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.
Keywordspartitions generating functions congruence relations
Unable to display preview. Download preview PDF.
- 1.G.E. Andrews, The theory of partitions, Cambridge Univ. Press, 1984.Google Scholar
- 2.G.E. Andrews, “Multiple q-series,” Houston J. Math. 7 (1981), 11–22.Google Scholar
- 3.G.E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its applications 71, Cambridge Univ. Press, 1999.Google Scholar
- 4.L. Bégin, J.-F. Fortin, P. Jacob, and P. Mathieu, “Fermionic characters for graded parafermions,” Nucl. Phys. B659 (2003), 365–386.Google Scholar
- 5.J.M. Borwein and P.B. Borwein, Pi and the AGM—A Study in Analytic Number Theory and Computational Complexity, Wiley, N.Y., 1987.Google Scholar
- 6.B. Feigin, M. Jimbo, S. Loktev, T. Miwa, and E. Mukhin, “Bosonic formulas for (k,l)-admissible partitions,” math.QA/0107054.Google Scholar
- 7.B. Feigin, M. Jimbo, and T. Miwa, “Vertex operator algebra arising from the minimal series M(3,p) and monomial basis,” math.QA/0012193.Google Scholar
- 8.B. Feigin, M. Jimbo, T. Miwa, E. Mukhinand, and Y. Takeyama, “Particle content of the (k,3)-configurations,” math.QA/0212348.Google Scholar
- 9.B. Feigin, M. Jimbo, T. Miwa, E. Mukhin, and Y. Takeyama, “Fermionic formulas for (k,3)-admissible configurations,” math.QA/0212347.Google Scholar
- 10.J.-F. Fortin, P. Jacob, and P. Mathieu, “Generating function for K-restricted jagged partitions,” The Electronic Journal of Combinatorics, 12 (2005), R12.Google Scholar
- 11.G.H. Hardy, Ramanujan, Cambridge Univ. Press, 1940.Google Scholar
- 12.H. Rademacher, Topics in Analytic Number Theory, Springer, Verlag, 1973.Google Scholar