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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. 3

%I

%S 0,2,0,12,40,270,1848,14840,133488,1334970,14684560,176214852,

%T 2290792920,32071101062,481066515720,7697064251760,130850092279648,

%U 2355301661033970,44750731559645088,895014631192902140

%N 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.

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

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

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

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

%e 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 *).

%p 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);

%Y Cf. A000166, A180188

%K nonn

%O 1,2

%A _Emeric Deutsch_, Sep 06 2010

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Last modified June 19 23:39 EDT 2019. Contains 324222 sequences. (Running on oeis4.)