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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
2D\'epartement de physique, de g\'enie physique et d'optique, Universit\'e Laval, Qu\'ebec, G1K 7P4, Canada
Annals of Combinatorics 13 (1) pp.87-102 March, 2009
AMS Subject Classification: 11P81, 05A17
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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