A003483 -id:A003483 - cp's OEIS frontend

A247005 Number A(n,k) of permutations on [n] that are the k-th power of a permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 2, 3, 24, 1, 1, 1, 1, 4, 12, 120, 1, 1, 1, 2, 3, 16, 60, 720, 1, 1, 1, 1, 6, 9, 80, 270, 5040, 1, 1, 1, 2, 1, 24, 45, 400, 1890, 40320, 1, 1, 1, 1, 6, 4, 96, 225, 2800, 14280, 362880, 1, 1, 1, 2, 3, 24, 40, 576, 1575, 22400, 128520, 3628800, 1

Offset: 0

Alois P. Heinz, Sep 09 2014

Number of permutations p on [n] such that a permutation q on [n] exists with p=q^k.

			A(3,0) = 1: (1,2,3).
A(3,1) = 6: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
A(3,2) = 3: (1,2,3), (2,3,1), (3,1,2).
A(3,3) = 4: (1,2,3), (1,3,2), (2,1,3), (3,2,1).
Square array A(n,k) begins:
  1,    1,    1,    1,    1,    1,    1,    1,    1, ...
  1,    1,    1,    1,    1,    1,    1,    1,    1, ...
  1,    2,    1,    2,    1,    2,    1,    2,    1, ...
  1,    6,    3,    4,    3,    6,    1,    6,    3, ...
  1,   24,   12,   16,    9,   24,    4,   24,    9, ...
  1,  120,   60,   80,   45,   96,   40,  120,   45, ...
  1,  720,  270,  400,  225,  576,  190,  720,  225, ...
  1, 5040, 1890, 2800, 1575, 4032, 1330, 4320, 1575, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
William Y. C. Chen and Elena L. Wang, r-Enriched Permutations and an Inequality of Bóna-McLennan-White, arXiv:2502.04136 [math.CO], 2025. See pp. 3, 14.
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, Theorem 4.8.2.

Columns k=0-10 give: A000012, A000142, A003483, A103619, A103620, A215716, A215717, A215718, A247006, A247007, A247008.
Main diagonal gives A247009.
Cf. A247026 (the same for endofunctions), A155510.

with(combinat): with(numtheory): with(padic):
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(irem(j, mul(p^ordp(k, p), p=factorset(i)))=0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1, k), 0), j=0..n/i)))
    end:
A:= (n, k)-> `if`(k=0, 1, b(n$2, k)):
seq(seq(A(n, d-n), n=0..d), d=0..14);

multinomial[n_, k_List] := n!/Times @@ (k!); b[, 1, ] = 1; b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[If[Mod[j, Product[ p^IntegerExponent[k, p], {p, FactorInteger[i][[All, 1]]}]] == 0, (i - 1)!^j*multinomial[n, Join[{n-i*j}, Array[i&, j]]]/j!*b[n-i*j, i-1, k], 0], {j, 0, n/i}]]]; A[n_, k_] := If[k == 0, 1, b[n, n, k]]; Table[A[n, d - n], {d, 0, 14}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 14 2017, after Alois P. Heinz *)

A102759 Number of partitions of n-set in which number of blocks of size 2k is even (or zero) for every k.

Original entry on oeis.org

1, 1, 1, 2, 8, 27, 82, 338, 1647, 7668, 37779, 210520, 1276662, 7985200, 51302500, 358798144, 2677814900, 20309850311, 160547934756, 1344197852830, 11666610870142, 104156661915427, 962681713955130, 9238216839975106, 91508384728188792, 930538977116673878

Offset: 0

Vladeta Jovovic, Feb 10 2005, Aug 05 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A003483, A006950, A130219 - A130223.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
       add(`if`(irem(i, 2)=1 or irem(j, 2)=0, multinomial(
       n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 08 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[i, 2] == 1 || Mod[j, 2] == 0, multinomial[n, Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1], 0], {j, 0, n/i}]]] ; a[n_] := b[n, n]; Table[ a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 16 2015, after Alois P. Heinz *)

N=31; x='x+O('x^N);
Vec(serlaplace(exp(sinh(x))*prod(k=1,N,cosh(x^(2*k)/(2*k)!))))
/* gives: [1, 1, 1, 2, 8, 27, 82, 338, 1647, 7668, ...] , Joerg Arndt, Jan 03 2011 */

E.g.f. for offset 2: exp(sinh(x))*Product_{k>=1} cosh(x^(2*k)/(2*k)!). - Geoffrey Critzer, Jan 02 2011

Offset changed to 0 and two 1's prepended by Alois P. Heinz, Mar 08 2015

A103619 Number of permutations of n elements admitting a cube root.

Original entry on oeis.org

1, 1, 2, 4, 16, 80, 400, 2800, 22400, 181440, 1814400, 19958400, 218803200, 2844441600, 39822182400, 556972416000, 8911558656000, 151496497152000, 2579172973977600, 49004286505574400, 980085730111488000, 19584861165821952000, 430866945648082944000

Offset: 0

Vladeta Jovovic, Feb 11 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 149, Eq. 4.8.2.

Cf. A003483, A103620, A102687, A215716, A215717, A215718.

Column k=3 of A247005.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 3))<>0, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

CoefficientList[Series[(1-x^3)^(1/3)/(1-x) * Product[1/3*E^(1/3*x^(3*m)/m) + 2/3*E^(-1/6*x^(3*m)/m) * Cos[1/6*3^(1/2)*x^(3*m)/m],{m,1,20}],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Sep 13 2014 *)

E.g.f.: (1-x^3)^(1/3)/(1-x)*Product(1/3*exp(1/3*x^(3*m)/m)+2/3*exp(-1/6*x^(3*m)/m)*cos(1/6*3^(1/2)*x^(3*m)/m), m = 1 .. infinity).

A103620 Number of permutations of n elements admitting a fourth root.

Original entry on oeis.org

1, 1, 1, 3, 9, 45, 225, 1575, 11130, 100170, 897750, 9875250, 108523800, 1410809400, 18332414100, 274986211500, 4127136413400, 70161319027800, 1192076391706200, 22649451442417800, 430247983427262000, 9035207651972502000

Offset: 0

Vladeta Jovovic, Feb 11 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 149, Eq. 4.8.2.

Cf. A003483, A103619, A102687, A215716, A215717, A215718.

Column k=4 of A247005.

with(combinat): with(numtheory): with(padic):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(irem(j, mul(p^ordp(4, p), p=factorset(i)))=0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 09 2014

CoefficientList[Series[((1+x)/(1-x))^(1/2) * Product[1/2*Cos[1/2*x^(2*m)/m] + 1/2*Cosh[1/2*x^(2*m)/m],{m,1,20}],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Sep 13 2014 *)

E.g.f.: ((1+x)/(1-x))^(1/2)*Product(1/2*cos(1/2*x^(2*m)/m)+1/2*cosh(1/2*x^(2*m)/m), m = 1 .. infinity).

A215716 Number of permutations on n points admitting a fifth root.

Original entry on oeis.org

1, 1, 2, 6, 24, 96, 576, 4032, 32256, 290304, 2612736, 28740096, 344881152, 4483454976, 62768369664, 878757175296, 14060114804736, 239021951680512, 4302395130249216, 81745507474735104, 1553164642019966976, 32616457482419306496, 717562064613224742912

Offset: 0

Eric M. Schmidt, Aug 22 2012

nonn

a(n) is the number of permutations of n points such that for all positive m, the number of (5m)-cycles is a multiple of 5.

Eric M. Schmidt, Table of n, a(n) for n = 0..200
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.

Cf. A003483, A103619, A103620, A215717, A215718.

Column k=5 of A247005.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 5))<>0, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, GCD[i, 5]] != 0, 0, (i-1)!^j*multinomial[n, Prepend[Table[i, {j}], n-i*j]]/j!*b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 21 2016, after Alois P. Heinz *)

{ A215716_list(numterms) = Vec(serlaplace((1 - x^5 + O(x^numterms))^(1/5)/(1-x) * prod(m=1, numterms\5, exp5(x^(5*m)/(5*m), numterms\(5*m)+1)))); }
{ exp5(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^5) + O(x^prec)), x, y); }

E.g.f.: (1 - x^5)^(1/5)/(1 - x) * Product(E_5(x^(5m)/(5m)), m = 1 .. infinity), where E_5(x) = 1 + x^5/5! + x^10/10! + ... .

A215717 Number of permutations on n points admitting a sixth root.

Original entry on oeis.org

1, 1, 1, 1, 4, 40, 190, 1330, 8680, 52920, 340200, 6237000, 76211520, 1098857760, 11677585920, 109679169600, 1497396700800, 41977644508800, 783593969558400, 15973899557616000, 263524120417958400, 3733362595368806400, 64262934423790502400

Offset: 0

Eric M. Schmidt, Aug 23 2012

nonn

a(n) is the number of permutations of n points such that for all positive m, the number of m-cycles is a multiple of gcd(m, 6).

Eric M. Schmidt, Table of n, a(n) for n = 0..200
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.

Cf. A003483, A103619, A103620, A215716, A215718.

Column k=6 of A247005.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 6))<>0, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, GCD[i, 6]] != 0, 0, (i-1)!^j*multinomial[n, Prepend[Table[i, {j}], n-i*j]]/j!*b[n-i*j, i - 1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 21 2016, after Alois P. Heinz *)

{ A215717_list(numterms) = Vec(serlaplace(prod(m=1, numterms, expthin(gcd(m, 6), x^m/m, numterms\m+1))) + O(x^numterms)); }
{ expthin(j, y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^j) + O(x^prec)), x, y); }

E.g.f.: prod(m>=1, E_(gcd(m,6))(x^m/m) ), where E_j(x) = 1 + x^j/j! + x^(2j)/(2j)! + ... .

A215718 Number of permutations on n points admitting a seventh root.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 4320, 34560, 311040, 3110400, 34214400, 410572800, 5337446400, 69386803200, 1040802048000, 16652832768000, 283098157056000, 5095766827008000, 96819569713152000, 1936391394263040000, 38727827885260800000, 852012213475737600000

Offset: 0

Eric M. Schmidt, Aug 23 2012

nonn

a(n) is the number of permutations of n points such that for all positive m, the number of (7*m)-cycles is a multiple of 7.

Eric M. Schmidt, Table of n, a(n) for n = 0..200
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 148-149, Thms. 4.8.2 and 4.8.3.

Cf. A003483, A103619, A103620, A215716, A215717.

Column k=7 of A247005.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 7))<>0, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[j, GCD[i, 7]] != 0, 0, (i-1)!^j*multinomial[n, Prepend[Table[i, {j}], n-i*j]]/j!*b[n-i*j, i - 1]], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Oct 21 2016, after Alois P. Heinz *)

{ A215718_list(numterms) = Vec(serlaplace((1 - x^7 + O(x^numterms))^(1/7)/(1-x) * prod(m=1, numterms\7, exp7(x^(7*m)/(7*m), numterms\(7*m)+1)))); }
{ exp7(y, prec) = subst(serconvol(exp(x + O(x^prec)), 1/(1-x^7) + O(x^prec)), x, y); }

E.g.f.: ((1 - x^7)^(1/7)/(1 - x)) * Product_{m>=1} E_7(x^(7*m)/(7*m)), where E_7(x) = 1 + x^7/7! + x^14/14! + ... .

A246945 Decimal expansion of the coefficient e^G appearing in the asymptotic expression of the probability that a random n-permutation is a square, as sqrt(2/Pi)*e^G/sqrt(n).

Original entry on oeis.org

1, 2, 2, 1, 7, 7, 9, 5, 1, 5, 1, 9, 2, 5, 3, 6, 8, 3, 3, 9, 6, 4, 8, 5, 2, 9, 8, 4, 4, 5, 6, 3, 6, 1, 2, 1, 2, 7, 8, 8, 8, 1, 0, 1, 4, 8, 1, 4, 6, 9, 7, 7, 2, 8, 6, 8, 3, 8, 6, 3, 9, 6, 2, 9, 7, 0, 9, 2, 3, 3, 0, 4, 0, 3, 0, 0, 4, 8, 9, 3, 7, 3, 9, 9, 9, 6, 6, 2, 9, 8, 4, 3, 6, 7, 7, 8, 7, 9, 8, 7, 5, 8, 6, 7, 0

Offset: 1

Jean-François Alcover, Sep 08 2014

			G = 0.2003084150040401276417752235643787366634879653405876198956293474890714...
e^G = 1.22177951519253683396485298445636121278881014814697728683863962970923...
sqrt(2/Pi)*e^G = 0.974839011877335012323657925154410019528043463671159620094...

See A003483.

Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 41.
Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO]

Cf. A003483, A249673.

evalf(1/(product(sech(1/(2*k)), k=1..infinity)), 120) # Vaclav Kotesovec, Sep 20 2014

digits = 42; m0 = 10^4; dm = 1000; tail[m_] := (406425600*PolyGamma[1, m] - 2822400*PolyGamma[3, m] + 9408*PolyGamma[5, m] - 17*PolyGamma[7, m])/3251404800; Clear[g]; g[m_] := g[m] = Sum[Log[Cosh[1/(2*k)]], {k, 1, m - 1}] + tail[m] // N[#, digits + 10] &; g[m0] ; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits + 5] != RealDigits[g[m - dm], 10, digits + 5], Print["m = ", m]; m = m + dm]; G = g[m]; RealDigits[E^G, 10, digits ] // First
Block[{$MaxExtraPrecision = 1000}, Do[Print[N[Exp[Sum[(-1)^(n + 1)*Zeta[2*n]^2*(1 - 1/2^(2*n))/n/Pi^(2*n), {n, 1, m}]], 120]], {m, 100, 150}]] (* Vaclav Kotesovec, Sep 20 2014 *)

e^G = prod_{k>=1} cosh(1/(2k)).

G = Sum_{n>=1} (-1)^(n+1) * Zeta(2*n)^2 * (1-1/2^(2*n)) / (n * Pi^(2*n)). - Vaclav Kotesovec, Sep 20 2014

More terms from Vaclav Kotesovec, Sep 20 2014

A060307 Number of degree-4n permutations without odd cycles and such that number of cycles of size 2k is even (or zero) for every k.

Original entry on oeis.org

1, 3, 1365, 8534295, 204893714025, 15735481638151275, 2760485970394430603325, 1006427270776555103089989375, 659316841888260316767029819420625, 740198799422691022278446846884066321875, 1306298536067264588818106780684613899555353125

Offset: 0

Vladeta Jovovic, Mar 28 2001, Aug 10 2007

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A003483.

Cf. A013302.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 and irem(j, 2)=0, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(4*n$2):
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 09 2015

nn = 40; Select[Range[0, nn]! CoefficientList[Series[Product[Cosh[x^(2 i)/(2 i)], {i, 1, nn}], {x, 0, nn}], x], # > 0 &] (* Geoffrey Critzer, Jan 16 2012 *)

E.g.f.: Product_{k >= 1} cosh x^(2k)/(2k).

A212599 Number of functions on n labeled points to themselves (endofunctions) such that the number of cycles of f that have each even size is even.

Original entry on oeis.org

1, 1, 3, 18, 160, 1875, 27126, 466186, 9275064, 209654325, 5307031000, 148720701426, 4570816040352, 152874605142727, 5527634477245440, 214862754390554250, 8934811701563214976, 395788795274021394729, 18606559519007667893376, 925222631836457779380370, 48518852386696450625510400

Offset: 0

Geoffrey Critzer, May 22 2012

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A003483, A246951, A116956.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(j, igcd(i, 2))<>0, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> add(b(j, j)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 08 2014

nn=20;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];p=Product[Cosh[t^(2i)/(2i)],{i,1,nn}];Range[0,nn]! CoefficientList[Series[((1+t)/(1-t))^(1/2) p,{x,0,nn}],x]

E.g.f.: ((1+T(x))/(1-T(x)))^(1/2) * Product_{i>=1} cosh(T(x)^(2*i)/(2*i)) where T(x) is the e.g.f. for A000169.

Maple program fixed by Vaclav Kotesovec, Sep 13 2014

cp's OEIS Frontend

A247005 Number A(n,k) of permutations on [n] that are the k-th power of a permutation; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A102759 Number of partitions of n-set in which number of blocks of size 2k is even (or zero) for every k.

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A103619 Number of permutations of n elements admitting a cube root.

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A103620 Number of permutations of n elements admitting a fourth root.

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A215716 Number of permutations on n points admitting a fifth root.

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A215717 Number of permutations on n points admitting a sixth root.

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A215718 Number of permutations on n points admitting a seventh root.

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