On the parity of generalized partition functions, III

Fethi Ben Saïd[1]; Jean-Louis Nicolas[2]; Ahlem Zekraoui[1]

  • [1] Université de Monastir Faculté des Sciences de Monastir Avenue de l’Environement 5000 Monastir, Tunisie
  • [2] Université de Lyon 1 Institut Camile Jordan, UMR 5208 Batiment Doyen Jean Braconnier 21 Avenue Claude Bernard F-69622 Villeurbanne, France

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 1, page 51-78
  • ISSN: 1246-7405

Abstract

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Improving on some results of J.-L. Nicolas [15], the elements of the set 𝒜 = 𝒜 ( 1 + z + z 3 + z 4 + z 5 ) , for which the partition function p ( 𝒜 , n ) (i.e. the number of partitions of n with parts in 𝒜 ) is even for all n ≥ 6 are determined. An asymptotic estimate to the counting function of this set is also given.

How to cite

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Ben Saïd, Fethi, Nicolas, Jean-Louis, and Zekraoui, Ahlem. "On the parity of generalized partition functions, III." Journal de Théorie des Nombres de Bordeaux 22.1 (2010): 51-78. <http://eudml.org/doc/116400>.

@article{BenSaïd2010,
abstract = {Improving on some results of J.-L. Nicolas [15], the elements of the set $\{\mathcal\{A\}\}=\{\mathcal\{A\}\}(1+z+z^3+z^4+z^5)$, for which the partition function $p(\{\mathcal\{A\}\},n)$ (i.e. the number of partitions of $n$ with parts in $\{\mathcal\{A\}\}$) is even for all $n\ge 6$ are determined. An asymptotic estimate to the counting function of this set is also given.},
affiliation = {Université de Monastir Faculté des Sciences de Monastir Avenue de l’Environement 5000 Monastir, Tunisie; Université de Lyon 1 Institut Camile Jordan, UMR 5208 Batiment Doyen Jean Braconnier 21 Avenue Claude Bernard F-69622 Villeurbanne, France; Université de Monastir Faculté des Sciences de Monastir Avenue de l’Environement 5000 Monastir, Tunisie},
author = {Ben Saïd, Fethi, Nicolas, Jean-Louis, Zekraoui, Ahlem},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Partitions; periodic sequences; order of a polynomial; orbits; $2$-adic numbers; counting function; Selberg-Delange formula; partitions; 2-adic numbers; counting functions},
language = {eng},
number = {1},
pages = {51-78},
publisher = {Université Bordeaux 1},
title = {On the parity of generalized partition functions, III},
url = {http://eudml.org/doc/116400},
volume = {22},
year = {2010},
}

TY - JOUR
AU - Ben Saïd, Fethi
AU - Nicolas, Jean-Louis
AU - Zekraoui, Ahlem
TI - On the parity of generalized partition functions, III
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2010
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 51
EP - 78
AB - Improving on some results of J.-L. Nicolas [15], the elements of the set ${\mathcal{A}}={\mathcal{A}}(1+z+z^3+z^4+z^5)$, for which the partition function $p({\mathcal{A}},n)$ (i.e. the number of partitions of $n$ with parts in ${\mathcal{A}}$) is even for all $n\ge 6$ are determined. An asymptotic estimate to the counting function of this set is also given.
LA - eng
KW - Partitions; periodic sequences; order of a polynomial; orbits; $2$-adic numbers; counting function; Selberg-Delange formula; partitions; 2-adic numbers; counting functions
UR - http://eudml.org/doc/116400
ER -

References

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  2. N. Baccar and F. Ben Saïd, On sets such that the partition function is even from a certain point on. International Journal of Number Theory 5 n o 3 (2009), 407–428. Zbl1214.11117MR2529082
  3. N. Baccar, F. Ben Saïd and A. Zekraoui, On the divisor function of sets with even partition functions. Acta Math. Hungarica 112 (1-2) (2006), 25–37. Zbl1121.11071MR2251128
  4. F. Ben Saïd, On a conjecture of Nicolas-Sárközy about partitions. Journal of Number Theory 95 (2002), 209–226. Zbl1041.11066MR1924098
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  7. F. Ben Saïd and J.-L. Nicolas, Sur une application de la formule de Selberg-Delange. Colloquium Mathematicum 98 n o 2 (2003), 223–247. Zbl1051.11049MR2033110
  8. F. Ben Saïd and J.-L. Nicolas, Even partition functions. Séminaire Lotharingien de Combinatoire 46 (2002), B 46i (http//www.mat.univie.ac.at/ slc/). Zbl1042.11008MR1921679
  9. F. Ben Saïd, H. Lahouar and J.-L. Nicolas, On the counting function of the sets of parts such that the partition function takes even values for n large enough. Discrete Mathematics 306 (2006), 1089–1096. Zbl1109.05019MR2245637
  10. P. M. Cohn, Algebra, Volume 1, Second Edition. John Wiley and Sons Ltd, 1988). MR663370
  11. H. Halberstam and H.-E. Richert, Sieve methods. Academic Press, New York, 1974. Zbl0298.10026MR424730
  12. H. Lahouar, Fonctions de partitions à parité périodique. European Journal of Combinatorics 24 (2003), 1089–1096. Zbl1049.11110MR2024560
  13. R. Lidl and H. Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, revised edition, 1994. Zbl0820.11072MR1294139
  14. J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy, On the parity of additive representation functions. J. Number Theory 73 (1998), 292–317. Zbl0921.11050MR1657968
  15. J.-L. Nicolas, On the parity of generalized partition functions II. Periodica Mathematica Hungarica 43 (2001), 177–189. Zbl0980.11049MR1830575

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