A053505 -id:A053505 - cp's OEIS frontend

A052501 Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5.

1, 1, 1, 1, 1, 25, 145, 505, 1345, 3025, 78625, 809425, 4809025, 20787625, 72696625, 1961583625, 28478346625, 238536558625, 1425925698625, 6764765838625, 189239120970625, 3500701266525625, 37764092547420625, 288099608198025625

Offset: 0

N. J. A. Sloane, Jan 15 2000; encyclopedia(AT)pommard.inria.fr, Jan 25 2000

The number of degree-n permutations of order exactly p (where p is prime) satisfies a(n) = a(n-1) + (1 + a(n-p))*(n-1)!/(n-p)! with a(n)=0 if p>n. Also a(n) = Sum_{j=1..floor(n/p)} (n!/(j!*(n-p*j)!*(p^j))).

These are the telephone numbers T^(5)n of [Artioli et al., p. 7]. - _Eric M. Schmidt, Oct 12 2017

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, Simonetta Pagnutti, Motzkin Numbers: an Operational Point of View, arXiv:1703.07262 [math.CO], 2017.
Tomislav Došlic, Darko Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019). - _N. J. A. Sloane_, May 01 2012
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 26
M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388.

Column k=5 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^5/5) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019

spec := [S,{S=Set(Union(Cycle(Z,card=1),Cycle(Z,card=5)))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

max = 30; CoefficientList[ Series[ Exp[x + x^5/5], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Feb 15 2012, after e.g.f. *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x + x^5/5) )) \\ G. C. Greubel, May 14 2019

m = 30; T = taylor(exp(x + x^5/5), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^5/5).
a(n+5) = a(n+4) + (24 +50*n +35*n^2 +10*n^3 +n^4)*a(n), with a(0)= ... = a(4) = 1.
a(n) = a(n-1) + a(n-5)*(n-1)!/(n-5)!.
a(n) = Sum_{j = 0..floor(n/5)} n!/(5^j * j! * (n-5*j)!).
a(n) = A059593(n) + 1.

A001471 Number of degree-n permutations of order exactly 3.

Original entry on oeis.org

0, 0, 0, 2, 8, 20, 80, 350, 1232, 5768, 31040, 142010, 776600, 4874012, 27027728, 168369110, 1191911840, 7678566800, 53474964992, 418199988338, 3044269834280, 23364756531620, 199008751634000, 1605461415071822

Offset: 0

N. J. A. Sloane and J. H. Conway

a(n) is the number of non-symmetric permutation matrices A of dimension n such that A^2 is the transpose of A. - Torlach Rush, Jul 09 2020

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).

T. D. Noe, Table of n, a(n) for n = 0..100
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Column k=3 of A057731.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^3/3) )); [Factorial(n-1)*b[n]-1: n in [1..m]]; // G. C. Greubel, May 14 2019

a[n_] := HypergeometricPFQ[{1/3-n/3, 2/3-n/3, -n/3}, {}, -9] - 1; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Oct 19 2011 *)
nxt[{n_,a_,b_,c_}]:={n+1,b,c,c+(1+a)(n-1)(n-2)}; NestList[nxt,{3,0,0,0},25][[;;,2]] (* Harvey P. Dale, Mar 09 2024 *)

a(n)=sum(j=1,n\3, n!/(j!*(n-3*j)!*(3^j))) \\ Charles R Greathouse IV, Jun 21 2017

first(n)=my(v=vector(n+1)); for(i=3,n, v[i+1]=v[i] + (1+v[i-2])*(i-1)*(i-2)); v \\ Charles R Greathouse IV, Jul 10 2020

m = 30; T = taylor(exp(x + x^3/3) -exp(x), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

From Henry Bottomley, Jan 26 2001: (Start)
a(n) = a(n-1) + (1 + a(n-3))*(n-1)(n-2).
a(n) = Sum_{j=1..floor(n/3)} n!/(j!*(n-3*j)!*(3^j)).
a(n) = A001470(n) - 1. (End)
E.g.f.: exp(x + x^3/3) - exp(x).

A053496 Number of degree-n permutations of order dividing 6.

Original entry on oeis.org

1, 1, 2, 6, 18, 66, 396, 2052, 12636, 91548, 625176, 4673736, 43575192, 377205336, 3624289488, 38829340656, 397695226896, 4338579616272, 54018173703456, 641634784488288, 8208962893594656, 113809776294348576, 1526808627197721792, 21533423236302943296

Offset: 0

N. J. A. Sloane, Jan 15 2000

nonn

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A053495-A053505, A005388, A261317.

Column k=6 of A008307.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^3/3 +x^6/6) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, May 14 2019

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*a(n-j), j=[1, 2, 3, 6])))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

a[n_] := a[n] = If[n<0, 0, If[n == 0, 1, Sum[Product[n-i, {i, 1, j-1}]*a[n-j], {j, {1, 2, 3, 6}}]]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 03 2014, after Alois P. Heinz *)
With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^3/3 +x^6/6], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 14 2019 *)

my(x='x+O('x^30)); Vec(serlaplace( exp(x+x^2/2+x^3/3+x^6/6) )) \\ G. C. Greubel, May 14 2019

m = 30; T = taylor(exp(x +x^2/2 +x^3/3 +x^6/6), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # G. C. Greubel, May 14 2019

E.g.f.: exp(x +x^2/2 +x^3/3 +x^6/6).

D-finite with recurrence a(n) -a(n-1) +(-n+1)*a(n-2) -(n-1)*(n-2)*a(n-3) -(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)=0. - R. J. Mathar, Jul 04 2023

A001473 Number of degree-n permutations of order exactly 4.

Original entry on oeis.org

0, 0, 0, 6, 30, 180, 840, 5460, 30996, 209160, 1290960, 9753480, 69618120, 571627056, 4443697440, 40027718640, 346953934320, 3369416698080, 31421601510336, 328430320909920, 3331475969159520, 37124416523261760

Offset: 1

N. J. A. Sloane and J. H. Conway

nonn

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).

G. C. Greubel, Table of n, a(n) for n = 1..565
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Column k=4 of A057731.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x + x^2/2 +x^4/4) -Exp(x+x^2/2) )); [0,0,0] cat [Factorial(n+3)*b[n]: n in [1..m-4]]; // G. C. Greubel, May 14 2019

Rest@With[{m = 30}, CoefficientList[Series[Exp[x +x^2/2 +x^4/4] - Exp[x +x^2/2], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, May 14 2019 *)

my(x=xx+O(xx^33)); concat([0,0,0], Vec(serlaplace(-exp(x+1/2*x^2) +exp(x+1/2*x^2+1/4*x^4)))) \\ Michel Marcus, Dec 12 2014

m = 30; T = taylor(exp(x +x^2/2 +x^4/4) - exp(x+x^2/2), x, 0, m); a=[factorial(n)*T.coefficient(x, n) for n in (0..m)]; a[1:] # G. C. Greubel, May 14 2019

E.g.f.: exp(x + x^2/2 + x^4/4) - exp(x + x^2/2).

More terms from Vladeta Jovovic, Apr 14 2001

A061121 Number of degree-n permutations of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 20, 240, 1470, 10640, 83160, 584640, 4496030, 42658440, 371762820, 3594871280, 38650622010, 396457108320, 4330689250160, 53963701424640, 641211774798510, 8205894865096280, 113786291585124060

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

nn=21;Range[0,nn]!CoefficientList[Series[(Exp[x^6/6]-1)Exp[x+x^2/2+x^3/3]+(Exp[x^2/2]-1)(Exp[x^3/3]-1)Exp[x],{x,0,nn}],x]//Rest  (* Geoffrey Critzer, Feb 04 2013 *)

E.g.f.: exp(x)-exp(x+1/2*x^2)-exp(x+1/3*x^3)+exp(x+1/2*x^2+1/3*x^3+1/6*x^6).

A061128 Number of degree-n permutations of order exactly 30.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 120960, 2661120, 27941760, 536215680, 6901614720, 90084234240, 1540714855680, 33110649411840, 554845922991360, 8393918663370240, 141081442901118720, 2869353360741853440

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

E.g.f.: - exp(x) + exp(x + 1/2*x^2) + exp(x + 1/3*x^3) + exp(x + 1/5*x^5) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) - exp(x + 1/2*x^2 + 1/5*x^5 + 1/10*x^10) - exp(x + 1/3*x^3 + 1/5*x^5 + 1/15*x^15) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/5*x^5 + 1/6*x^6 + 1/10*x^10 + 1/15*x^15 + 1/30*x^30).

A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 10, 3, 2, 1, 1, 26, 9, 4, 1, 1, 1, 76, 21, 16, 1, 2, 1, 1, 232, 81, 56, 1, 6, 1, 1, 1, 764, 351, 256, 25, 18, 1, 2, 1, 1, 2620, 1233, 1072, 145, 66, 1, 4, 1, 1, 1, 9496, 5769, 6224, 505, 396, 1, 16, 3, 2, 1, 1, 35696, 31041, 33616, 1345, 2052, 1, 56, 9, 4, 1, 1

Offset: 1

N. J. A. Sloane

Solutions to x^k = 1 in Symm_n (the symmetric group of degree n).

			Array begins:
  1,   1,    1,    1,    1,     1,    1,     1, ...
  1,   2,    1,    2,    1,     2,    1,     2, ...
  1,   4,    3,    4,    1,     6,    1,     4, ...
  1,  10,    9,   16,    1,    18,    1,    16, ...
  1,  26,   21,   56,   25,    66,    1,    56, ...
  1,  76,   81,  256,  145,   396,    1,   256, ...
  1, 232,  351, 1072,  505,  2052,  721,  1072, ...
  1, 764, 1233, 6224, 1345, 12636, 5761, 11264, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
J. D. Dixon, B. Mortimer, Permutation Groups, Springer (1996), Exercise 1.2.13.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
M. B. Kutler, C. R. Vinroot, On q-Analogs of Recursions for the Number of Involutions and Prime Order Elements in Symmetric Groups, JIS 13 (2010) #10.3.6, eq (5) for primes k.

Rows give A000034, A284517, A284518.
Columns give A000012, A000085, A001470, A001472, A052501, A053496, A053497, A053498, A053499, A053500, A053501, A053502, A053503, A053504, A053505.
Main diagonal gives A074759. - Alois P. Heinz, Feb 14 2013

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=numtheory[divisors](k))))
    end:
seq(seq(A(1+d-k, k), k=1..d), d=1..12); # Alois P. Heinz, Feb 14 2013
# alternative
A008307 := proc(n,m)
    local x,d ;
    add(x^d/d, d=numtheory[divisors](m)) ;
    exp(%) ;
    coeftayl(%,x=0,n) ;
    %*n! ;
end proc:
seq(seq(A008307(1+d-k,k),k=1..d),d=1..12) ; # R. J. Mathar, Apr 30 2017

t[n_ /; n >= 0, k_ /; k >= 0] := t[n, k] = Sum[(n!/(n - d + 1)!)*t[n - d, k], {d, Divisors[k]}]; t[, ] = 1; Flatten[ Table[ t[n - k, k], {n, 0, 12}, {k, 1, n}]] (* Jean-François Alcover, Dec 12 2011, after given formula *)

T(n+1,k) = Sum_{d|k} (n)_(d-1)*T(n-d+1,k), where (n)_i = n!/(n - i)! = n*(n - 1)*(n - 2)*...*(n - i + 1) is the falling factorial.

E.g.f. for n-th row: Sum_{n>=0} T(n,k)*t^n/n! = exp(Sum_{d|k} t^d/d).

More terms from Vladeta Jovovic, Apr 13 2001

A005388 Number of degree-n permutations of order a power of 2.

Original entry on oeis.org

1, 1, 2, 4, 16, 56, 256, 1072, 11264, 78976, 672256, 4653056, 49810432, 433429504, 4448608256, 39221579776, 1914926104576, 29475151020032, 501759779405824, 6238907914387456, 120652091860975616, 1751735807564578816, 29062253310781161472, 398033706586943258624

Offset: 0

N. J. A. Sloane and J. H. Conway

Differs from A053503 first at n=32. - Alois P. Heinz, Feb 14 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.2.10.

Alois P. Heinz, Table of n, a(n) for n = 0..200
J. M. Møller, Euler characteristics of equivariant subcategories, arXiv preprint arXiv:1502.01317, 2015. See page 20.
L. Moser and M. Wyman, On solutions of x^d = 1 in symmetric groups, Canad. J. Math., 7 (1955), 159-168.
A. Recski, Enumerating partitional matroids, Preprint.
A. Recski & N. J. A. Sloane, Correspondence, 1975

Cf. A000085, A001470, A001472, A053495, A053496, A053497, A053498, A053499.

Cf. A053500, A053501, A053502, A053503, A053504, A053505.

R:=PowerSeriesRing(Rationals(), 40);
f:= func< x | Exp( (&+[x^(2^j)/2^j: j in [0..14]]) ) >;
Coefficients(R!(Laplace( f(x) ))); // G. C. Greubel, Nov 17 2022

a:= proc(n) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..2^j-1)*a(n-2^j), j=0..ilog2(n))))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 14 2013

max = 23; CoefficientList[ Series[ Exp[ Sum[x^2^m/2^m, {m, 0, max}]], {x, 0, max}], x]*Range[0, max]! (* Jean-François Alcover, Sep 10 2013 *)

def f(x): return exp(sum(x^(2^j)/2^j for j in range(15)))
def A005388_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).egf_to_ogf().list()
A005388_list(40) # G. C. Greubel, Nov 17 2022

E.g.f.: exp(Sum_{m>=0} x^(2^m)/2^m).

E.g.f.: 1/Product_{k>=1} (1 - x^(2*k-1))^(mu(2*k-1)/(2*k-1)), where mu() is the Moebius function. - Seiichi Manyama, Jul 06 2024

A061132 Number of degree-n even permutations of order dividing 10.

Original entry on oeis.org

1, 1, 1, 1, 4, 40, 190, 610, 1660, 13420, 174700, 1326700, 30818800, 342140800, 2534931400, 16519411000, 143752426000, 4842417082000, 73620307162000, 687934401562000, 17165461784680000, 308493094924720000, 4585953613991980000, 53843602355379220000

Offset: 0

Vladeta Jovovic, Apr 14 2001

			For n=4 the a(4)=4 solutions are (1), (1, 2)(3, 4), (1, 3)(2, 4), (1, 4)(2, 3) (permutations in cyclic notation). - _Luis Manuel Rivera Martínez_, Jun 18 2019

J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).

Alois P. Heinz, Table of n, a(n) for n = 0..491
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128, A000704, A061129-A061132, A048099, A051695, A061133-A061135.

With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^5/5 + x^10/10] + 1/2 Exp[x - x^2/2 + x^5/5 - x^10/10], {x, 0, nn}], x]* Range[0, nn]!] (* Luis Manuel Rivera Martínez, Jun 18 2019 *)

my(x='x+O('x^25)); Vec(serlaplace(1/2*exp(x + 1/2*x^2 + 1/5*x^5 + 1/10*x^10) + 1/2*exp(x - 1/2*x^2 + 1/5*x^5 - 1/10*x^10))) \\ Michel Marcus, Jun 18 2019

E.g.f.: 1/2*exp(x + 1/2*x^2 + 1/5*x^5 + 1/10*x^10) + 1/2*exp(x - 1/2*x^2 + 1/5*x^5 - 1/10*x^10).

A061133 Number of degree-n even permutations of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 210, 5040, 37800, 201600, 2044350, 25530120, 213993780, 1692490800, 19767998250, 232823791200, 2235629476080, 23171222430720, 294649445112750, 4300403589581400, 55176842335916700, 660577269463243440

Offset: 1

Vladeta Jovovic, Apr 14 2001

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135.

E.g.f.: exp(x) - 1/2*exp(x + 1/2*x^2) - 1/2*exp(x - 1/2*x^2) - exp(x + 1/3*x^3) + 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) + 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).

cp's OEIS Frontend

A052501 Number of permutations sigma such that sigma^5=Id; degree-n permutations of order dividing 5.

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A001471 Number of degree-n permutations of order exactly 3.

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A053496 Number of degree-n permutations of order dividing 6.

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A001473 Number of degree-n permutations of order exactly 4.

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A061121 Number of degree-n permutations of order exactly 6.

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A061128 Number of degree-n permutations of order exactly 30.

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A008307 Table T(n,k) giving number of permutations of [1..n] with order dividing k, read by antidiagonals.

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