A165961 Number of circular permutations of length n without 3-sequences.

1, 5, 20, 102, 627, 4461, 36155, 328849, 3317272, 36757822, 443846693, 5800991345, 81593004021, 1228906816941, 19733699436636, 336554404751966, 6075478765948135, 115734570482611885, 2320148441078578447, 48827637296350480457, 1076313671861962141616

Offset

3,2

Comments

Circular permutations are permutations whose indices are from the ring of integers modulo n. 3-sequences are of the form i,i+1,i+2. Sequence gives number of permutations of [n] starting with 1 and having no 3-sequences.

a(n) is also the number of permutations of length n-1 without consecutive fixed points (cf. A180187). - David Scambler, Mar 27 2011

References

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. - From N. J. A. Sloane, Sep 15 2012 [broken link]

Links

Michael De Vlieger, Table of n, a(n) for n = 3..450

Wayne M. Dymacek and Isaac Lambert, Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.6.

Kyle Parsons, Arithmetic progressions in permutations, Thesis, 2011.

Formula

Let b(n) be the sequence A002628. Then for n > 5, this sequence satisfies a(n) = b(n-1) - b(n-3) + a(n-3).

a(n) = Sum_{k=0..n/2} binomial(n-k,k)*d(n-k-1), where d(j)=A000166(j) are the derangement numbers. - Emeric Deutsch, Sep 07 2010

Example

For n=4 the a(4)=5 solutions are (0,1,3,2), (0,2,1,3), (0,2,3,1), (0,3,1,2) and (0,3,2,1).

Maple

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n) options operator, arrow: sum(binomial(n-k, k)*d[n-k-1], k = 0 .. floor((1/2)*n)) end proc: seq(a(n), n = 3 .. 23); # Emeric Deutsch, Sep 07 2010

Mathematica

a[n_] := Sum[Binomial[n-k, k] Subfactorial[n-k-1], {k, 0, n/2}];
a /@ Range[3, 21] (* Jean-François Alcover, Oct 29 2019 *)

Crossrefs

Cf. A002628, A165960, A165962.

Cf. A000166, A180186, - Emeric Deutsch, Sep 07 2010

A column of A216718. - N. J. A. Sloane, Sep 15 2012

Sequence in context: A110595 A369488 A092640 * A276314 A292358 A259275

Adjacent sequences: A165958 A165959 A165960 * A165962 A165963 A165964

Keyword

nonn

Author

Isaac Lambert, Oct 01 2009

Extensions

More terms from Emeric Deutsch, Sep 07 2010

Edited by N. J. A. Sloane, Apr 04 2011

Status

approved