Geodesic
A generalization of conjugation of integer partitions
Seamus Albion ; Theresia Eisenkölbl ; Ilse Fischer1 ; Moritz Gangl ; Hans Höngesberg ; Christian Krattenthaler ; Martin Rubey
1Universität Wien, Fakultät für Mathematik
The electronic journal of combinatorics, Tome 32 (2025) no. 2
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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.
DOI : 10.37236/13489
Classification : 05A17, 05A19, 05A15, 05A30
Mots-clés : involutions on the set of partitions

Seamus Albion    ; Theresia Eisenkölbl    ; Ilse Fischer  1   ; Moritz Gangl    ; Hans Höngesberg    ; Christian Krattenthaler    ; Martin Rubey 

1 Universität Wien, Fakultät für Mathematik 
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     author = {Seamus  Albion and Theresia  Eisenk\"olbl and Ilse Fischer and Moritz  Gangl and Hans  H\"ongesberg and Christian  Krattenthaler and Martin  Rubey},
     title = {A generalization of conjugation of integer partitions},
     journal = {The electronic journal of combinatorics},
     year = {2025},
     volume = {32},
     number = {2},
     doi = {10.37236/13489},
     zbl = {1565.05008},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/13489/}
}
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AU  - Martin  Rubey
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%A Christian  Krattenthaler
%A Martin  Rubey
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Seamus  Albion; Theresia  Eisenkölbl; Ilse Fischer; Moritz  Gangl; Hans  Höngesberg; Christian  Krattenthaler; Martin  Rubey. A generalization of conjugation of integer partitions. The electronic journal of combinatorics, Tome 32 (2025) no. 2. doi: 10.37236/13489

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