A Note on the Factorization of Permutations into Cycles

Abstract

We find conditions on $k, n\in \N$, where $3\leq k\leq n$ forwhich a permutation in $S_n$ can be written as a product ofdistinct $k$-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.