Abstract
By jagged partitions we refer to an ordered collection of non-negative integers (n1, n2,..., n m ) with n m ≥ p for some positive integer p, further subject to some weakly decreasing conditions that prevent them for being genuine partitions. The case analyzed in greater detail here corresponds to p = 1 and the following conditions n i ≥ ni+1−1 and n i ≥ ni+2. A number of properties for the corresponding partition function are derived, including rather remarkable congruence relations. An interesting application of jagged partitions concerns the derivation of generating functions for enumerating partitions with special restrictions, a point that is illustrated with various examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G.E. Andrews, The theory of partitions, Cambridge Univ. Press, 1984.
G.E. Andrews, “Multiple q-series,” Houston J. Math. 7 (1981), 11–22.
G.E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its applications 71, Cambridge Univ. Press, 1999.
L. Bégin, J.-F. Fortin, P. Jacob, and P. Mathieu, “Fermionic characters for graded parafermions,” Nucl. Phys. B659 (2003), 365–386.
J.M. Borwein and P.B. Borwein, Pi and the AGM—A Study in Analytic Number Theory and Computational Complexity, Wiley, N.Y., 1987.
B. Feigin, M. Jimbo, S. Loktev, T. Miwa, and E. Mukhin, “Bosonic formulas for (k,l)-admissible partitions,” math.QA/0107054.
B. Feigin, M. Jimbo, and T. Miwa, “Vertex operator algebra arising from the minimal series M(3,p) and monomial basis,” math.QA/0012193.
B. Feigin, M. Jimbo, T. Miwa, E. Mukhinand, and Y. Takeyama, “Particle content of the (k,3)-configurations,” math.QA/0212348.
B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, and Y. Takeyama, “Fermionic formulas for (k,3)-admissible configurations,” math.QA/0212347.
J.-F. Fortin, P. Jacob, and P. Mathieu, “Generating function for K-restricted jagged partitions,” The Electronic Journal of Combinatorics, 12 (2005), R12.
G.H. Hardy, Ramanujan, Cambridge Univ. Press, 1940.
H. Rademacher, Topics in Analytic Number Theory, Springer, Verlag, 1973.
Author information
Authors and Affiliations
Corresponding author
Additional information
2000 Mathematics Subject Classification: Primary—05A15, 05A17, 05A19
Rights and permissions
About this article
Cite this article
Fortin, JF., Jacob, P. & Mathieu, P. Jagged Partitions. Ramanujan J 10, 215–235 (2005). https://doi.org/10.1007/s11139-005-4848-8
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11139-005-4848-8