A000166 -id:A000166 - cp's OEIS frontend

A165961 Number of circular permutations of length n without 3-sequences.

1, 5, 20, 102, 627, 4461, 36155, 328849, 3317272, 36757822, 443846693, 5800991345, 81593004021, 1228906816941, 19733699436636, 336554404751966, 6075478765948135, 115734570482611885, 2320148441078578447, 48827637296350480457, 1076313671861962141616

Offset: 3

Isaac Lambert, Oct 01 2009

nonn

Circular permutations are permutations whose indices are from the ring of integers modulo n. 3-sequences are of the form i,i+1,i+2. Sequence gives number of permutations of [n] starting with 1 and having no 3-sequences.

a(n) is also the number of permutations of length n-1 without consecutive fixed points (cf. A180187). - David Scambler, Mar 27 2011

			For n=4 the a(4)=5 solutions are (0,1,3,2), (0,2,1,3), (0,2,3,1), (0,3,1,2) and (0,3,2,1).

Wayne M. Dymacek, Isaac Lambert and Kyle Parsons, Arithmetic Progressions in Permutations, http://math.ku.edu/~ilambert/CN.pdf, 2012. - From N. J. A. Sloane, Sep 15 2012 [broken link]

Michael De Vlieger, Table of n, a(n) for n = 3..450
Wayne M. Dymacek and Isaac Lambert, Permutations Avoiding Runs of i, i+1, i+2 or i, i-1, i-2, Journal of Integer Sequences, Vol. 14 (2011), Article 11.1.6.
Kyle Parsons, Arithmetic progressions in permutations, Thesis, 2011.

Cf. A002628, A165960, A165962.
Cf. A000166, A180186, - Emeric Deutsch, Sep 07 2010
A column of A216718. - N. J. A. Sloane, Sep 15 2012

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: a := proc (n) options operator, arrow: sum(binomial(n-k, k)*d[n-k-1], k = 0 .. floor((1/2)*n)) end proc: seq(a(n), n = 3 .. 23); # Emeric Deutsch, Sep 07 2010

a[n_] := Sum[Binomial[n-k, k] Subfactorial[n-k-1], {k, 0, n/2}];
a /@ Range[3, 21] (* Jean-François Alcover, Oct 29 2019 *)

Let b(n) be the sequence A002628. Then for n > 5, this sequence satisfies a(n) = b(n-1) - b(n-3) + a(n-3).

a(n) = Sum_{k=0..n/2} binomial(n-k,k)*d(n-k-1), where d(j)=A000166(j) are the derangement numbers. - Emeric Deutsch, Sep 07 2010

More terms from Emeric Deutsch, Sep 07 2010

Edited by N. J. A. Sloane, Apr 04 2011

A002628 Number of permutations of length n without 3-sequences.

Original entry on oeis.org

1, 1, 2, 5, 21, 106, 643, 4547, 36696, 332769, 3349507, 37054436, 446867351, 5834728509, 82003113550, 1234297698757, 19809901558841, 337707109446702, 6094059760690035, 116052543892621951, 2325905946434516516, 48937614361477154273, 1078523843237914046247

Offset: 0

N. J. A. Sloane

nonn

a(n) = sum of row n of A180185. - Emeric Deutsch, Sep 06 2010

			a(4) = 21 because only 1234, 2341, and 4123 contain 3-sequences. - _Emeric Deutsch_, Sep 06 2010

Jackson, D. M.; Reilly, J. W. Permutations with a prescribed number of p-runs. Ars Combinatoria 1 (1976), number 1, 297-305.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..450
D. M. Jackson and R. C. Read, A note on permutations without runs of given length, Aequationes Math. 17 (1978), no. 2-3, 336-343.
J. Riordan, Permutations without 3-sequences, Bull. Amer. Math. Soc., 51 (1945), 745-748.

Column k=0 of A047921.
Cf. A165960, A165961, A165962. - Isaac Lambert, Oct 07 2009
Cf. A000166, A180185. - Emeric Deutsch, Sep 06 2010

seq(coeff(convert(series(add(m!*((t-t^3)/(1-t^3))^m,m=0..50),t,50), polynom), t,n),n=0..25); # Pab Ter, Nov 06 2005
d[-1]:= 0: for n from 0 to 51 do d[n] := n*d[n-1]+(-1)^n end do: a:= proc(n) add(binomial(n-k, k)*(d[n-k]+d[n-k-1]), k = 0..floor((1/2)*n)) end proc: seq(a(n), n = 0..25); # Emeric Deutsch, Sep 06 2010
# third Maple program:
a:= proc(n) option remember; `if`(n<5,
      [1$2, 2, 5, 21][n+1], (n-3)*a(n-1)+(3*n-6)*a(n-2)+
      (4*n-12)*a(n-3)+(3*n-12)*a(n-4)+(n-5)*a(n-5))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 21 2019

d[0] = 1; d[n_] := d[n] = n d[n - 1] + (-1)^n;
T[n_, k_] := If[n == 0 && k == 0, 1, If[k <= n/2, Binomial[n - k, k] d[n + 1 - k]/(n - k), 0]];
a[n_] := Sum[T[n, k], {k, 0, Quotient[n, 2]}];
a /@ Range[0, 25] (* Jean-François Alcover, May 23 2020 *)

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)*(d(n-k) + d(n-k-1)) for n>0, where d(j) = A000166(j) are the derangement numbers. - Emeric Deutsch, Sep 06 2010

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Nov 06 2005

a(0)=1 prepended by Alois P. Heinz, Jul 21 2019

A033815 Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).

Original entry on oeis.org

1, 1, 14, 426, 24024, 2170680, 287250480, 52370755920, 12585067447680, 3854801333416320, 1465957162768492800, 677696237345719468800, 374281829360322587827200, 243388909697235614324812800, 184070135024053703140543027200, 160192129141963141211280644352000

Offset: 0

N. J. A. Sloane

Also turns up as the solution to Problem #18, p. 326 of Alan Tucker's Applied Combinatorics, 4th ed, Wiley NY 2002 [Tucker's `n' is the `2n' here]. - John L Leonard, Sep 15 2003
Number of acyclic orientations of the Turán graph T(2n,n). - Alois P. Heinz, Jan 13 2016
n-th term of the n-th forward differences of n!. - Alois P. Heinz, Feb 22 2019

R. P. Stanley, Enumerative Combinatorics I, Chap.2, Exercise 10, p. 89.

Reinhard Zumkeller, Table of n, a(n) for n = 0..200
Leo Chao, Paul DesJarlais and John L Leonard, A binomial identity, via derangements, Math. Gaz. 89 (2005), 268-270.
Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.

Main diagonal of array in A068106 and of A047920.
Cf. A000142, A002119, A116854, A267383.
Column k=2 of A372326.

a033815 n = a116854 (2 * n + 1) (n + 1)
-- Reinhard Zumkeller, Aug 31 2014

A033815 := proc(n) local i; add(binomial(n, i)*(-1)^i*(2*n - i)!, i = 0 .. n) end;
# second Maple program:
A:= proc(n, k) A(n, k):= `if`(k=0, n!, A(n+1, k-1) -A(n, k-1)) end:
a:= n-> A(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 22 2019

a[n_] := (2n)!*Hypergeometric1F1[-n, -2n, -1]; Table[a[n], {n, 0, 14}] (* Jean-François Alcover, Jun 13 2012, after Vladimir Reshetnikov *)

a(n) = A002119(n)*n!*(-1)^n.
D-finite with recurrence a(n) = 2n*(2n-1)*a(n-1) + n*(n-1)*a(n-2).
a(n) = Sum_{i=0..n} binomial(n, i)*(-1)^i*(2*n-i)!.
From John L Leonard, Sep 15 2003: (Start)
a(n) = Sum_{i=0..n} C(n, i)*(2n-i)!*Sum_{j=0..2n-i} (-1)^j/j!.
a(n) = n!*Sum_{i=0..n} C(n, i)*n!/(n-i)!*Sum_{j=0..n-i} (-1)^j*C(n-i, j)*(n-j)!/i!. (End)
a(n) = Sum_{k=0..n} binomial(n,k)*A000166(n+k). - Vladeta Jovovic, Sep 04 2006
a(n) = A116854(2*n+1,n+1). - Reinhard Zumkeller, Aug 31 2014
a(n) = A267383(2n,n). - Alois P. Heinz, Jan 13 2016
a(n) ~ sqrt(Pi) * 2^(2*n + 1) * n^(2*n + 1/2) / exp(2*n + 1/2). - Vaclav Kotesovec, Feb 18 2017
a(n) = n!*exp(-1/2)*((-1)^n * BesselI(n+1/2,1/2)*Pi^(1/2) + BesselK(n+1/2,1/2)/Pi^(1/2) ). - Mark van Hoeij, Jul 15 2022

A087208 Expansion of e.g.f. exp(x)/(1-x^2).

Original entry on oeis.org

1, 1, 3, 7, 37, 141, 1111, 5923, 62217, 426457, 5599531, 46910271, 739138093, 7318002277, 134523132927, 1536780478171, 32285551902481, 418004290062513, 9879378882159187, 142957467201379447, 3754163975220491061, 60042136224579367741, 1734423756551866870183

Offset: 0

Vladeta Jovovic, Oct 19 2003

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A000166, A000522, A002747, A010050.

With[{nn=20},CoefficientList[Series[Exp[x]/(1-x^2),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 11 2017 *)

a(n) = Sum_{k=0..floor(n/2)} n!/(n-2*k)!.
a(n) = n*(n-1)*a(n-2) + 1. - Vladeta Jovovic, Aug 24 2004
a(n) = (A000522(n) + (-1)^n*A000166(n))/2. - Vladeta Jovovic, Aug 24 2004
a(n) = Sum_{k=0..n} binomial(n, k)*(1+(-1)^k)k!/2. Binomial transform of A010050 (with interpolated zeros). - Paul Barry, Sep 14 2004
a(n) = Sum_{k=0..n} P(n, k)[1, 0, 1, 0, 1, 0, ...](k). - Ross La Haye, Aug 29 2005
a(n) = (1/(2*exp(1))) * (Integral_{t=0..2} t^n*exp(1-abs(1-t)) dt + Integral_{t=0..oo} ((2+t)^n + (-t)^n) * exp(-t) dt). - Groux Roland, Jan 15 2011
E.g.f.: 1/U(0) where U(k) = 1 - x^2/(1 - 1/(1 + x*(k+1)/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Oct 16 2012
If n is even then a(n) ~ n!*(e/2 + 1/(2*e)) = 1.543080634815243... * n!, if n is odd then a(n) ~ n!*(e/2 - 1/(2*e)) = 1.175201193643801... * n!. - Vaclav Kotesovec, Nov 20 2012
Conjecture: a(n) -a(n-1) -n*(n-1)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 29 2013
From Peter Bala, Sep 05 2022: (Start)
The e.g.f. A(x) satisfies the differential equation (x^2 - 1)*A'(x) + (1 + 2*x - x^2)*A(x) = 0 with A(0) = 1. Mathar's recurrence above follows from this.
For k a positive integer, reducing the sequence modulo k produces a purely periodic sequence whose period divides k. For example, modulo 5 the sequence becomes [1, 1, 3, 2, 2, 1, 1, 3, 2, 2, ...] of period 5. (End)

Definition clarified by Harvey P. Dale, Aug 11 2017

A103816 Numerator of Sum_{k=1..n} (-1)^(k+1)/k!.

Original entry on oeis.org

0, 1, 1, 2, 5, 19, 91, 177, 3641, 28673, 28319, 2523223, 27526069, 109339663, 4239014627, 59043418019, 26718637649, 14052333488521, 238063061452591, 158218865944829, 7358312808534631, 124213980448686521, 11277840764547411113, 67527236643922308689

Offset: 0

N. J. A. Sloane, Apr 02 2005

Numerator of (n! - A000166(n))/n!.

Numerator of 1 - A053557/A053556.

			0, 1, 1/2, 2/3, 5/8, 19/30, 91/144, 177/280, 3641/5760, 28673/45360, 28319/44800, ...

Alois P. Heinz, Table of n, a(n) for n = 0..250
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Cf. A053556 (denominators).

b:= proc(n) b(n):=`if`(n<2, 1-n, (n-1)*(b(n-1)+b(n-2))) end:
a:= n-> numer((n!-b(n))/n!):
seq(a(n), n=0..30);  # Alois P. Heinz, May 15 2013

Table[Numerator[Sum[ -(-1)^k/k!, {k, n}]], {n, 0, 22}] (* Robert G. Wilson v *)
Table[Numerator[1 - Subfactorial[n]/n!], {n, 0, 23}] (* Jean-François Alcover, Feb 11 2014 *)
Join[{0},Accumulate[Times@@@Partition[Riffle[1/Range[30]!,{1,-1},{2,-1,2}],2]]//Numerator] (* Harvey P. Dale, Apr 18 2023 *)

from math import factorial
from fractions import Fraction
def A103816(n): return sum(Fraction(1 if k&1 else -1,factorial(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, Jul 31 2023

The Aitken delta-squared process leaves the sequence S(n) = Sum_{k=1..n} (-1)^(k+1)/k! essentially unchanged: S(n+3) = (S(n)*S(n+2) - (S(n+1))^2)/(S(n) + S(n+2) - 2*S(n+1)).

Numerators of coefficients in expansion of (1 - exp(-x)) / (1 - x). - Ilya Gutkovskiy, May 24 2022

More terms from Robert G. Wilson v, Oct 13 2005

A124625 Even numbers sandwiched between 1's.

Original entry on oeis.org

1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 1, 10, 1, 12, 1, 14, 1, 16, 1, 18, 1, 20, 1, 22, 1, 24, 1, 26, 1, 28, 1, 30, 1, 32, 1, 34, 1, 36, 1, 38, 1, 40, 1, 42, 1, 44, 1, 46, 1, 48, 1, 50, 1, 52, 1, 54, 1, 56, 1, 58, 1, 60, 1, 62, 1, 64, 1, 66, 1, 68, 1, 70, 1, 72, 1, 74, 1, 76, 1, 78, 1, 80, 1, 82, 1, 84

Offset: 0

N. J. A. Sloane, Jun 13 2007

Interleaving of A000012 and A005843.
Created to simplify the definition of A129952.
a(n) = abs(A009531(n-1)).
Starting (1, 2, 1, 4,...): square (1 + x - x^2 - x^3 + x^4 + x^5 - ...) = (1 + 2x - x^2 - 4x^3 + x^4 + 6x^5 - ...).
With a(3) taken as 0, a(n+2) = n^k+1 mod 2*n, n>=1, for any k>=2, also for k=n. - Wolfdieter Lang, Dec 21 2011
Also !(n+2) mod n for n>0 where !n is a subfactorial number (A000166). - Michel Lagneau, Sep 05 2012
Greatest common divisor of n-1 and (n-1) mod 2. - Bruno Berselli, Mar 07 2017

Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A000012 (all 1's), A005843 (even numbers), A009531, A093178, A152271.

Magma
```
&cat[[1, 2*k]: k in [0..42]];
```

A124625:=n->(n-(n-2)*(-1)^n)/2; seq(A124625(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Join[{1},Riffle[2Range[0,50],1]] (* Harvey P. Dale, Nov 02 2011 *)

{for(n=0, 85, print1(if(n%2>0, n-1, 1), ","))}

print([(n-1)**(n%2) for n in range(0, 86)]) # Karl V. Keller, Jr., Jul 26 2020

a(n) = 1 for even n, a(n) = n-1 for odd n.
a(2*k) = 1, a(2*k+1) = 2*k.
G.f.: (1 - x^2 + 2*x^3)/((1 - x)^2*(1 + x)^2).
a(n) = (n - (n - 2)*(-1)^n)/2. - Bruno Berselli, May 06 2011
E.g.f.: 1 + x^2*U(0)/2 where U(k) = 1 + 2*x*(k+1)/(2*k + 3 - x*(2*k+3)/(x + 4*(k+2)*(k+1)/U(k+1))) (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Oct 20 2012
a(n) = 2*floor(n/2) - (n-1)*((n-1) mod 2). - Wesley Ivan Hurt, Oct 19 2013
a(n) = (n-1)^((1-(-1)^n)/2). - Wesley Ivan Hurt, Mar 21 2015
a(n) = (n-1) - a(a(n-1))*a(n-1), a(0) = 0. - Eli Jaffe, Jun 07 2016
E.g.f.: (x + 1)*cosh(x) - sinh(x). - Ilya Gutkovskiy, Jun 07 2016
a(n) = (-1)^n mod n for n > 0. - Franz Vrabec, Mar 06 2020
a(n) = (n-1)^(n mod 2). - Karl V. Keller, Jr., Aug 01 2020

More terms from Klaus Brockhaus, Jun 16 2007

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

A145877 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 4, 0, 2, 15, 3, 0, 6, 76, 20, 0, 0, 24, 455, 105, 40, 0, 0, 120, 3186, 714, 420, 0, 0, 0, 720, 25487, 5845, 2688, 1260, 0, 0, 0, 5040, 229384, 52632, 22400, 18144, 0, 0, 0, 0, 40320, 2293839, 525105, 223200, 151200, 72576, 0, 0, 0, 0, 362880, 25232230

Offset: 1

Emeric Deutsch, Oct 27 2008

Row sums are the factorials (A000142).
Sum(T(n,k), k=2..n) = A000166(n) (the derangement numbers).
T(n,1) = A002467(n).
T(n,n) = (n-1)! (A000142).
Sum(k*T(n,k),k=1..n) = A028417(n).
For the statistic "length of the longest cycle", see A126074.

			T(4,2)=3 because we have 3412=(13)(24), 2143=(12)(34) and 4321=(14)(23).
Triangle starts:
      1;
      1,    1;
      4,    0,    2;
     15,    3,    0,    6;
     76,   20,    0,    0, 24;
    455,  105,   40,    0,  0, 120;
   3186,  714,  420,    0,  0,   0, 720;
  25487, 5845, 2688, 1260,  0,   0,   0, 5040;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Cf. A000142, A000166, A002467, A028417, A126074.

T(2n,n) gives A110468(n-1) (for n>0). - Alois P. Heinz, Apr 21 2017

F:=proc(k) options operator, arrow: (1-exp(-x^k/k))*exp(-(sum(x^j/j, j = 1 .. k-1)))/(1-x) end proc: for k to 16 do g[k]:= series(F(k),x=0,16) end do: T:= proc(n,k) options operator, arrow: factorial(n)*coeff(g[k],x,n) end proc: for n to 11 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

Rest[Transpose[Table[Range[0, 16]! CoefficientList[
      Series[(Exp[x^n/n] -1) (Exp[-Sum[x^k/k, {k, 1, n}]]/(1 - x)), {x, 0, 16}],x], {n, 1, 8}]]] // Grid (* Geoffrey Critzer, Mar 04 2011 *)

E.g.f. for column k is (1-exp(-x^k/k))*exp( -sum(j=1..k-1, x^j/j ) ) / (1-x). - Vladeta Jovovic

A177251 Number of permutations of [n] having no adjacent 3-cycles, i.e., no cycles of the form (i, i+1, i+2).

Original entry on oeis.org

1, 1, 2, 5, 22, 114, 697, 4923, 39612, 357899, 3588836, 39556420, 475392841, 6187284605, 86701097310, 1301467245329, 20835850494474, 354382860600678, 6381494425302865, 121290065781743383, 2426510081356069016, 50969474697328055063, 1121571023472780698152

Offset: 0

Emeric Deutsch, May 07 2010

nonn

			a(4)=22 because the only permutations of {1,2,3,4} having adjacent 3-cycles are (123)(4) and (1)(234).

Seiichi Manyama, Table of n, a(n) for n = 0..449
R. A. Brualdi and Emeric Deutsch, Adjacent q-cycles in permutations, arXiv:1005.0781 [math.CO], 2010.

Cf. A000166, A008290, A177248, A177249, A177250, A177252, A177253.

Cf. A370324, A370569.

A177251:= func< n | (&+[(-1)^j*Factorial(n-2*j)/Factorial(j): j in [0..Floor(n/3)]]) >;
[A177251(n): n in [0..30]]; // G. C. Greubel, Apr 28 2024

a := proc (n) options operator, arrow: sum((-1)^j*factorial(n-2*j)/factorial(j), j = 0 .. floor((1/3)*n)) end proc: seq(a(n), n = 0 .. 22);

a[n_] := Sum[(-1)^j*(n - 2*j)!/j!, {j, 0, n/3}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 17 2017 *)

def A177251(n): return sum((-1)^j*factorial(n-2*j)/factorial(j) for j in range(1+n//3))
[A177251(n) for n in range(31)] # G. C. Greubel, Apr 28 2024

a(n) = Sum_{j=0..floor(n/3)} (-1)^j*(n-2*j)!/j!.
a(n) = A177250(n,0).
a(n) - n*a(n-1) = 2*a(n-3) + 3*(-1)^(n/3) if 3 | n, otherwise a(n) - n*a(n-1) = 2*a(n-3).
lim_{n -> oo} a(n)/n! = 1.
The o.g.f. g(z) satisfies z^2*(1+z^3)*g'(z) - (1+z^3)(1-z-2z^3)g(z) + 1 - 2z^3 = 0; g(0)=1.
G.f.: hypergeometric2F0([1,1], [], x/(1+x^3))/(1+x^3). - Mark van Hoeij, Nov 08 2011
D-finite with recurrence a(n) = n*a(n-1) + a(n-3) + (n-3)*a(n-4) + 2*a(n-6). - R. J. Mathar, Jul 26 2022
G.f.: Sum_{k>=0} k! * x^k / (1+x^3)^(k+1). - Seiichi Manyama, Feb 20 2024

Crossreferences corrected by Emeric Deutsch, May 09 2010

A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 36, 42, 50, 60, 70, 82, 96, 110, 126, 144, 163, 184, 208, 234, 264, 298, 336, 380, 430, 486, 550, 622, 702, 792, 892, 1002, 1125, 1260, 1408, 1572, 1752, 1950, 2168, 2408, 2672

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
  (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
  (821)  (543)  (643)   (653)   (753)    (763)    (863)
         (921)  (A21)   (743)   (843)    (853)    (953)
                (5431)  (B21)   (C21)    (943)    (A43)
                        (5432)  (6432)   (D21)    (E21)
                        (6431)  (6531)   (6532)   (7532)
                                (7431)   (7432)   (7631)
                                (54321)  (7531)   (8432)
                                         (8431)   (8531)
                                         (64321)  (9431)
                                                  (65321)
                                                  (74321)

The non-strict version is A001522 (unproved, ranked by A352827 or A352874).
The version for permutations is A002467, complement A000166.
The reverse version is A096765 (or A025147 shifted right once).
The non-strict reverse version is A238395, ranked by A352872.
The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A352875, complement A238351.
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]>0&]],{n,0,30}]

G.f.: Sum_{n>=1} q^(n*(3*n-1)/2)*Product_{k=1..n-1} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

A007060 Number of ways n married couples can sit in a row without any spouses next to each other.

Original entry on oeis.org

1, 0, 8, 240, 13824, 1263360, 168422400, 30865121280, 7445355724800, 2287168006717440, 871804170613555200, 403779880746418176000, 223346806774106790297600, 145427383048755178635264000, 110105698060190464791596236800, 95914116314126658718742347776000, 95252504853751428295192341381120000

Offset: 0

David Roberts Keeney (David.Roberts.Keeney(AT)directory.Reed.edu)

Limit_{n->oo} a(n)/(2n)! = 1/e.
Also the number of (directed) Hamiltonian paths of the n-cocktail party graph. - Eric W. Weisstein, Dec 16 2013
Also the number of ways to label the cells of a 2 X n grid such that no vertically adjacent cells have adjacent labels. - Sela Fried, May 29 2023

			For n = 2, the a(2) = 8 solutions for the couples {1,2} and {3,4} are {1324, 1423, 2314, 2413, 3142, 3241, 4132, 4231}.

Andrew Woods, Table of n, a(n) for n = 0..100
G. Almkvist, Letter to N. J. A. Sloane, Apr. 1992.
Eric Weisstein's World of Mathematics, Cocktail Party Graph.
Eric Weisstein's World of Mathematics, Hamiltonian Path.

Cf. A000166, A000806, A114938, A177840, A053983, A053984, A173841, A193624, A229430, A330266.

seq(add((-1)^i*binomial(n, i)*2^i*(2*n-i)!, i=0..n),n=0..20);

Table[Sum[(-1)^i Binomial[n,i] (2 n - i)! 2^i, {i, 0, n}], {n, 0, 20}]
Table[(2 n)! Hypergeometric1F1[-n, -2 n, -2], {n, 0, 20}]

a(n)=sum(k=0, n, binomial(n, k)*(-1)^(n-k)*(n+k)!*2^(n-k)) \\ Charles R Greathouse IV, May 11 2016

from sympy import binomial, subfactorial
def a(n): return sum([(-1)**(n - k)*binomial(n, k)*subfactorial(2*k) for k in range(n + 1)]) # Indranil Ghosh, Apr 28 2017

a(n) = (Pi*BesselI(n+1/2,1)*(-1)^n+BesselK(n+1/2,1))*exp(-1)*(2/Pi)^(1/2)*2^n*n!. - Mark van Hoeij, Nov 12 2009
a(n) = (-1)^n*2^n*n!*A000806(n), n>0. - Vladeta Jovovic, Nov 19 2009
a(n) = n!*hypergeom([-n, n+1],[],1/2)*(-2)^n. - Mark van Hoeij, Nov 13 2009
a(n) = 2^n * A114938(n). - Toby Gottfried, Nov 22 2010
a(n) = 2*n((2*n-1)*a(n-1) + (2*n-2)*a(n-2)), n > 1. - Aaron Meyerowitz, May 14 2014
From Peter Bala, Mar 06 2015: (Start)
a(n) = Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)*A000166(2*k).
For n >= 1, Integral_{x = 0..1} (x^2 - 1)^n*exp(x) dx = a(n)*e - A177840(n). Hence lim_{n->oo} A177840(n)/a(n) = e. (End)
a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n + 1/2) / exp(2*n+1). - Vaclav Kotesovec, Mar 09 2016
a(n) = A173841(2n). - David Radcliffe, Sep 09 2025

More terms from Michel ten Voorde, Apr 11 2001

cp's OEIS Frontend

A165961 Number of circular permutations of length n without 3-sequences.

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A002628 Number of permutations of length n without 3-sequences.

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A033815 Number of standard permutations of [ a_1..a_n b_1..b_n ] (b_i is not immediately followed by a_i, for all i).

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A087208 Expansion of e.g.f. exp(x)/(1-x^2).

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A103816 Numerator of Sum_{k=1..n} (-1)^(k+1)/k!.

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A124625 Even numbers sandwiched between 1's.

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