A207819 Number of permutations of [n] with a fixed point and/or a succession.

0, 1, 1, 6, 20, 106, 618, 4358, 34836, 313592, 3135988, 34498646, 414007634, 5382362086, 75356174332, 1130382058576, 18086649408624, 307480839465174, 5534775895914982, 105162728081809146, 2103289132221173216, 44169707042511725964, 971745847021319655464, 22350404337704558809666, 536415027665581568375190, 13410494347081333360291850

Offset: 0

Author

Jon Perry, Jan 10 2013

Keywords

• nonn

Comments

A succession of a permutation p is the appearance of [k,k+1], e.g. in 23541, 23 is a succession.

Examples

			For n=4 the only permutations that do not count are 2143, 2413, 3142 and 4321, so a(4) = 4!-4 = 20.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..30

Crossrefs

Cf. A000166, A002467, A180191, A201452, A207821, A209322.

Programs

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] = n! - F[Range[n]]; Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 05 2019, using Robert Israel's code for A209322 *)

PARI

A207819(n)={my(p,c);sum(k=1,n!,p=numtoperm(n,k);(c=(p[1]==1)) || for(j=2,n,p[j]!=j & p[j]-1!=p[j-1] & next; c++; break);c)} \\ M. F. Hasler, Jan 13 2013

Formula

a(n) = n! - A209322(n). - Robert Israel, Mar 27 2017

Extensions

Values a(1..10) double-checked by M. F. Hasler, Jan 13 2013

a(11)-a(14) from Alois P. Heinz, Jan 15 2013

a(15)-a(20) from Robert Israel, Mar 27 2017

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

Terms a(24) onward from Max Alekseyev, Apr 03 2025