A002619 -id:A002619 - cp's OEIS frontend

A370068 Number of nonequivalent directed unicursal star polygons (no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon up to rotations.

0, 0, 0, 0, 2, 1, 10, 47, 350, 3005, 28722, 302519, 3471738, 43181993, 578730766, 8317664191, 127644961618, 2083638325661, 36055062511490, 659316772258655, 12705552903848466, 257373902883624297, 5467702595346969530, 121562217391867941767

Offset: 1

Andrew Howroyd, Feb 23 2024

nonn

Directed means that the direction of travel is significant.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Wikipedia, Unicursal hexagram.

Cf. A002619 (if edges may join adjacent vertices), A231091 (undirected), A326411, A370459.

Q(n,k)={subst(serlaplace(polcoef((1 - 2*x - x^2)/((1 + x)*(1 + (1 - y)*x + y*x^2)) + O(x^n), n-1)), y, k)}
E(r,d)={eulerphi(d)*Q(r,d) + 2*(-1)^r}
a370068(n)={if(n<3, 0, sumdiv(n,d,eulerphi(d)*E(n/d,d))/n)}

A047919 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d)/n if k|n else 0, where U(n,k) = A047916(n,k) (1<=k<=n).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 20, 2, 4, 6, 0, 0, 108, 6, 0, 0, 0, 0, 0, 714, 4, 4, 0, 40, 0, 0, 0, 4992, 6, 0, 30, 0, 0, 0, 0, 0, 40284, 4, 16, 0, 0, 380, 0, 0, 0, 0, 362480, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628790, 4, 8, 60, 312, 0, 3768, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

Divide n-th row of array A047918 by n.

Cf. A002024.

a047919 n k = a047919_tabl !! (n-1) !! (k-1)
a047919_row n = a047919_tabl !! (n-1)
a047919_tabl = zipWith (zipWith div) a047918_tabl a002024_tabl
-- Reinhard Zumkeller, Mar 19 2014

U[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; a[n_, k_] := Sum[If[Divisible[n, k], MoebiusMu[d]*U[n, k/d], 0], {d, Divisors[k]}]; row[n_] := Table[a[n, k], {k, 1, n}]/n; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Nov 21 2012, after A047918 *)

Offset corrected by Reinhard Zumkeller, Mar 19 2014

A384630 Number of self-inverse double cosets in Z_n\S_n/Z_n.

Original entry on oeis.org

1, 1, 2, 3, 6, 14, 34, 98, 294, 952, 3246, 11698, 43732, 170752, 689996, 2888034, 12458784, 55406422, 253142182, 1187934740, 5712033368, 28131119956, 141645386202, 728841303696, 3827217750492, 20499431084644, 111876916526070, 621831335167486, 3516904353610572

Offset: 1

Ludovic Schwob, Jun 05 2025

nonn

Z_n is the cyclic group of order n, seen as a subgroup of the symmetric group S_n.

Cosets in S_n/Z_n are in bijection with cycles obtained by connecting cyclically n equally spaced points on a circle. Double cosets in Z_n\S_n/Z_n are in bijection with cycles up to rotation.

Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See Proposition 4.1 p. 9.

Cf. A000142 (cycles), A002619 (cycles up to rotation), A384631 (self-inverse polygons).

# From Proposition 4.1 in the reference:
from sympy import factorial,divisors,totient
def A384630(n):
    s = 0
    if n%2==0:
        for d in divisors(n//2):
            if d%2==0:
                s += totient(d)*(d//2)**(n//2//d)*factorial(n//d)//factorial(n//2//d)
            else:
                s += totient(d)*sum(factorial(n//d)*d**i//2**i//factorial(i)//factorial(n//d-2*i) for i in range(n//2//d+1))
    else:
        for d in divisors(n):
            s += totient(d)*sum(factorial(n//d)*d**i//2**i//factorial(i)//factorial(n//d-2*i) for i in range(n//d//2+1))
    return s//n

A047917 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk!/n if k|n else 0 (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 2, 0, 2, 2, 2, 0, 6, 4, 0, 0, 0, 24, 2, 6, 8, 0, 0, 120, 6, 0, 0, 0, 0, 0, 720, 4, 8, 0, 48, 0, 0, 0, 5040, 6, 0, 36, 0, 0, 0, 0, 0, 40320, 4, 20, 0, 0, 384, 0, 0, 0, 0, 362880, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 4, 12, 64, 324, 0, 3840, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

			1; 1,1; 2,0,2; 2,2,0,6; 4,0,0,0,24; 2,6,8,0,0,120; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

Divide n-th row of A047916 by n.
Row sums give A061417.
Cf. A002024.

a047917 n k = a047917_tabl !! (n-1) !! (k-1)
a047917_row n = a047917_tabl !! (n-1)
a047917_tabl = zipWith (zipWith div) a047916_tabl a002024_tabl
-- Reinhard Zumkeller, Mar 19 2014

a[n_, k_] := If[ Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!/n, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]](* Jean-François Alcover, Feb 17 2012 *)

Offset corrected by Reinhard Zumkeller, Mar 19 2014

A325956 Number of cyclic permutations of [n] with symmetry order s=1.

Original entry on oeis.org

1, 0, 0, 4, 20, 108, 714, 4992, 40284, 362480, 3628790, 39912648, 479001588, 6226974684, 87178287120, 1307673722880, 20922789887984, 355687417715904, 6402373705727982, 121645100223034480, 2432902008175589664, 51090942167993548700, 1124000727777607679978, 25852016738803204991232

Offset: 1

Michel Marcus, Sep 10 2019

nonn

Saeed Zakeri, Cyclic Permutations: Degrees and Combinatorial Types, arXiv:1909.03300 [math.DS], 2019. See Table 1 p. 8.

Cf. A002619, A064852.

Table[(1/n) DivisorSum[n, MoebiusMu[#] EulerPhi[#] #^(n/#)*(n/#)! &], {n, 24}] (* Michael De Vlieger, May 07 2021 *)

a(n) = sumdiv(n, d, moebius(d)*eulerphi(d)*d^(n/d)*(n/d)!)/n;

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

a(n) = n*A064852(n). - Andrew Howroyd, May 07 2021

a(1)=1 prepended by Andrew Howroyd, May 07 2021

cp's OEIS Frontend

A370068 Number of nonequivalent directed unicursal star polygons (no edge joins adjacent vertices) that can be formed by connecting the vertices of a regular n-gon up to rotations.

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A047919 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d)/n if k|n else 0, where U(n,k) = A047916(n,k) (1<=k<=n).

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A384630 Number of self-inverse double cosets in Z_n\S_n/Z_n.

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A047917 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k!/n if k|n else 0 (1<=k<=n).

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A325956 Number of cyclic permutations of [n] with symmetry order s=1.

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Mathematica

PARI

Formula

Extensions

A047917 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk!/n if k|n else 0 (1<=k<=n).