A060725 -id:A060725 - cp's OEIS frontend

A000090 Expansion of e.g.f. exp((-x^3)/3)/(1-x).

1, 1, 2, 4, 16, 80, 520, 3640, 29120, 259840, 2598400, 28582400, 343235200, 4462057600, 62468806400, 936987251200, 14991796019200, 254860532326400, 4587501779660800, 87162533813555200, 1743250676271104000, 36608259566534656000, 805381710463762432000

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 3-cycle.

			a(3) = 4 because the permutations in S_3 that contain no 3-cycles are the trivial permutation and the 3 transpositions.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
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).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Christian G. Bower, Table of n, a(n) for n = 0..100
Simon Plouffe, Exact formulas for integer sequences
L. W. Shapiro & N. J. A. Sloane, Correspondence, 1976

Cf. A000142, A000138, A000266, A060725.

seq(coeff(convert(series(exp((-x^3)/3)/(1-x),x,50),polynom),x,i)*i!,i=0..30);# series expansion A000090:=n->n!*add((-1)^i/(i!*3^i),i=0..floor(n/3));seq(A000090(n),n=0..30); # formula (Pab Ter)

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x^3/3]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 28 2012 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^3 / 3) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */

a(n) = n! * Sum_{i=0..floor(n/3)} (-1)^i / (i! * 3^i); a(n)/n! ~ Sum_{i >= 0} (-1)^i / (i! * 3^i) = e^(-1/3); a(n) ~ e^(-1/3) * n!; a(n) ~ e^(-1/3) * (n/e)^n * sqrt(2 * Pi * n). - Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), here k=3, n>=0. - Simon Plouffe from old notes, 1993
E.g.f.: E(x) = exp(-x^3/3)/(1-x)=G(0)/((1-x)^2); G(k) = 1 - x/(1 - x^2/(x^2 + 3*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 11 2012

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

Entry improved by comments from Michael Somos, Jul 28 2009

A000138 Expansion of e.g.f. exp(-x^4/4)/(1-x).

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 540, 3780, 31500, 283500, 2835000, 31185000, 372972600, 4848643800, 67881013200, 1018215198000, 16294848570000, 277012425690000, 4986223662420000, 94738249585980000, 1894745192712372000, 39789649046959812000, 875372279033115864000

Offset: 0

N. J. A. Sloane

a(n) is the number of permutations in the symmetric group S_n whose cycle decomposition contains no 4-cycle.

			a(4) = 18 because in S_4 the permutations with no 4-cycle are the complement of the six 4-cycles so a(4) = 4! - 6 = 18.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 85.
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).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

T. D. Noe, Table of n, a(n) for n = 0..100
Simon Plouffe, Exact formulas for integer sequences

Cf. A000142, A000090, A000266, A060725, A060726, A060727.

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x^4/4]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 28 2012 *)

{a(n) = if( n<0, 0, n! * polcoeff( exp( -(x^4/4) + x*O(x^n)) / (1 - x), n))} /* Michael Somos, Jul 28 2009 */

a(n) = n! * Sum_{i=0..floor(n/4)} (-1)^i / (i! * 4^i); a(n)/n! ~ Sum_{i >= 0} (-1)^i / (i! * 4^i) = e^(-1/4); a(n) ~ e^(-1/4) * n!; a(n) ~ e^(-1/4) * (n/e)^n * sqrt(2*Pi*n). - Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001
a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), here k=4, n>=0. Simon Plouffe, from old notes, 1993
E.g.f.: exp(-x^4/4)/(1-x) = 1/G(0); G(k) = 1 - x/(1 - (x^3)/(x^3 - 4*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Feb 28 2012

Entry improved by comments from Michael Somos, Jul 28 2009

Name corrected by Joerg Arndt, May 27 2011

A060726 For n >= 1, a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 600, 4200, 33600, 302400, 3024000, 33264000, 405820800, 5275670400, 73859385600, 1107890784000, 17726252544000, 301346293248000, 5419293175296000, 102966570330624000, 2059331406612480000

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

nonn

This is the expansion of exp ((-x^6)/6) /(1-x).

			a(6) = 600 because in S_6 the permutations with no 6-cycle are the complement of the 120 6-cycles so a(6) = 6! - 120 = 600.

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Harry J. Smith, Table of n, a(n) for n = 0..100
Plouffe, Simon, Exact formulas for integer sequences

Cf. A000142, A000266, A000090, A000138, A060725, A060726, A060727.

for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 6^i)), i=0..floor(n/6))) od:

a(n)={n! * sum(i=0, n\6, (-1)^i / (i! * 6^i))} \\ Harry J. Smith, Jul 10 2009

The formula for a(n) is: a(n) = n! * Sum_{i=0..floor(n/6)} ((-1)^i /(i! * 6^i)) by this formula we have as n -> infinity: a(n)/n! ~ Sum_{i>= 0} (-1)^i /(i! * 6^i) = e^(-1/6) or a(n) ~ e^(-1/6) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/6) * (n/e)^n * sqrt(2 * Pi * n)

a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), k=6, n >= 0. - Simon Plouffe, Feb 18 2011

More terms from James Sellers, Apr 24 2001

A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 15, 6, 3, 75, 30, 15, 435, 225, 45, 15, 3045, 1575, 315, 105, 24465, 12180, 3150, 420, 105, 220185, 109620, 28350, 3780, 945, 2200905, 1100925, 274050, 47250, 4725, 945, 24209955, 12110175, 3014550, 519750, 51975, 10395, 290529855

Offset: 0

Vladeta Jovovic, Feb 05 2006

Row n has 1+floor(n/2) terms. Row sums yield the factorials (A000142). Sum(k*T(n,k),k>0)=n!/2 for n>=2. - Emeric Deutsch, Feb 17 2006

			T(3,1) = 3 because we have (1)(23), (12)(3) and (13)(2).
Triangle begins:
    1;
    1;
    1,   1;
    3,   3;
   15,   6,   3;
   75,  30,  15;
  435, 225,  45,  15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A008290, A000266, A000090, A088436, A000138, A060725, A060726, A086659, A105422, A105114.

G:= exp((y-1)*x^2/2)/(1-x): Gser:= simplify(series(G,x=0,15)): P[0]:=1: for n from 1 to 12 do P[n]:= n!*coeff(Gser,x^n) od: for n from 0 to 12 do seq(coeff(y*P[n], y^j), j=1..1+floor(n/2)) od;  # yields sequence in triangular form - Emeric Deutsch, Feb 17 2006

d = Exp[-x^2/2!]/(1 - x);f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Transpose[Table[Range[0, 10]!CoefficientList[Series[x^(2 k)/(2^k k!) d, {x, 0, 10}], x], {k, 0, 5}]]]]  (* Geoffrey Critzer, Nov 29 2011 *)

E.g.f.: exp((y-1)*x^2/2)/(1-x). More generally, e.g.f. for number of permutations of n elements with k m-cycles is exp((y-1)*x^m/m)/(1-x).

T(n,k) = n!/(2^k*k!) * Sum_{j=0..floor(n/2)-k} (-1/2)^j/j!. - Alois P. Heinz, Nov 30 2011

More terms from Emeric Deutsch, Feb 17 2006

A060727 For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 75613824000, 1134207360000, 18147317760000, 308504401920000, 5553079234560000, 105508505456640000, 2110170109132800000, 44288746761093120000, 974352428744048640000

Offset: 0

Avi Peretz (njk(AT)netvision.net.il), Apr 22 2001

nonn

This is the expansion of exp ((-x^7)/7)/(1-x).

			a(7) = 4320 because in S_7 the permutations with no 7-cycle are the complement of the 720 7-cycles so a(7) = 7! - 720 = 4320.

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, page 93, problem 7.

Harry J. Smith, Table of n, a(n) for n = 0..100

Cf. A000142, A000090, A000266, A000138, A060725, A060726, A060727.

for n from 0 to 30 do printf(`%d,`, n! * sum(( (-1)^i /(i! * 7^i)), i=0..floor(n/7))) od:

{ for (n=0, 100, write("b060727.txt", n, " ", n! * sum(i=0, n\7, (-1)^i / (i! * 7^i))); ) } \\ Harry J. Smith, Jul 10 2009

The formula for a(n) is: a(n) = n! * sum i=0 ... [ n/7 ]( (-1)^i /(i! * 7^i)) by this formula we have as n -> infinity: a(n)/n! ~ sum i >= 0 (-1)^i /(i! * 7^i) = e^(-1/7) or a(n) ~ e^(-1/7) * n! and using Stirling's formula in A000142: a(n) ~ e^(-1/7) * (n/e)^n * sqrt(2 * Pi * n)

a(n,k) = n!*floor(floor(n/k)!*k^floor(n/k)/exp(1/k) + 1/2)/(floor(n/k)!*k^floor(n/k)), k=7, n>=0. - Simon Plouffe, Feb 18 2011

More terms from James Sellers, Apr 24 2001

A122974 Triangle T(n,k), the number of permutations on n elements that have no cycles of length k.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 9, 15, 16, 18, 44, 75, 80, 90, 96, 265, 435, 520, 540, 576, 600, 1854, 3045, 3640, 3780, 4032, 4200, 4320, 14833, 24465, 29120, 31500, 32256, 33600, 34560, 35280, 133496, 220185, 259840, 283500, 290304, 302400, 311040, 317520, 322560

Offset: 1

Dennis P. Walsh, Oct 27 2006

Read as sequence, a(n) is the number of permutations on j elements with no cycles of length i where j=round((2*n)^.5) and i=n-C(j,2).

T(n,k) generalizes several sequences already in the On-Line Encyclopedia, such as A000166, the number of permutations on n elements with no fixed points and A000266, the number of permutations on n elements with no transpositions (i.e., no 2-cycles). See the cross references for further examples.

			T(3,2)=3 since there are exactly 3 permutations of 1,2,3 that have no cycles of length 2, namely, (1)(2)(3),(1 2 3) and (2 1 3).
Triangle T(n,k) begins:
      0;
      1,     1;
      2,     3,     4;
      9,    15,    16,    18;
     44,    75,    80,    90,    96;
    265,   435,   520,   540,   576,   600;
   1854,  3045,  3640,  3780,  4032,  4200,  4320;
  14833, 24465, 29120, 31500, 32256, 33600, 34560, 35280;
  ...

Alois P. Heinz, Rows for n = 1..141, flattened
Dennis P. Walsh, The Number of Permutations with No k-Cycles.

Cf. T(n, 1)=A000166 for n=>1 T(n, 2)=A000266 for n=>2 T(n, 3)=A000090 for n=>3 T(n, 4)=A000138 for n=>4 T(n, 5)=A060725 for n=>5 T(n, 6)=A060726 for n=>6 T(n, 7)=A060727 for n=>7.

T(n,n) gives A094304(n+1).

seq((round((2*n)^.5))!*sum((-1/(n-binomial(round((2*n)^.5),2)))^r/r!,r=0..floor(round((2*n)^.5)/(n-binomial(round((2*n)^.5),2)))),n=1..66);
# second Maple program:
T:= proc(n, k) option remember; `if`(n=0, 1, add(`if`(j=k, 0,
      T(n-j, k)*binomial(n-1, j-1)*(j-1)!), j=1..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Nov 24 2019

T[n_, k_] := T[n, k] = If[n==0, 1, Sum[If[j==k, 0, T[n - j, k] Binomial[n - 1, j - 1] (j - 1)!], {j, 1, n}]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 08 2019, after Alois P. Heinz *)

T(n,k)=n!*sum r=0..floor(n/k)((-1/k)^r/r!) E.G.F: exp(-x^k/k)/(1-x) a(n)=(round((2*n)^.5))!*sum((-1/(n-binomial(round((2*n)^.5),2)))^r/r!,r=0..floor(round((2*n)^.5)/(n-binomial(round((2*n)^.5),2)))).

T(n,k) = n! - A293211(n,k). - Alois P. Heinz, Nov 24 2019

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A000090 Expansion of e.g.f. exp((-x^3)/3)/(1-x).

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A000138 Expansion of e.g.f. exp(-x^4/4)/(1-x).

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A060726 For n >= 1, a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 6-cycle.

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A114320 Triangle T(n,k) = number of permutations of n elements with k 2-cycles.

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A060727 For n >= 1 a(n) is the number of permutations in the symmetric group S_n such that their cycle decomposition contains no 7-cycle.

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A122974 Triangle T(n,k), the number of permutations on n elements that have no cycles of length k.

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