A238942 Number of squarefree permutations of 1,...,n that are fixed under reverse-complement, up to symmetry.

1, 1, 1, 0, 3, 0, 7, 0, 32, 0, 113, 0, 606, 0, 2340, 0, 19941, 0, 122868, 0, 1086205, 0, 6508459, 0, 85816852, 0

Offset

1,5

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.

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). The number of squarefree permutations of 1,...,n up to symmetry is given in the sequence A238937.

A permutation s is fixed under reverse-complement if s=c(r(s)) or, equivalently, s=r(c(s)). This sequence gives the number of squarefree permutations that are fixed under reverse-complement.

Links

Table of n, a(n) for n=1..26.

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) = 2*A238937(n) - A221989(n)/2.

For n > 1, a(2*n) = 0 as in any permutation of size 2*n fixed under reverse-complement, the two factors of length 2 at the center form a square. - Max Alekseyev, Mar 22 2023

Crossrefs

Cf. A221989, A238937.

Sequence in context: A266437 A077896 A359061 * A046269 A153346 A011080

Adjacent sequences: A238939 A238940 A238941 * A238943 A238944 A238945

Keyword

nonn,more

Author

Olexandr Konovalov et al., Mar 07 2014

Extensions

a(2) corrected and a(18)-a(26) added by Max Alekseyev, Mar 24 2023

Status

approved