A238937 Number of squarefree permutations of {1, ..., n} up to symmetry.

1, 1, 2, 3, 10, 26, 105, 278, 1011, 3804, 17065, 78012, 406795, 2192844, 13318687, 79804728, 533838106, 3605026566, 27263032020, 203967032072, 1693025508202, 14125460316054, 129839854469553

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

Table of n, a(n) for n=1..23.

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(n) = (A221989(n) + 2*A238942(n)) / 4.

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

Crossrefs

Cf. A221989, A238942.

Sequence in context: A005225 A211208 A303836 * A278088 A052929 A151415

Adjacent sequences: A238934 A238935 A238936 * A238938 A238939 A238940

Keyword

nonn,more

Author

Olexandr Konovalov et al., Mar 07 2014

Extensions

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

Status

approved