OFFSET
0,13
COMMENTS
Equivalently, T(n,k) is the number of orbits of k-element subsets of the vertices of a regular n-gon under the usual action of the dihedral group D_n, or under the action of Euclidean plane isometries. Note that each row of the table is symmetric and unimodal. - Austin Shapiro, Apr 20 2009
Also, the number of k-chords in n-tone equal temperament, up to (musical) transposition and inversion. Example: there are 29 tetrachords, 38 pentachords, 50 hexachords in the familiar 12-tone equal temperament. Called "Forte set-classes," after Allen Forte who first cataloged them. - Jon Wild, May 21 2004
REFERENCES
Martin Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246.
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.
LINKS
Washington Bomfim, Rows n = 0..130, flattened
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
G. Gori, S. Paganelli, A. Sharma, P. Sodano, and A. Trombettoni, Bell-Paired States Inducing Volume Law for Entanglement Entropy in Fermionic Lattices, arXiv:1405.3616 [cond-mat.stat-mech], 2014. See Section V.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10(8) (1979), 964-999.
S. Karim, J. Sawada, Z. Alamgirz, and S. M. Husnine, Generating bracelets with fixed content, Theoretical Computer Science, 475 (2013), 103-112.
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n, k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From N. J. A. Sloane, Nov 26 2012
A. L. Patterson, Ambiguities in the X-Ray Analysis of Crystal Structures, Phys. Rev. 65 (1944), 195 - 201 (see Table I). [From N. J. A. Sloane, Mar 14 2009]
Richard H. Reis, A formula for C(T) in Gupta's paper, Indian J. Pure and Appl. Math., 10(8) (1979), 1000-1001.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35(5) (2004), 629-638.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35(5) (2004), 629-638.
FORMULA
T(0,0) = 1. If n > 0, T(n,k) = binomial(floor(n/2) - (k mod 2) * (1 - (n mod 2)), floor(k/2)) / 2 + Sum_{d|n, d|k} (phi(d)*binomial(n/d, k/d)) / (2*n). - Washington Bomfim, Jun 30 2012 [edited by Petros Hadjicostas, May 29 2019]
From Freddy Barrera, Apr 21 2019: (Start)
T(n,k) = T(n, n-k) for each k < n (Theorem 2 of H. Gupta). (End)
G.f. for column k >= 1: (x^k/2) * ((1/k) * Sum_{m|k} phi(m)/(1 - x^m)^(k/m) + (1 + x)/(1 - x^2)^floor((k/2) + 1)). (This formula is due to Herbert Kociemba.) - Petros Hadjicostas, May 25 2019
Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = (1/2) * ((x + 1) * (x*y + 1) / (1 - x^2 * (y^2 + 1)) + 1 - Sum_{d >= 1} (phi(d)/d) * log(1 - x^d * (1 + y^d))). - Petros Hadjicostas, Jun 13 2019
EXAMPLE
Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
1;
1, 1;
1, 1, 1;
1, 1, 1, 1;
1, 1, 2, 1, 1;
1, 1, 2, 2, 1, 1;
1, 1, 3, 3, 3, 1, 1;
1, 1, 3, 4, 4, 3, 1, 1;
1, 1, 4, 5, 8, 5, 4, 1, 1;
1, 1, 4, 7, 10, 10, 7, 4, 1, 1;
1, 1, 5, 8, 16, 16, 16, 8, 5, 1, 1;
1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1;
1, 1, 6, 12, 29, 38, 50, 38, 29, 12, 6, 1, 1;
...
MAPLE
A052307 := proc(n, k)
local hk, a, d;
if k = 0 then
return 1 ;
end if;
hk := k mod 2 ;
a := 0 ;
for d in numtheory[divisors](igcd(k, n)) do
a := a+ numtheory[phi](d)*binomial(n/d-1, k/d-1) ;
end do:
%/k + binomial(floor((n-hk)/2), floor(k/2)) ;
%/2 ;
end proc: # R. J. Mathar, Sep 04 2011
MATHEMATICA
Table[If[m*n===0, 1, 1/2*If[EvenQ[n], If[EvenQ[m], Binomial[n/2, m/2], Binomial[(n-2)/2, (m-1)/2 ]], If[EvenQ[m], Binomial[(n-1)/2, m/2], Binomial[(n-1)/2, (m-1)/2]]] + 1/2*Fold[ #1 +(EulerPhi[ #2]*Binomial[n/#2, m/#2])/n &, 0, Intersection[Divisors[n], Divisors[m]]]], {n, 0, 12}, {m, 0, n}] (* Wouter Meeussen, Aug 05 2002, Jan 19 2009 *)
PROG
(PARI)
B(n, k)={ if(n==0, return(1)); GCD = gcd(n, k); S = 0;
for(d = 1, GCD, if((k%d==0)&&(n%d==0), S+=eulerphi(d)*binomial(n/d, k/d)));
return (binomial(floor(n/2)- k%2*(1-n%2), floor(k/2))/2 + S/(2*n)); }
n=0; k=0; for(L=0, 8645, print(L, " ", B(n, k)); k++; if(k>n, k=0; n++))
/* Washington Bomfim, Jun 30 2012 */
(Python)
from sympy import binomial as C, totient, divisors, gcd
def T(n, k): return 1 if n==0 else C((n//2) - k%2 * (1 - n%2), (k//2))/2 + sum(totient(d)*C(n//d, k//d) for d in divisors(gcd(n, k)))/(2*n)
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 23 2017
CROSSREFS
KEYWORD
AUTHOR
Christian G. Bower, Nov 15 1999
STATUS
approved