A000940 Number of n-gons with n vertices. (Formerly M1260 N0482)

1, 2, 4, 12, 39, 202, 1219, 9468, 83435, 836017, 9223092, 111255228, 1453132944, 20433309147, 307690667072, 4940118795869, 84241805734539, 1520564059349452, 28963120073957838, 580578894859915650, 12217399235411398127, 269291841184184374868, 6204484017822892034404

Offset

3,2

Comments

Number of inequivalent undirected Hamiltonian cycles in complete graph on n labeled nodes under action of dihedral group of order 2n acting on nodes.

References

J. H. Kwak and J. Lee, Enumeration of graph coverings, surface branched coverings and related group theory, in Combinatorial and Computational Mathematics (Pohang, 2000), ed. S. Hong et al., World Scientific, Singapore 2001, pp. 97-161.

R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Ludovic Schwob, Table of n, a(n) for n = 3..500 (terms 3..100 from T. D. Noe).

S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353.

S. W. Golomb and L. R. Welch, On the enumeration of polygons, Amer. Math. Monthly, 67 (1960), 349-353. [Annotated scanned copy]

Samuel Herman and Eirini Poimenidou, Orbits of Hamiltonian Paths and Cycles in Complete Graphs, arXiv:1905.04785 [math.CO], 2019.

E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence, except he has a(6)=7 instead of 12)

R. C. Read, Letter to N. J. A. Sloane, 1992

R. C. Read, Combinatorial problems in theory of music, Discrete Math. 167 (1997), 543-551.

N. J. A. Sloane, Illustration of initial terms [Annotated page from Golomb-Welch article]

Venta Terauds and J. Sumner, Circular genome rearrangement models: applying representation theory to evolutionary distance calculations, arXiv preprint arXiv:1712.00858 [q-bio.PE], 2017.

Venta Terauds and Jeremy Sumner, A new algebraic approach to genome rearrangement models, arXiv:2012.11665 [q-bio.PE], 2020.

Formula

For formula see Maple lines.

a(p) = ((((p-1)! + 1)/p) + p - 2 + (2^((p-1)/2)*((p-1)/2)!))/4 for prime p. See A007619. - Ian Mooney, Oct 05 2022

a(n) ~ sqrt(2*Pi)/4 * n^(n-3/2) / e^n. - Ludovic Schwob, Nov 03 2022

Example

Label the vertices of a regular n-gon 1,2,...,n.
For n=3,4,5 representatives for the polygons counted here are:
  (1,2,3,1),
  (1,2,3,4,1), (1,2,4,3,1),
  (1,2,3,4,5,1), (1,2,3,5,4,1), (1,2,4,5,3,1), (1,3,5,2,4,1).
For n=6:
  (1,2,3,4,5,6,1), (1,2,3,4,6,5,1), (1,2,3,5,6,4,1),
  (1,2,3,6,5,4,1), (1,2,4,3,6,5,1), (1,2,4,6,3,5,1),
  (1,2,4,6,5,3,1), (1,2,5,3,6,4,1), (1,2,5,4,6,3,1),
  (1,2,5,6,3,4,1), (1,2,6,4,5,3,1), (1,3,5,2,6,4,1).

Maple

with(numtheory);
# for n odd:
Sd:=proc(n) local t1, d; t1:=2^((n-1)/2)*n^2*((n-1)/2)!; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
# for n even:
Se:=proc(n) local t1, d; t1:=2^(n/2)*n*(n+6)*(n/2)!/4; for d from 1 to n do if n mod d = 0 then t1:=t1+phi(n/d)^2*d!*(n/d)^d; fi; od: t1/(4*n^2); end;
A000940:=n-> if n mod 2 = 0 then Se(n) else Sd(n); fi;

Mathematica

a[n_] := (t1 = If[OddQ[n], 2^((n - 1)/2)*n^2*((n - 1)/2)!, 2^(n/2)*n*(n + 6)*(n/2)!/4]; For[ d = 1 , d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[n/d]^2*d!*(n/d)^d]]; t1/(4*n^2)); Table[a[n], {n, 3, 25}] (* Jean-François Alcover, Jun 19 2012, after Maple *)

Prog

(PARI) a(n)={if(n<3, 0, (2^(n\2-2)*(n\2)!*n*if(n%2, 4*n, n + 6) + sumdiv(n, d, eulerphi(n/d)^2*d!*(n/d)^d))/(4*n^2))} \\ Andrew Howroyd, Sep 09 2018
(Python)
from sympy import factorial, divisors, totient
def A000940(n): return 1 if n == 3 else ((sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n, generator=True))+(1<<(k:=n>>1)-2)*n*(n<<2 if n&1 else (n+6))*factorial(k))>>2)//n//n # Chai Wah Wu, Nov 07 2022

Crossrefs

Cf. A000939, A007619. Bisections give A094156, A094157.

For permutation classes under various symmetries see A089066, A262480, A002619.

Sequence in context: A268069 A215071 A180487 * A008404 A170815 A170814

Adjacent sequences: A000937 A000938 A000939 * A000941 A000942 A000943

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 05 2004

Status

approved