# 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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