A generalization of conjugation of integer partitions
(20 pages)
English Abstract.
We exhibit, for any positive integer )parameter s, an
involution on the set of integer partitions of n. These
involutions show the joint symmetry of the distributions of the
following two statistics. The first counts the number of parts of
a partition
divisible by s, whereas the second counts the number of cells in the
Ferrers diagram of a partition
whose leg length is zero and whose arm length has remainder s-1 when
dividing by s. In particular, for s=1 this involution is just
conjugation.
Additionally, we provide explicit expressions for the bivariate
generating functions.
Our primary motivation to construct these involutions is that we
know only of two other ``natural'' bijections on integer
partitions of a given size, one of which is the Glaisher-Franklin
bijection sending the set of parts divisible by s, each divided
by s, to the set of parts occurring at least s times.
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