A180189 Number of permutations of [n] having exactly 1 circular succession. A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

0, 2, 0, 12, 40, 270, 1848, 14840, 133488, 1334970, 14684560, 176214852, 2290792920, 32071101062, 481066515720, 7697064251760, 130850092279648, 2355301661033970, 44750731559645088, 895014631192902140, 18795307255050944520, 413496759611120779902

Offset

1,2

Comments

For example, p=(4,1,2,5,3) has 2 circular successions: (1,2) and (3,4).

Links

Table of n, a(n) for n=1..22.

S. M. Tanny, Permutations and successions, J. Combinatorial Theory, Series A, 21 (1976), 196-202.

Formula

a(n) = n*(n-1)*d(n-2), where d(j)=A000166(j) are the derangement numbers.

a(n) = A180188(n,1).

E.g.f.: x^2 * exp(-x) / (1 - x). - Ilya Gutkovskiy, Oct 11 2021

a(n) = 2 * A000387(n). - Alois P. Heinz, Oct 11 2021

D-finite with recurrence (-n+2)*a(n) +n*(n-3)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

Example

a(4)=12 because we have 1*243, 142*3, 13*42, 31*24, 3142*, 431*2, 213*4, 4213*, 2*314, 2431*, 42*31, and 3*421 (the circular succession is marked *).

Maple

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: seq(n*(n-1)*d[n-2], n = 1 .. 22);

Crossrefs

Cf. A000166, A000387, A180188.

Sequence in context: A134831 A074001 A228621 * A182437 A326860 A013316

Adjacent sequences: A180186 A180187 A180188 * A180190 A180191 A180192

Keyword

nonn

Author

Emeric Deutsch, Sep 06 2010

Status

approved