A Note on the Factorization of Permutations into Cycles

Yıl 2017, Cilt: 22 Sayı: 22, 125 - 132, 11.07.2017

J. Elder O. Vega

https://doi.org/10.24330/ieja.325937

Öz

We find conditions on $k, n\in \N$, where $3\leq k\leq n$ for
which a permutation in $S_n$ can be written as a product of
distinct $k$-cycles in $S_{n+i}\setminus S_n$, for some $i\in \N$.
This result generalizes a problem that was solved in 2010 in an episode of the television show Futurama:
 the so-called Futurama Theorem.

Anahtar Kelimeler

Permutation, product of cycles

Kaynakça

• R. Evans, L. Huang and T. Nguyen, Keeler's theorem and products of distinct transpositions, Amer. Math. Monthly, 121(2) (2014), 136-144.
• R. Evans and L. Huang, Mind switches in Futurama and Stargate, Math. Mag., 87(4) (2014), 252-262.
• H. Georgiev, The Futurama theorem explained, The Commutator, 2 (2010), 18-20.
• T. Phillips, Math in the Media, Amer. Math. Soc., Original math on Futurama, (2010). http://www.ams.org/news/math-in-the-media/10-2010-media.
• Previous Nominees & Winners of the Writers Guild Awards (last accessed on 08/15/16). http://awards.wga.org/wga-awards/previous-nominees-winners.
• S. Singh, The Simpsons and their mathematical secrets, Bloomsbury, NY, 2013.
• The prisoner of Benda, The Infosphere, the Futurama Wiki (last accessed on 08/15/16). http://theinfosphere.org/The_Prisoner_of_Benda.