A155521 Smallest fixed point summed over all non-derangement permutations of {1,2,...,n}.

0, 1, 1, 7, 31, 191, 1331, 10655, 95887, 958879, 10547659, 126571919, 1645434935, 23036089103, 345541336531, 5528661384511, 93987243536671, 1691770383660095, 32143637289541787, 642872745790835759, 13500327661607550919

Offset: 0

Author

Emeric Deutsch, Apr 21 2009

Keywords

• nonn

Comments

a(n) is also the number of permutations of {1,2,...,n,n+1} having at least 2 fixed points. Example: a(3)=7 because we have 1234, 1243, 1324, 1432, 2134, 4231, and 3214.

Examples

			a(3)=7 because the non-derangements of {1,2,3} are 123, 132, 213, 321 with smallest fixed points 1, 1, 3, 2.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Emeric Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792 [math.CO], 2009.

Crossrefs

Cf. A000166, A047920.

Programs

Maple

a[0] := 0: for n to 25 do a[n] := (n+1)*a[n-1]+n*(-1)^(n+1) end do: seq(a[n], n = 0 .. 21);

Mathematica

CoefficientList[Series[(1-(1+x^2)*E^(-x))/(1-x)^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 20 2012 *)

Formula

a(n) = (n+1)*a(n-1) +n*(-1)^(n+1); a(0)=0.

E.g.f.: (1-(1+x^2)*exp(-x))/(1-x)^2.

a(n) = (n+1)!+(-1)^n-2(n+1)*d(n),

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

a(n) ~ n!*n*(1-2/e). - Vaclav Kotesovec, Oct 20 2012

a(n) = Sum_{k=0..n-1} (k+1) * A047920(n-1,k). - Alois P. Heinz, Sep 01 2021

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