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Jagged Partitions and Lattice Paths
P. Jacob1, 2 and P. Mathieu2
1Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, UK
patrick.jacob@hotmail.com
2D\'epartement de physique, de g\'enie physique et d'optique, Universit\'e Laval, Qu\'ebec, G1K 7P4, Canada
pmathieu@phy.ulaval.ca
Annals of Combinatorics 13 (1) pp.87-102 March, 2009
AMS Subject Classification: 11P81, 05A17
Abstract:
A lattice-path description of $K$-restricted jagged partitions is presented. The corresponding lattice paths can have peaks only at even $x$ coordinates and the maximal value of the height cannot be larger than $K-1$. Its weight is twice that of the corresponding jagged partitions. The equivalence is demonstrated at the level of generating functions. A bijection is given between $K$-restricted jagged partitions and partitions restricted by the following frequencies conditions: $f_{2j-1}$ is even and $f_j+f_{j+1}\leq K-1$, where $f_j$ is the number of occurrences of the part $j$ in the partition. Bijections are given between paths and these restricted partitions and between paths and partitions with successive ranks in a prescribed interval.
Keywords: partitions, jagged partitions, lattice paths, generating functions

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