A110143 -id:A110143 - cp's OEIS frontend

A241584 Number of orbits of Sym(n)^2 under action of Iso_1(n).

Original entry on oeis.org

1, 4, 11, 43, 155, 761, 4043, 27190

Offset: 1

N. J. A. Sloane, May 12 2014

Mathieu Bogaerts and Peter Dukes, Semidefinite programming for permutation codes, Discrete Math. 326 (2014), 34--43. MR3188985.

Cf. A241585, A110143.

Proposition 3.2 of Bogaerts et al. gives an explicit formula.

A241585 Number of orbits of Sym(n)^2 under action of Iso_1*(n).

Original entry on oeis.org

1, 3, 8, 28, 93, 425, 2151, 14016

Offset: 1

N. J. A. Sloane, May 12 2014

Mathieu Bogaerts and Peter Dukes, Semidefinite programming for permutation codes, Discrete Math. 326 (2014), 34--43. MR3188985.

Cf. A241584, A110143.

Proposition 3.3 of Bogaerts et al. gives an explicit formula.

A309682 G.f.: C(x)C(2x^2)C(3x^3)*..., where C(x) is the g.f. for A000108.

Original entry on oeis.org

1, 1, 4, 10, 33, 81, 282, 762, 2599, 7979, 27343, 89371, 315256, 1078498, 3857048, 13651786, 49475282, 178736186, 655247192, 2401663838, 8883371016, 32906649488, 122619768860, 457836275272, 1716620421629, 6449729802639, 24308647131627, 91800114425437

Offset: 0

Vaclav Kotesovec, Aug 12 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1669

Cf. A025174, A110143, A322204, A326985.

C:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
b:= proc(n, i) option remember; `if`(n=0 or i=1,
      C(n), add(C(j)*i^j*b(n-i*j, i-1), j=0..n/i))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 23 2019

nmax = 30; CoefficientList[Series[Product[Sum[CatalanNumber[k]*j^k*x^(j*k), {k, 0, nmax/j}], {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Product[(1 - Sqrt[1 - 4*k*x^k])/(2*k*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n / n^(3/2), where c = 1/(2*sqrt(Pi)) * Product_{k>=1} (2^k*(2^(k-1) - sqrt(4^(k-1) - k))/k) = 0.711438694828613555153724789...

A327150 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

Original entry on oeis.org

1, 1, 1, 9, 22, 77, 400, 2624, 20747, 183544, 1826374, 20045348, 240262047, 3120641718, 43665293393, 654731266933, 10472819759734, 178001257647196, 3203520381407270, 60859480965537820, 1217072840308660049

Offset: 0

Derek Lim, Aug 23 2019

nonn

			For n = 3, representatives of the n=9 orbits are (e,e), (e,(123)), (e,(132)), ((123),e), ((132),e), ((123),(123)), ((123),(132)), ((132),(123)), ((132),(132)), where e is the identity.

Derek Lim, Table of n, a(n) for n = 0..61
MathOverflow, A general formula for the number of conjugacy classes of S_n×S_n acted on by S_n

Cf. A000702, A110143, A327014, A327151.

G:= AlternatingGroup(n);; Size(G)*Sum(List(ConjugacyClasses(G), K -> 1/Size(K)));

a(n) = (n!/2) * Sum_{K conjugacy class in A_n} 1/|K|.

A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 2, 6, 0, 4, 11, 4, 24, 0, 5, 10, 14, 12, 120, 0, 6, 31, 62, 34, 48, 720, 0, 7, 28, 60, 84, 120, 240, 5040, 0, 8, 66, 102, 490, 228, 552, 1440, 40320, 0, 9, 60, 299, 292, 708, 912, 3120, 10080, 362880, 0, 10, 120, 282, 722, 4396, 2136, 4752, 20880, 80640, 3628800

Offset: 0

Peter Luschny, Apr 17 2016

S(n,k) = Sum_{p in P(n,k)} n!/Aut(p) are the Stirling cycle numbers A132393.

			Triangle starts:
[1]
[0, 1]
[0, 2, 2]
[0, 3, 2, 6]
[0, 4, 11, 4, 24]
[0, 5, 10, 14, 12, 120]
[0, 6, 31, 62, 34, 48, 720]
[0, 7, 28, 60, 84, 120, 240, 5040]

Cf. A110143 (row sums), A132393, A271708.

def A271707(n,k):
    P = Partitions(n, length=k)
    return sum(p.aut() for p in P)
for n in (0..10): print([A271707(n,k) for k in (0..n)])

A327015 Number of equivalence classes of pairs of permutations in S_n where two pairs are equivalent if they are simultaneously conjugate to each other or simultaneously conjugate to each other after a reversal of one pair.

Original entry on oeis.org

1, 1, 3, 8, 28, 98, 518, 3096, 23415, 201795, 1973189, 21347935, 253282652, 3263902430

Offset: 0

Derek Lim, Aug 13 2019

			For n = 2, representatives of the a(2) = 3 classes are: (e,e), (e, (12)), ((12),(12)), where e is identity.

Amit Harlev, Charles R. Johnson, and Derek Lim, The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach, arXiv:1908.03647 [math.SP], 2019.

Cf. A110143, A327014.

a(n) = (A110143(n) + A327014(n)) / 2.

A327151 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation, such that both permutations in a representative pair are of the same conjugacy class in A_n.

Original entry on oeis.org

1, 1, 1, 3, 8, 23, 82, 452, 2369, 18356, 143308, 1396222, 13000455, 152886068

Offset: 0

Derek Lim, Aug 23 2019

			For n = 3, representatives of the a(3) = 3 orbits are: (e,e), ((123),(123)), ((132),(132)), where e is the identity.

Cf. A110143, A327014, A327150.

A338001 Irregular triangle read by rows, a refinement of A271708.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 6, 2, 3, 0, 24, 8, 4, 3, 4, 0, 120, 8, 12, 6, 6, 4, 5, 0, 720, 48, 16, 48, 18, 6, 18, 8, 8, 5, 6, 0, 5040, 48, 48, 240, 18, 24, 12, 72, 12, 8, 24, 10, 10, 6, 7, 0, 40320, 384, 96, 192, 1440, 36, 36, 24, 36, 360, 32, 12, 32, 16, 96, 15, 10, 30, 12, 12, 7, 8

Offset: 0

Peter Luschny, Nov 13 2020

Row n of the triangle gives the sizes of the centralizers of any permutation of cycle type given by the partitions of n with max. part k.

T(n, k) divides n! if k > 0 and in this case the n!/T(n, k) give, up to order, the rows of A036039.

			Triangle rows start:
0: [1];
1: [0], [1];
2: [0], [2],    [2];
3: [0], [6],    [2],           [3];
4: [0], [24],   [8, 4],        [3],              [4];
5: [0], [120],  [8, 12],       [6, 6],           [4],         [5];
6: [0], [720],  [48, 16, 48],  [18, 6, 18],      [8, 8],      [5],      [6];
7: [0], [5040], [48, 48, 240], [18, 24, 12, 72], [12, 8, 24], [10, 10], [6], [7];
.
For n = 4 the partition of 4 with cycle type [2, 2] has centralizer size 8, and the partition [2, 1, 1] has centralizer size 4. Therefore in column 2 in the above triangle the pair [8, 4] appears.

S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.

Cf. A271708, A110143 (row sums), A052810 (row length), A126074, A036039.

def A338001(n):
    R = []
    for k in (0..n):
        P = Partitions(n, max_part=k, inner=[k])
        q = [p.aut() for p in P]
        R.append(q if q != [] else [0])
    return flatten(R)
for n in (0..7): print(A338001(n))

cp's OEIS Frontend

A241584 Number of orbits of Sym(n)^2 under action of Iso_1(n).

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A241585 Number of orbits of Sym(n)^2 under action of Iso_1*(n).

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A309682 G.f.: C(x)*C(2*x^2)*C(3*x^3)*..., where C(x) is the g.f. for A000108.

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Maple

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A327150 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation.

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GAP

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A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]*j[1]!*...*n^j[n]*j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.

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Sage

A327015 Number of equivalence classes of pairs of permutations in S_n where two pairs are equivalent if they are simultaneously conjugate to each other or simultaneously conjugate to each other after a reversal of one pair.

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A327151 Number of orbits of the direct square of the alternating group A_n^2 where A_n acts by conjugation, such that both permutations in a representative pair are of the same conjugacy class in A_n.

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A338001 Irregular triangle read by rows, a refinement of A271708.

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SageMath

A309682 G.f.: C(x)C(2x^2)C(3x^3)*..., where C(x) is the g.f. for A000108.

A271707 Triangle read by rows, T(n,k) = Sum_{p in P(n,k)} Aut(p) where P(n,k) are the partitions of n with length k and Aut(p) = 1^j[1]j[1]!...n^j[n]j[n]! where j[m] is the number of parts in the partition p equal to m; for n>=0 and 0<=k<=n.