On the parity of generalized partition functions, III

Ben Saïd, Fethi; Nicolas, Jean-Louis; Zekraoui, Ahlem

Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 51-78.

Dans cet article, nous complétons les résultats de J.-L. Nicolas [15], en déterminant tous les éléments de l’ensemble $𝒜=𝒜(1+z+z^3+z^4+z^5)$ pour lequel la fonction de partition $p(𝒜,n)$ (c-à-d le nombre de partitions de $n$ en parts dans $𝒜$) est paire pour tout $n\ge 6$. Nous donnons aussi un équivalent asymptotique à la fonction de décompte de cet ensemble.

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 $n\ge 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

Classification : 11P81, 11N25, 11N37
Mots clés : Partitions, periodic sequences, order of a polynomial, orbits, $2$-adic numbers, counting function, Selberg-Delange formula.

[1] N. Baccar, Sets with even partition functions and 2-adic integers. Periodica Math. Hung. 55 (2) (2007), 177–193. | MR | Zbl

[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°3 (2009), 407–428. | MR | Zbl

[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. | MR | Zbl

[4] F. Ben Saïd, On a conjecture of Nicolas-Sárközy about partitions. Journal of Number Theory 95 (2002), 209–226. | MR | Zbl

[5] F. Ben Saïd, On some sets with even valued partition function. The Ramanujan Journal 9 (2005), 63–75. | MR | Zbl

[6] F. Ben Saïd and J.-L. Nicolas, Sets of parts such that the partition function is even. Acta Arithmetica 106 (2003), 183–196. | EuDML | MR | Zbl

[7] F. Ben Saïd and J.-L. Nicolas, Sur une application de la formule de Selberg-Delange. Colloquium Mathematicum 98 n° 2 (2003), 223–247. | EuDML | MR | Zbl

[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/). | EuDML | MR | Zbl

[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. | MR | Zbl

[10] P. M. Cohn, Algebra, Volume 1, Second Edition. John Wiley and Sons Ltd, 1988). | MR

[11] H. Halberstam and H.-E. Richert, Sieve methods. Academic Press, New York, 1974. | MR | Zbl

[12] H. Lahouar, Fonctions de partitions à parité périodique. European Journal of Combinatorics 24 (2003), 1089–1096. | MR | Zbl

[13] R. Lidl and H. Niederreiter, Introduction to finite fields and their applications. Cambridge University Press, revised edition, 1994. | MR | Zbl

[14] J.-L. Nicolas, I.Z. Ruzsa and A. Sárközy, On the parity of additive representation functions. J. Number Theory 73 (1998), 292–317. | MR | Zbl

[15] J.-L. Nicolas, On the parity of generalized partition functions II. Periodica Mathematica Hungarica 43 (2001), 177–189. | MR | Zbl