that can motivate an
array of investigations.

Kaynakça

  • G. E. Andrews, A. Dixit, A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions $omega(q)$, $nu(q)$, and $phi(q)$, Res. Number Theory 1 (2015) 1–9.
  • G. Andrews, F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18(2) (1988) 167–171.
  • F. Breuer, D. Eichhorn, B Kronholm, Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts, pre-print available at http://arxiv.org/abs/1508.00397.
  • K. Bringmann, K. Ono, The f(q) mock theta function conjecture and partition ranks, Invent. Math. 165(2) (2006) 243–266.
  • D. Ford, J. McKay, S. P. Norton, More on replicable functions, Comm. Algebra 22(13) (1994) 5175–5193.
  • F. G. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101(1) (1990) 1–17.
  • K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. 102(43) (2005) 15373–15376.
  • S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc. 93(2) (2006) 304–324.
Yıl 2016, , 177 - 186, 09.08.2016
https://doi.org/10.13069/jacodesmath.41075

Öz

Kaynakça

  • G. E. Andrews, A. Dixit, A. J. Yee, Partitions associated with the Ramanujan/Watson mock theta functions $omega(q)$, $nu(q)$, and $phi(q)$, Res. Number Theory 1 (2015) 1–9.
  • G. Andrews, F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18(2) (1988) 167–171.
  • F. Breuer, D. Eichhorn, B Kronholm, Polyhedral geometry, supercranks, and combinatorial witnesses of congruences for partitions into three parts, pre-print available at http://arxiv.org/abs/1508.00397.
  • K. Bringmann, K. Ono, The f(q) mock theta function conjecture and partition ranks, Invent. Math. 165(2) (2006) 243–266.
  • D. Ford, J. McKay, S. P. Norton, More on replicable functions, Comm. Algebra 22(13) (1994) 5175–5193.
  • F. G. Garvan, D. Kim, D. Stanton, Cranks and t-cores, Invent. Math. 101(1) (1990) 1–17.
  • K. Mahlburg, Partition congruences and the Andrews-Garvan-Dyson crank, Proc. Natl. Acad. Sci. 102(43) (2005) 15373–15376.
  • S. Treneer, Congruences for the coefficients of weakly holomorphic modular forms, Proc. London Math. Soc. 93(2) (2006) 304–324.
Toplam 8 adet kaynakça vardır.

Ayrıntılar

Bölüm Makaleler
Yazarlar

William J. Keith Bu kişi benim

Yayımlanma Tarihi 9 Ağustos 2016
Yayımlandığı Sayı Yıl 2016

Kaynak Göster

APA Keith, W. J. (2016). The part-frequency matrices of a partition. Journal of Algebra Combinatorics Discrete Structures and Applications, 3(3), 177-186. https://doi.org/10.13069/jacodesmath.41075
AMA Keith WJ. The part-frequency matrices of a partition. Journal of Algebra Combinatorics Discrete Structures and Applications. Ağustos 2016;3(3):177-186. doi:10.13069/jacodesmath.41075
Chicago Keith, William J. “The Part-Frequency Matrices of a Partition”. Journal of Algebra Combinatorics Discrete Structures and Applications 3, sy. 3 (Ağustos 2016): 177-86. https://doi.org/10.13069/jacodesmath.41075.
EndNote Keith WJ (01 Ağustos 2016) The part-frequency matrices of a partition. Journal of Algebra Combinatorics Discrete Structures and Applications 3 3 177–186.
IEEE W. J. Keith, “The part-frequency matrices of a partition”, Journal of Algebra Combinatorics Discrete Structures and Applications, c. 3, sy. 3, ss. 177–186, 2016, doi: 10.13069/jacodesmath.41075.
ISNAD Keith, William J. “The Part-Frequency Matrices of a Partition”. Journal of Algebra Combinatorics Discrete Structures and Applications 3/3 (Ağustos 2016), 177-186. https://doi.org/10.13069/jacodesmath.41075.
JAMA Keith WJ. The part-frequency matrices of a partition. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3:177–186.
MLA Keith, William J. “The Part-Frequency Matrices of a Partition”. Journal of Algebra Combinatorics Discrete Structures and Applications, c. 3, sy. 3, 2016, ss. 177-86, doi:10.13069/jacodesmath.41075.
Vancouver Keith WJ. The part-frequency matrices of a partition. Journal of Algebra Combinatorics Discrete Structures and Applications. 2016;3(3):177-86.