Ranks For Two Partition Quadruple Functions
Chris Jennings-Shaffer1
1 Department of Mathematics Oregon State University Corvallis, Oregon 97331, USA
Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 425-443.

L’auteur a récemment introduit deux fonctions de partitions entières qui satisfont des congruences du type Ramanujan modulo 3, 5, 7, et 13. On definit une statistique du type rang et obtient une amélioration de l’interprêtation combinatoire des congruences modulo 3, 5 et 7.

Recently the author introduced two new integer partition quadruple functions, which satisfy Ramanujan-type congruences modulo 3, 5, 7, and 13. Here we reprove the congruences modulo 3, 5, and 7 by defining a rank-type statistic that gives a combinatorial refinement of the congruences.

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DOI : 10.5802/jtnb.986
Classification : 11P81, 11P83
Mots clés : Number theory, partitions, vector partitions, congruences, ranks, cranks
Chris Jennings-Shaffer 1

1 Department of Mathematics Oregon State University Corvallis, Oregon 97331, USA
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Chris Jennings-Shaffer. Ranks For Two Partition Quadruple Functions. Journal de théorie des nombres de Bordeaux, Tome 29 (2017) no. 2, pp. 425-443. doi : 10.5802/jtnb.986. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.986/

[1] George E. Andrews The theory of partitions, Encyclopedia of Mathematics and Its Applications, 2, Addison-Wesley Publishing Company, 1976, xiv+255 pages

[2] George E. Andrews; Bruce C. Berndt Ramanujan’s lost notebook. Part III, Springer, 2012, xii+435 pages

[3] George E. Andrews; Frank G. Garvan Dyson’s crank of a partition, Bull. Am. Math. Soc., Volume 18 (1988) no. 2, pp. 167-171 | DOI

[4] A. Oliver L. Atkin; Henry Peter Francis Swinnerton-Dyer Some properties of partitions, Proc. Lond. Math. Soc., Volume 4 (1954), pp. 84-106 | DOI

[5] Song Heng Chan Generalized Lambert series identities, Proc. Lond. Math. Soc., Volume 91 (2005) no. 3, pp. 598-622 | DOI

[6] Freeman J. Dyson Some guesses in the theory of partitions, Eureka, Volume 8 (1944), pp. 10-15

[7] Frank G. Garvan New combinatorial interpretations of Ramanujan’s partition congruences mod 5,7 and 11, Trans. Am. Math. Soc., Volume 305 (1988) no. 1, pp. 47-77

[8] Chris Jennings-Shaffer Two partition functions with congruences modulo 3, 5, 7, and 13 (to appear in Ann. Comb.)

[9] Chris Jennings-Shaffer Some Smallest Parts Functions from Variations of Bailey’s Lemma (2015) (https://arxiv.org/abs/1506.05344)

[10] Sander Zwegers Mock theta functions, Utrecht University (2003) (Ph. D. Thesis)

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