A087639 -id:A087639 - cp's OEIS frontend

A007838 Number of permutations of n elements with distinct cycle lengths.

1, 1, 1, 5, 14, 74, 474, 3114, 24240, 219456, 2231280, 23753520, 288099360, 3692907360, 51677246880, 775999798560, 12364465397760, 208583679951360, 3770392002048000, 71251563061002240, 1421847102467635200, 29861872557056870400, 655829140087057305600

Offset: 0

Arnold Knopfmacher

nonn

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms, 2nd ed., Birkhäuser, Boston, 1982.

Seiichi Manyama, Table of n, a(n) for n = 0..450 (terms 0..200 from Vincenzo Librandi)
Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO], 2006.
A. Knopfmacher and R. Warlimont, Counting permutations and polynomials with a restricted factorization pattern, Australasian J. of Combinatorics, 13 (1996), 151-162.
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388.
A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Examples 8.10 and 11.8 (pdf, ps)

Cf. A000142, A080130, A087639, A088994, A317166.

p := product((1+x^m/m), m=1..100): s := series(p,x,100): for i from 1 to 100 do printf(`%.0f,`,i!*coeff(s,x,i)) od:
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> n!*b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 23 2022

max = 20; p = Product[(1 + x^m/m), {m, 1, max}]; s = Series[p, {x, 0, max}]; CoefficientList[s, x]*Range[0, max]! (* Jean-François Alcover, Oct 05 2011, after Maple *)

{a(n)=if(n<0, 0, n!*polcoeff( prod(k=1, n, 1+x^k/k, 1+x*O(x^n)), n))} /* Michael Somos, Sep 19 2006 */

E.g.f.: Product_{m >= 1} (1+x^m/m).
a(n) = Sum_{k=1..n} (n-1)!/(n-k)!*b(k)*a(n-k), where b(k) = Sum_{d divides k} (-d)^(1-k/d) and a(0) = 1. - Vladeta Jovovic, Oct 13 2002
Asymptotics: a(n) ~ n!(e^{-g} + e^{-g}/n + O((log n)/n^2)), where g is the Euler gamma.
E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*j^k)). - Ilya Gutkovskiy, May 27 2018

More terms from James Sellers, Dec 24 1999

A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

Original entry on oeis.org

1, 1, 0, 2, 8, 24, 144, 720, 8448, 64512, 576000, 5529600, 74972160, 887546880, 11285084160, 168318259200, 2843121254400, 44790578380800, 747955947110400, 13937735643955200, 287117441217331200, 5838778006909747200, 120976472421826560000, 2712639152754878054400

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 01 2003

nonn

a(n) is the number of n-permutations composed only of odd cycles of distinct length. - Geoffrey Critzer, Mar 08 2013

Also the number of permutations p of [n] with unique (functional) square root, i.e., there exists a unique permutation g such that g^2 = p. - Keith J. Bauer, Jan 08 2024

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000142, A007838, A007841, A087639, A088335, A294506, A305199, A309319.

b:= proc(n, i) option remember; `if`(((i+1)/2)^2n, 0, (i-1)!*
       b(n-i, i-2)*binomial(n, i))))
    end:
a:= n-> b(n, n-1+irem(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 01 2017

nn=20;Range[0,nn]!CoefficientList[Series[Product[1+x^(2i-1)/(2i-1),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Mar 08 2013 *)

{a(n)=n!*polcoeff( prod(k=1, n, 1+(k%2)*x^k/k, 1+x*O(x^n)), n)} /* Michael Somos, Sep 19 2006 */

E.g.f.: Product_{m >= 1} (1+x^(2*m-1)/(2*m-1)). - Vladeta Jovovic, Nov 05 2003
a(n) ~ exp(-gamma/2) * n! / sqrt(2*n), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 23 2019
a(n) = n! - A088335(n). - Alois P. Heinz, Jan 27 2020

More terms from Vladeta Jovovic, Nov 03 2003

a(0)=1 prepended by Seiichi Manyama, Nov 01 2017

A294506 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)).

Original entry on oeis.org

1, 1, 2, 8, 32, 184, 1184, 9008, 74752, 726528, 7583232, 87931392, 1092516864, 14863589376, 215094226944, 3358032635904, 55181218873344, 970561417248768, 17945595514847232, 351221170194874368, 7186120683011702784, 155103171658691641344

Offset: 0

Seiichi Manyama, Nov 01 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..449
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 5).

Cf. A007838, A007841, A087639, A088994, A309319.

nmax = 30; CoefficientList[Series[1/Product[(1-x^(2*k-1)/(2*k-1)), {k, 1, Floor[nmax/2] + 1}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Nov 02 2017 *)

a(n) ~ 2*exp(-gamma/2) * sqrt(2*n) * n! / Pi, where gamma is the Euler-Mascheroni constant A001620 [Lehmer, 1972]. - Vaclav Kotesovec, Jul 23 2019

A309319 E.g.f.: 1/Product_{k>0} (1 - x^(2k)/(2k)) (even powers only).

Original entry on oeis.org

1, 1, 12, 300, 15960, 1232280, 157006080, 25418352960, 5859886032000, 1655203620470400, 604893737678630400, 261278195494386470400, 140231830875916632652800, 86107922772424330377600000, 63316800257542340301112320000, 52666943508290765740968161280000

Offset: 0

Vaclav Kotesovec, Jul 23 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..225
D. H. Lehmer, On reciprocally weighted partitions, Acta Arithmetica XXI (1972), 379-388 (Theorem 6).

Cf. A007838, A007841, A087639, A088994, A294506.

nmax = 20; Table[(CoefficientList[Series[1/Product[(1 - x^(2*k)/(2*k)), {k, 1, 2*nmax}], {x, 0, 2*nmax}], x]*Range[0, 2*nmax]!)[[2 n + 1]], {n, 0, nmax}]

a(n) ~ exp(-gamma/2) * (2*n)! / sqrt(n) [Lehmer, 1972], where gamma is the Euler-Mascheroni constant A001620.

cp's OEIS Frontend

A007838 Number of permutations of n elements with distinct cycle lengths.

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A088994 Number of permutations in the symmetric group S_n such that the size of their centralizer is odd.

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Maple

Mathematica

PARI

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A294506 E.g.f.: 1/Product_{k>0} (1-x^(2*k-1)/(2*k-1)).

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Author

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Crossrefs

Programs

Mathematica

Formula

A309319 E.g.f.: 1/Product_{k>0} (1 - x^(2*k)/(2*k)) (even powers only).

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Programs

Mathematica

Formula

A294506 E.g.f.: 1/Product_{k>0} (1-x^(2k-1)/(2k-1)).

A309319 E.g.f.: 1/Product_{k>0} (1 - x^(2k)/(2k)) (even powers only).