A209322 Number of derangements of [n] with no succession.

1, 0, 1, 0, 4, 14, 102, 682, 5484, 49288, 492812, 5418154, 64993966, 844658714, 11822116868, 177292309424, 2836140479376, 48206588630826, 867597809813018, 16482372327022854, 329612875955466784, 6921235129197714036, 152254880756288024536, 3501612401180417830334

Offset

0,5

Comments

A derangement is a permutation with no fixed points. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

Links

Table of n, a(n) for n=0..23.

Formula

a(n) = n! - A207819(n).

Example

For n=4 we have 2143, 2413, 3142 and 4321, so a(4) = 4.

Maple

F:= proc(S) add(G(S minus {s}, s-1), s = S minus {nops(S)}) end proc:
G:= proc(S, t) option remember;
if S = {} then return 1 fi;
add(procname(S minus {s}, s-1), s = S minus {t, nops(S)})
end proc:
1, seq(F({$1..n}), n=1..19); # Robert Israel, Mar 02 2017

Mathematica

F[{}] = 1; F[S_] := Sum[G[S ~Complement~ {s}, s-1], {s, S ~Complement~ {Length[S]}}];
G[{}, _] = 1; G[S_, t_] := G[S, t] = Sum[G[S ~Complement~ {s}, s-1], {s, S ~Complement~ {t, Length[S]}}];
Table[a[n] = F[Range[n]]; Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 05 2019, after Robert Israel *)

Crossrefs

Cf. A000166, A002467, A180191, A207819, A207821, A209325, A209326.

Sequence in context: A316414 A268689 A299199 * A110302 A282617 A005743

Adjacent sequences: A209319 A209320 A209321 * A209323 A209324 A209325

Keyword

nonn

Author

Jon Perry, Jan 19 2013

Extensions

a(11)-a(14) from Alois P. Heinz, Jan 19 2013

a(15)-a(20) from Robert Israel, Mar 02 2017

a(21)-a(23) from Alois P. Heinz, Jul 04 2021

Status

approved