OEIS A000090: Permutations avoiding 3-cycles

A000090 counts the permutations of n elements with no 3-cycles in their cycle decomposition. For example, with n = 3 there are 4 such permutations (out of 6), and with n = 4 there are 16. The exponential generating function is exp(-x^3/3) divided by (1 - x), i.e. exp(sum_{k≠3} x^k/k). As n grows, the count a(n) is asymptotically e^{-1/3} n!, revealing the surprising appearance of the constant e in a purely discrete setting. This sequence sits among rich connections in the OEIS, including links to derangements and other restricted-cycle-length families. Explore its terms, generating function, and asymptotics to see how a simple “no 3-cycles” rule unlocks a web of structure in permutations.

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