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A238942 The number of squarefree permutations of 1,...,n that are fixed under reverse-complement. 2
1, 2, 1, 0, 3, 0, 7, 0, 32, 0, 113, 0, 606, 0, 2340, 0, 19941 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..17.

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


For n>2, A221989(n) = 4*A238937(n) - 2*a(n).


Cf. A221989, A238937.

Sequence in context: A131047 A143714 A004172 * A082754 A063173 A120111

Adjacent sequences:  A238939 A238940 A238941 * A238943 A238944 A238945




Alexander Konovalov et al., Mar 07 2014



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Last modified January 26 21:14 EST 2022. Contains 350600 sequences. (Running on oeis4.)