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A052186 Number of permutations of [n] with no strong fixed points. 14
1, 0, 1, 3, 14, 77, 497, 3676, 30677, 285335, 2928846, 32903721, 401739797, 5298600772, 75092880273, 1138261010851, 18378421938366, 314928827507717, 5708689036074089, 109145365739197964, 2195167574579322013, 46331767712354136479, 1023970009016490622478 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

A strong fixed point is a fixed point (or splitter) p(k)=k such that p(i)<k for i<k and p(j)>k for j>k.

Equals INVERTi transform of the factorials, n starting with 0. Triangle A144108 has row sums = n! with left border = A052186. - Gary W. Adamson, Sep 11 2008

REFERENCES

Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 49

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..450

J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]

FORMULA

G.f.: F(x)/(1+x*F(x)), F(x) = Sum_{n >= 0} n!*x^n.

a(0)=1, a(1)=0, a(n) = (n-2)*a(n-1) + Sum_{k=0..n-1} a(k)*a(n-1-k) + Sum_{k=0..n-2} a(k)*a(n-2-k) if n>1. - Michael Somos, Oct 11 2006

G.f.: 1/(1-x^2/(1-3x-4x^2/(1-5x-9x^2/(1-7x-16x^2/(1-9x-25x^2/(1-... (continued fraction). - Paul Barry, Dec 09 2009

If p[i]=stirling1(i,1) and if A is the Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=(-1)^(n-1) det A. - Milan Janjic, May 08 2010

From Gary W. Adamson, Jul 22 2011: (Start)

a(n) = upper left term in (-1)*M^(n+1), M = an infinite square production matrix in which a column of (-1)'s is prepended to Pascal's triangle as follows:

-1, 1, 0, 0, 0, 0,...

-1, 1, 1, 0, 0, 0,...

-1, 1, 2, 1, 0, 0,...

-1, 1, 3, 3, 1, 0,...

-1, 1, 4, 6, 4, 1,...

...(End)

G.f.: A(x) =1/(1/G(0) + x) ; G(k)= 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))) ; (continued fraction). - Sergei N. Gladkovskii, Dec 29 2011

G.f.: A(x)=1/x =1/(1+x)*(1+x/((1+x)*G(0)-x)) ; G(k)= 1 + x*(k+1) - x*(k+2)/G(k+1) ; (continued fraction Euler's kind,1-step ). - Sergei N. Gladkovskii, Dec 29 2011

G.f.: 1/(G(0)+x) where G(k) = 1  - x*(k+1)/(1 - x*(k+1)/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012

G.f.: 1/(1-W(0)) where W(k) = x*(2*k+1)-1 - x^2*(k+1)^2/W(k+1); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 19 2012

G.f.:  1/(G(0) + x), where G(k)= 1 + x*k - x*(k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jul 03 2013

a(n) ~ n! * (1 - 2/n + 1/n^2 - 1/n^3 - 9/n^4 - 59/n^5 - 474/n^6 - 4560/n^7 - 50364/n^8 - 625385/n^9 - 8622658/n^10), for coefficients see A256168. - Vaclav Kotesovec, Mar 16 2015

MAPLE

t1 := add(n!*x^n, n=0..100): F := series(t1/(1+x*t1), x, 100): for i from 0 to 20 do printf(`%d, `, coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009

# second Maple program:

a:= proc(n) a(n):= -`if`(n<0, 1, add(a(n-i-1)*i!, i=0..n)) end:

seq(a(n), n=0..25);  # Alois P. Heinz, May 21 2013

MATHEMATICA

m = 20; CoefficientList[ Series[ 1 / (x + 1/Sum[ n!*x^n, {n, 0, m}]), {x, 0, m}], x] (* Jean-Fran├žois Alcover, Aug 30 2011, after M. Somos *)

nmax = 25; Rest[CoefficientList[Assuming[Element[x, Reals], Series[-1/(ExpIntegralEi[1/x]/E^(1/x) + 1), {x, 0, nmax+1}]], x]] (* Vaclav Kotesovec, Aug 05 2015 *)

PROG

(PARI) {a(n)=if(n<0, 0, polcoeff( 1/ (x+1/sum(k=0, n, k!*x^k, x*O(x^n))), n))} /* Michael Somos, Oct 11 2006 */

CROSSREFS

Cf. A006932, A201684, A256168.

Cf. A144108, A000142. - Gary W. Adamson, Sep 11 2008

Column k=0 of A186373.

Sequence in context: A198656 A240402 A048779 * A228656 A244507 A074538

Adjacent sequences:  A052183 A052184 A052185 * A052187 A052188 A052189

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Feb 04 2000

EXTENSIONS

Better description from James A. Sellers, Mar 13 2000

STATUS

approved

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Last modified December 10 11:55 EST 2016. Contains 279001 sequences.