OFFSET

1,3

COMMENTS

A permutation is squarefree if it does not contain two consecutive factors of length two or more that are in the same relative order. For example, the permutation 243156 is squarefree, while the permutation 631425 contains the square 3142 (indeed, 31 is order-isomorphic to 42). Squarefree permutations exist of any length, and their numbers are given in the sequence A221989.

This sequence gives the number of squarefree permutations of 1,...,n up to symmetry. There are two kinds of symmetries involved: the reverse of a permutation s = i_1 i_2 ... i_n is the permutation r(s) = i_n ... i_2 i_1, and the complement of s is the permutation c(s) = (n+1-i_1) (n+1-i_2) ... (n+1-i_n). "Up to symmetry" means that if a permutation s has been already counted, then c(s), r(s) and c(r(s))=r(c(s)) are not counted.

LINKS

Ian Gent, Sergey Kitaev, Alexander Konovalov, Steve Linton and Peter Nightingale, S-crucial and bicrucial permutations with respect to squares, arXiv:1402.3582 [math.CO], 2014 and J. Int. Seq. 18 (2015) 15.6.5.

FORMULA

For n > 1, a(2*n) = A221989(2*n) / 4. - Max Alekseyev, Mar 22 2023

CROSSREFS

KEYWORD

nonn,more

AUTHOR

Olexandr Konovalov et al., Mar 07 2014

EXTENSIONS

a(18)-a(23) from Max Alekseyev, Mar 23 2023

STATUS

approved