A005225 -id:A005225 - cp's OEIS frontend

A346057 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).

1, -1, -1, 2, 3, 14, -55, 62, -637, 338, -3861, 335312, -4499803, 43490108, -246353731, 2189950310, -47336985225, 1224524919590, -21516426400621, 346681988108648, -4499477383730851, 69294602646065900, -1418045089870455795, 45246859024830444566

Offset: 0

Seiichi Manyama, Jul 02 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..451

Cf. A000587, A005225, A330199, A346055, A346058.

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, exp(1-exp(x^k/k)))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=1, N, 1-exp(x^k/k)))))

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, 1/(d!*(k/d)^d))*x^k))))

a(n) = if(n==0, 1, -(n-1)!*sum(k=1, n, k*sumdiv(k, d, 1/(d!*(k/d)^d))*a(n-k)/(n-k)!));

E.g.f.: exp( Sum_{k>=1} (1 - exp(x^k/k)) ).
E.g.f.: exp( -Sum_{k>=1} A005225(k)*x^k/k! ).
a(n) = -(n-1)! * Sum_{k=1..n} k * (Sum_{d|k} 1/(d! * (k/d)^d)) * a(n-k)/(n-k)! for n > 0.

A364967 Number T(n,k) of permutations of [n] for which the difference between the longest and the shortest cycle length is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

Original entry on oeis.org

1, 1, 2, 3, 3, 10, 6, 8, 25, 45, 20, 30, 176, 60, 250, 90, 144, 721, 861, 770, 1344, 504, 840, 6406, 1778, 7980, 6300, 8736, 3360, 5760, 42561, 23283, 38808, 75348, 45360, 66240, 25920, 45360, 436402, 84150, 363680, 456120, 708048, 378000, 572400, 226800, 403200

Offset: 0

Alois P. Heinz, Aug 14 2023

T(0,0) = 1 by convention.

			T(4,0) = 10: (1)(2)(3)(4), (12)(34), (13)(24), (14)(23), (1234), (1243), (1324), (1342), (1423), (1432).
T(4,1) = 6: (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4), (14)(2)(3).
T(4,2) = 8: (1)(234), (1)(243), (123)(4), (132)(4), (124)(3), (142)(3), (134)(2), (143)(2).
Triangle T(n,k) begins:
      1;
      1;
      2;
      3,     3;
     10,     6,     8;
     25,    45,    20,    30;
    176,    60,   250,    90,   144;
    721,   861,   770,  1344,   504,   840;
   6406,  1778,  7980,  6300,  8736,  3360,  5760;
  42561, 23283, 38808, 75348, 45360, 66240, 25920, 45360;
  ...

Alois P. Heinz, Rows n = 0..150, flattened
Wikipedia, Permutation

Row sums give A000142.
Column k=0 gives A005225 (for n>=1).
T(n+1,n-1) gives A001048(n) (for n>=1).
Cf. A126074, A145877, A364971, A365229.

b:= proc(n, l, m) option remember; `if`(n=0, x^(m-l), add(
     b(n-j, min(l, j), max(m, j))*binomial(n-1, j-1)*(j-1)!, j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=0..12);

b[n_, l_, m_] := b[n, l, m] = If[n == 0, x^(m - l), Sum[b[n - j, Min[l, j], Max[m, j]]*Binomial[n - 1, j - 1]*(j - 1)!, {j, 1, n}]];
T[n_] := CoefficientList[b[n, n, 0], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 08 2023, after Alois P. Heinz *)

T(n,k) == 0 (mod k!).

Sum_{k=0..max(0,n-2)} T(n,k)/k! = A365229(n).

A087905 a(n) = n! * Sum_{d|n} (d/n)^d.

Original entry on oeis.org

1, 3, 8, 36, 144, 1010, 5760, 50400, 416640, 4250232, 43545600, 553106400, 6706022400, 95865541200, 1410695430144, 22720842144000, 376610217984000, 6888030445296000, 128047474114560000, 2587520533615041024

Offset: 1

Vladeta Jovovic, Oct 14 2003

nonn

G. C. Greubel, Table of n, a(n) for n = 1..445

Cf. A061095, A038041, A057625, A038048, A005225.

a[n_]:= n!*DivisorSum[n, (#/n)^# &]; Array[a, 50] (* G. C. Greubel, May 16 2018 *)

{a(n)= n!*sumdiv(n, d, (d/n)^d)};
for(n=1, 30, print1(a(n), ", ")) \\ G. C. Greubel, May 16 2018

E.g.f.: Sum_{k>0} x^k/(k-x^k).

A317329 Number of permutations of [n] with equal lengths of increasing runs.

Original entry on oeis.org

1, 2, 2, 7, 2, 82, 2, 1456, 1515, 50774, 2, 3052874, 2, 199364414, 136835794, 19451901825, 2, 2510158074714, 2, 370671075758054, 132705620239756, 69348874393843334, 2, 15772160279898993782, 613498040952503, 4087072509293134292962, 705927677748508225534

Offset: 1

Alois P. Heinz, Jul 25 2018

nonn

			a(4) = 7: 1234, 1324, 1423, 2314, 2413, 3412, 4321.

Alois P. Heinz, Table of n, a(n) for n = 1..400

Column k=1 of A317327.

Cf. A000040, A005225.

b:= proc(u, o, t, d) option remember; `if`(u+o=0, 1,
      `if`(t=d, add(b(u-j, o+j-1, 1, d), j=1..u),
       add(b(u+j-1, o-j, t+1, d), j=1..o)))
    end:
a:= proc(n) option remember; `if`(n=1, 1, 2+add(
      b(n, 0, d$2), d=numtheory[divisors](n) minus {1, n}))
    end:
seq(a(n), n=1..35);

b[u_, o_, t_, d_] := b[u, o, t, d] = If[u + o == 0, 1,
     If[t == d, Sum[b[u - j, o + j - 1, 1, d], {j, 1, u}],
     Sum[b[u + j - 1, o - j, t + 1, d], {j, 1, o}]]];
a[n_] := a[n] = If[n == 1, 1, 2 + Sum[b[n, 0, d, d], {d, Divisors[n] ~Complement~ {1, n}}]];
Array[a, 35] (* Jean-François Alcover, Aug 01 2021, after Alois P. Heinz *)

a(n) = 2 <=> n in { A000040 }.

A133118 Number of partitions of n-set with 3 block sizes.

Original entry on oeis.org

60, 315, 2268, 14742, 72180, 464640, 2676366, 16400098, 94209206, 673282610, 4095231104, 29371828846, 197547348216, 1513916607683, 10904464442572, 87070803499372, 673555061736062, 5718121102062336, 47028289679340734, 418812093667530755, 3680961843042545490, 34161428275433710485

Offset: 6

Vladeta Jovovic, Sep 18 2007

Alois P. Heinz, Table of n, a(n) for n = 6..300

Cf. A038041, A088142, A122404, A005225, A005772.

Column k=3 of A208437.

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[multinomial[n, Prepend[Table[i, {j}], n - i*j]]/j!*b[n - i*j, i - 1]*If[j == 0, 1, x], {j, 0, n/i}]]];
a[n_] := Coefficient[b[n, n], x, 3];
Array[a, 22, 6] (* Jean-François Alcover, May 24 2019, after Alois P. Heinz in A208437 *)

We obtain e.g.f. for number of partitions of n-set with m block sizes if we substitute x(i) with -Sum_{k>0} (1-exp(x^k/k!))^i in cycle index Z(S(m); x(1),x(2),...,x(n)) of symmetric group S(m) of degree m.

More terms from Max Alekseyev, Jun 17 2011

A133119 Number of permutations of [n] with 3 cycle lengths.

Original entry on oeis.org

120, 1050, 12712, 141876, 1418400, 17061660, 212254548, 2735287698, 37354035628, 581350330470, 8895742806480, 151305163230480, 2659183039338192, 50112909523522476, 976443721325014300, 20413628375979803370, 434137453618439716068

Offset: 6

Vladeta Jovovic, Sep 18 2007

Alois P. Heinz, Table of n, a(n) for n = 6..200

Cf. A005225, A005772, A038041, A088142.

Column k=3 of A218868.

We obtain e.g.f. for number of permutations of [n] with m cycle lengths if we substitute x(i) with -Sum_{k>0} ((1-exp(x^k/k))^i in cycle index Z(S(m); x(1),x(2),..,x(m)) of symmetric group S(m) of degree m.

More terms from Max Alekseyev, Feb 08 2010

A359951 Number of permutations of [n] such that the GCD of the cycle lengths is a prime.

Original entry on oeis.org

0, 0, 1, 2, 3, 24, 145, 720, 4725, 22400, 602721, 3628800, 67692625, 479001600, 12924021825, 103953833984, 2116670180625, 20922789888000, 959231402754625, 6402373705728000, 257071215652932681, 3242340687872000000, 142597230222616430625, 1124000727777607680000

Offset: 0

Alois P. Heinz, Jan 19 2023

nonn

			a(2) = 1: (12).
a(3) = 2: (123), (132).
a(4) = 3: (12)(34), (13)(24), (14)(23).
a(5) = 24: (12345), (12354), (12435), (12453), (12534), (12543), (13245), (13254), (13425), (13452), (13524), (13542), (14235), (14253), (14325), (14352), (14523), (14532), (15234), (15243), (15324), (15342), (15423), (15432).

Alois P. Heinz, Table of n, a(n) for n = 0..451
Wikipedia, Permutation

Cf. A000040, A000142, A005225, A079128, A214003, A346085, A346086.

b:= proc(n, g) option remember; `if`(n=0, `if`(isprime(g), 1, 0),
      add(b(n-j, igcd(j, g))*(n-1)!/(n-j)!, j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..23);

b[n_, g_] := b[n, g] = If[n == 0, If[PrimeQ[g], 1, 0], Sum[b[n - j, GCD[j, g]]*(n - 1)!/(n - j)!, {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)

a(n) = Sum_{prime p <= n} A346085(n,p).

a(p) = (p-1)! for prime p.

A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 3, 0, 6, 1, 0, 0, 0, 24, 1, 15, 40, 0, 0, 120, 1, 0, 0, 0, 0, 0, 720, 1, 105, 0, 1260, 0, 0, 0, 5040, 1, 0, 2240, 0, 0, 0, 0, 0, 40320, 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 1, 10395, 246400, 1247400, 0, 6652800, 0, 0, 0, 0, 0, 39916800

Offset: 1

Marko Riedel, Dec 17 2023

			Row n=6 is 1, 15, 40, 120 because there is one permutation of [6] consisting of six fixed points, there are 15 permutations consisting of three transpositions, there are forty permutations consisting of two three-cycles and there are one hundred and twenty permutations consisting of just one six-cycle (6!/6).
Triangular array starts:
[ 1] 1;
[ 2] 1,   1;
[ 3] 1,   0,    2;
[ 4] 1,   3,    0,    6;
[ 5] 1,   0,    0,    0,    24;
[ 6] 1,  15,   40,    0,     0, 120;
[ 7] 1,   0,    0,    0,     0,   0, 720;
[ 8] 1, 105,    0, 1260,     0,   0,   0, 5040;
[ 9] 1,   0, 2240,    0,     0,   0,   0,    0, 40320;
[10] 1, 945,    0,    0, 72576,   0,   0,    0,     0, 362880;

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, pages 120-122.

Marko Riedel, math.stackexchange.com, Number of permutations in the symmetric group.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles with analytic combinatorics.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles using the principle of inclusion-exclusion.

Cf. A005225 (row sums), A008290.

Cf. A123023 (column 2), A052502 (column 3), A060706 (column 4).

T:= (n, m)-> `if`(irem(n,m)=0, n!/m^(n/m)/(n/m)!, 0):
seq(seq(T(n, m), m = 1..n), n=1..15);

A368213[n_,k_]:=If[Divisible[n,k],n!/(k^(n/k)(n/k)!),0];
Table[A368213[n,k],{n,15},{k,n}] (* Paolo Xausa, Dec 18 2023 *)

def T(n, d): return factorial(n) // (d ** (n//d) * factorial(n//d))
for n in range(1, 19):
    print([T(n, d) if n % d == 0 else 0 for d in range(1, n+1)])
# Peter Luschny, Dec 17 2023

T(n, k) = n! / ( k^(n/k) * (n/k)! ) if k divides n otherwise 0.

A370602 a(n) = n! * Sum_{d|n} 1/((d-1)! * (n/d)^(d-1)).

Original entry on oeis.org

1, 4, 9, 40, 125, 1056, 5047, 51248, 383049, 4364020, 39916811, 576885552, 6227020813, 99634224704, 1334500527375, 23592657488416, 355687428096017, 7202890599354468, 121645100408832019, 2679832071577681040, 51612375654647808021, 1226182612423511392672

Offset: 1

Seiichi Manyama, Feb 23 2024

nonn

Cf. A370579, A370603.

Cf. A005225.

a(n) = n!*sumdiv(n, d, 1/((d-1)!*(n/d)^(d-1)));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=1, N, x^k*exp(x^k/k))))

a(n) = n * A005225(n).
If p is prime, a(p) = p + p!.
E.g.f.: Sum_{k>0} x^k * exp(x^k/k).

A343576 Number of permutations of [n] without fixed points and all cycles equal length.

Original entry on oeis.org

1, 0, 1, 2, 9, 24, 175, 720, 6405, 42560, 436401, 3628800, 48073795, 479001600, 7116730335, 88966701824, 1474541093025, 20922789888000, 400160588853025, 6402373705728000, 133991603578884051, 2457732174030848000, 55735573291977790575, 1124000727777607680000

Offset: 0

Gary Yane, Apr 20 2021

			a(4) = 9: (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (2,3,4,1), (2,4,1,3), (3,1,4,2), (3,4,2,1), (4,1,2,3), (4,3,1,2).

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

Cf. A000040, A001358, A005225, A261431.

a:= n-> `if`(n=0, 1, add(n!/d!*(d/n)^d, d=numtheory[divisors](n) minus {n})):
seq(a(n), n=0..23);  # Alois P. Heinz, Apr 20 2021

a(n) = if (n, sumdiv(n, d, if (dMichel Marcus, Apr 21 2021

a(n) = Sum_{d|n, d0, a(0) = 1.

a(n) = A261431(n) for n in { A000040, A001358 }.

cp's OEIS Frontend

A346057 Expansion of e.g.f. Product_{k>=1} exp(1 - exp(x^k/k)).

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A364967 Number T(n,k) of permutations of [n] for which the difference between the longest and the shortest cycle length is k; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

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Maple

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A087905 a(n) = n! * Sum_{d|n} (d/n)^d.

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Mathematica

PARI

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A317329 Number of permutations of [n] with equal lengths of increasing runs.

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Examples

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Maple

Mathematica

Formula

A133118 Number of partitions of n-set with 3 block sizes.

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Mathematica

Formula

Extensions

A133119 Number of permutations of [n] with 3 cycle lengths.

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Formula

Extensions

A359951 Number of permutations of [n] such that the GCD of the cycle lengths is a prime.

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Programs

Maple

Mathematica

Formula

A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

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