Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COLORED PERMUTATIONS WITH NO MONOCHROMATIC CYCLES

  • Kim, Dongsu (Department of Mathematical Sciences KAIST) ;
  • Kim, Jang Soo (Department of Mathematics Sungkyunkwan University) ;
  • Seo, Seunghyun (Department of Mathematics Education Kangwon National University)
  • Received : 2016.05.30
  • Published : 2017.07.01
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Abstract

An ($n_1,\;n_2,\;{\ldots},\;n_k$)-colored permutation is a permutation of $n_1+n_2+{\cdots}+n_k$ in which $1,\;2,\;{\ldots},\;n_1$ have color 1, and $n_1+1,\;n_1+2,\;{\ldots},\;n_1+n_2$ have color 2, and so on. We give a bijective proof of Steinhardt's result: the number of colored permutations with no monochromatic cycles is equal to the number of permutations with no fixed points after reordering the first $n_1$ elements, the next $n_2$ element, and so on, in ascending order. We then find the generating function for colored permutations with no monochromatic cycles. As an application we give a new proof of the well known generating function for colored permutations with no fixed colors, also known as multi-derangements.

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References

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