A256168 -id:A256168 - cp's OEIS frontend

A052186 Number of permutations of [n] with no strong fixed points.

1, 0, 1, 3, 14, 77, 497, 3676, 30677, 285335, 2928846, 32903721, 401739797, 5298600772, 75092880273, 1138261010851, 18378421938366, 314928827507717, 5708689036074089, 109145365739197964, 2195167574579322013, 46331767712354136479, 1023970009016490622478

Offset: 0

N. J. A. Sloane, Feb 04 2000

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

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

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.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Natasha Blitvić and Einar Steingrímsson, Permutations, moments, measures, arXiv:2001.00280 [math.CO], 2020.
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
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V. Strehl, The average number of splitters in a random permutation [Unpublished; included here with the author's permission.]

Cf. A006932, A201684, A256168.
Cf. A144108, A000142. - Gary W. Adamson, Sep 11 2008
Column k=0 of A186373.

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

m = 20; CoefficientList[ Series[ 1 / (x + 1/Sum[ n!*x^n, {n, 0, m}]), {x, 0, m}], x] (* Jean-François Alcover, Aug 30 2011, after Michael 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 *)

{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 */

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
a(n) = n! - Sum_{k=0..n-1} (n-k-1)!*a(k). - Pontus von Brömssen, Jul 10 2021
a(n) + A006932(n) = n!. - Pontus von Brömssen, Jul 10 2021

Better description from James Sellers, Mar 13 2000

A260503 Coefficients in an asymptotic expansion of sequence A003319.

Original entry on oeis.org

1, -2, -1, -5, -32, -253, -2381, -25912, -319339, -4388949, -66495386, -1100521327, -19751191053, -382062458174, -7924762051957, -175478462117633, -4132047373455024, -103115456926017761, -2718766185148876961, -75529218928863243200, -2205316818199975235447

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

			A003319(n) / n! ~ 1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Cf. A003319, A259472, A261214, A261239, A261253, A261254.

Cf. A256168, A260491, A260530, A260532, A260578.

Flatten[{1, Table[Sum[Assuming[Element[x,Reals], SeriesCoefficient[E^(2/x)*x^2 / ExpIntegralEi[1/x]^2,{x,0,k}]] * StirlingS2[n-1,k-1], {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -k! / (2 * (log(2))^(k+1)).
For n>0, Sum_{k=1..n} a(k) * Stirling1(n-1, k-1) = A259472(n). - Vaclav Kotesovec, Aug 03 2015
For n>0, a(n) = Sum_{k=1..n} A259472(k) * Stirling2(n-1, k-1). - Vaclav Kotesovec, Aug 03 2015

A260578 Coefficients in asymptotic expansion of sequence A259869.

Original entry on oeis.org

1, 0, -2, -6, -29, -196, -1665, -16796, -194905, -2549468, -37055681, -592013436, -10307671769, -194225544124, -3937581243201, -85460277981116, -1977127315636969, -48573021658496348, -1262954975286604673, -34650561545808167292, -1000438355724912080873

Offset: 0

Vaclav Kotesovec, Jul 29 2015

sign

For k > 1 is a(k) negative.

			A259869(n) / (n!/exp(1)) ~ 1 - 2/n^2 - 6/n^3 - 29/n^4 - 196/n^5 - 1665/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A259869, A256168, A260491, A260503, A260530, A260532, A260948, A260949, A260950.

nmax = 20; b = CoefficientList[Assuming[Element[x, Reals], Series[x^2*E^(2 + 2/x)/ExpIntegralEi[1 + 1/x]^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -k! / (2 * exp(1) * (log(2))^(k+1)).

A260491 Coefficients in asymptotic expansion of sequence A077607.

Original entry on oeis.org

1, -4, 0, -8, -76, -752, -8460, -107520, -1522124, -23717424, -402941324, -7407988448, -146479479308, -3099229422352, -69863683041868, -1671667534710720, -42318672085310540, -1130167625049525232, -31758424368739424780, -936840101208573355680

Offset: 0

Vaclav Kotesovec, Jul 27 2015

sign

For k > 2 is a(k) negative.

			A077607(n) / (-n!) ~ 1 - 4/n - 8/n^3 - 76/n^4 - 752/n^5 - 8460/n^6 - ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..126
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A077607, A111529, A256168, A260503, A260530, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[x^4*E^(2/x)/(ExpIntegralEi[1/x] - x*E^(1/x))^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ -k * k! / (4 * (log(2))^(k+2)).

A260532 Coefficients in asymptotic expansion of sequence A051295.

Original entry on oeis.org

1, 2, 7, 31, 165, 1025, 7310, 59284, 543702, 5618267, 65200918, 846462826, 12229783811, 195394019337, 3427472046792, 65526442181293, 1355785469986828, 30166624979467869, 717769036033944699, 18174105506247664633, 487655384740384445407, 13816406622559942660420

Offset: 0

Vaclav Kotesovec, Jul 28 2015

nonn

			A051295(n)/(n-1)! ~ 1 + 2/n + 7/n^2 + 31/n^3 + 165/n^4 + 1025/n^5 + 7310/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..134
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A051295, A134378, A256168, A260491, A260503, A260530, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x)*x / (ExpIntegralEi[1/x] - E^(1/x))^2, {x, 0, nmax+1}]], x]; Table[Sum[b[[k+1]] * StirlingS2[n, k-1], {k, 1, n+1}], {n, 0, nmax}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ 2 * (k-1)! / (log(2))^k.

a(n) = Sum_{k=0..n} A134378(k) * Stirling2(n, k). - Vaclav Kotesovec, Aug 04 2015

A260530 Coefficients in asymptotic expansion of sequence A051296.

Original entry on oeis.org

1, 2, 7, 35, 216, 1575, 13243, 126508, 1359437, 16312915, 217277446, 3194459333, 51557948291, 908431129702, 17376289236947, 358847480175063, 7959468559605624, 188702262366570387, 4760773506835189975, 127312428854513811012, 3596091234340397964321

Offset: 0

Vaclav Kotesovec, Jul 28 2015

nonn

			A051296(n) / n! ~ 1 + 2/n + 7/n^2 + 35/n^3 + 216/n^4 + 1575/n^5 + 13243/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132
Richard J. Martin, and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.

Cf. A051296, A256168, A260491, A260503, A260532, A260578.

nmax = 30; b = CoefficientList[Assuming[Element[x, Reals], Series[E^(2/x)*x^2 / (ExpIntegralEi[1/x] - 2*x*E^(1/x))^2, {x, 0, nmax}]], x]; Flatten[{1, Table[Sum[b[[k+1]]*StirlingS2[n-1, k-1], {k, 1, n}], {n, 1, nmax}]}] (* Vaclav Kotesovec, Aug 03 2015 *)

a(k) ~ k! / (2 * (log(2))^(k+1)).

A201684 a(n) = 2*A052186(n) - n!.

Original entry on oeis.org

1, -1, 0, 0, 4, 34, 274, 2312, 21034, 207790, 2228892, 25890642, 324477994, 4370180744, 63007469346, 968847653702, 15834053988732, 274170226919434, 5015004366420178, 96645631069563928, 1957433140982004026

Offset: 0

N. J. A. Sloane, Dec 03 2011

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178. See Table 1.

Cf. A052186, A256168.

a(n) ~ n! * (1 - 4/n + 2/n^2 - 2/n^3 - 18/n^4 - 118/n^5 - 948/n^6 - 9120/n^7 - 100728/n^8 - 1250770/n^9 - 17245316/n^10), for coefficients see 2*A256168. - Vaclav Kotesovec, Mar 17 2015

A260957 Coefficients in asymptotic expansion of sequence A225960.

Original entry on oeis.org

1, -1, 0, 2, 11, 97, 1105, 13905, 189633, 2803873, 44875599, 774931369, 14385283720, 286010393240, 6069158797378, 137001560841500, 3279733485742038, 83029219746406610, 2216965307395350215, 62282348739259200015, 1836840301301564071196, 56750954571528670989664

Offset: 0

Vaclav Kotesovec, Aug 05 2015

sign

			A225960(n)/(2*(n-1)!) ~ 1 - 1/n + 2/n^3 + 11/n^4 + 97/n^5 + 1105/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..132

Cf. A225960, A256168.

a(k) ~ k! / (2 * (log(2))^(k+1)). - Vaclav Kotesovec, Aug 05 2015

A305275 Coefficients in asymptotic expansion of sequence A302557.

Original entry on oeis.org

1, 0, 2, 6, 35, 256, 2187, 21620, 243947, 3098528, 43799819, 682540780, 11630529643, 215190967544, 4296657514283, 92083313483300, 2108244638675035, 51350077108834832, 1325682930813985547, 36157047428501464220, 1038793351537388253211, 31354977545074731373512

Offset: 0

Vaclav Kotesovec, Aug 18 2018

nonn

			A302557(n) / (exp(-1) * n!) ~ 1 + 2/n^2 + 6/n^3 + 35/n^4 + 256/n^5 + 2187/n^6 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..75

Cf. A256168, A260491, A260503, A260530, A260532, A260578, A260948, A260949, A260950, A302557.

a(k) ~ k! / (2 * exp(1) * (log(2))^(k+1)).

cp's OEIS Frontend

A052186 Number of permutations of [n] with no strong fixed points.

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A260503 Coefficients in an asymptotic expansion of sequence A003319.

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A260578 Coefficients in asymptotic expansion of sequence A259869.

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A260491 Coefficients in asymptotic expansion of sequence A077607.

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A260532 Coefficients in asymptotic expansion of sequence A051295.

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A260530 Coefficients in asymptotic expansion of sequence A051296.

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A201684 a(n) = 2*A052186(n) - n!.

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A260957 Coefficients in asymptotic expansion of sequence A225960.

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