OFFSET
1,5
COMMENTS
Turning over the necklace is not allowed (the group is cyclic not dihedral). T(n,k) = T(k,n) follows immediately from the formula. - N. J. A. Sloane, May 03 2014
T(n, k) is the number of equivalence classes of k-tuples of residues modulo n, identifying those that differ componentwise by a constant and those that differ by a permutation. - Álvar Ibeas, Sep 21 2021
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233--238. MR1691428 (2000c:13006)
A. Elashvili, M. Jibladze and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173--188. MR1719140 (2000j:05009). See p. 174. - N. J. A. Sloane, Aug 06 2014
N. J. A. Sloane, A Note on Modular Partitions and Necklaces
FORMULA
T(n,k) = (Sum_{d | gcd(n,k)} phi(d)*binomial((n+k)/d, n/d))/(n+k). [Corrected by N. J. A. Sloane, May 03 2014]
EXAMPLE
The table starts:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, ...
1, 2, 4, 5, 7, 10, 12, 15, 19, 22, 26, 31, ...
1, 3, 5, 10, 14, 22, 30, 43, 55, 73, 91, 116, ...
1, 3, 7, 14, 26, 42, 66, 99, 143, 201, 273, 364, ...
1, 4, 10, 22, 42, 80, 132, 217, 335, 504, 728, 1038, ...
...
MAPLE
# Maple program for the table - N. J. A. Sloane, May 03 2014:
with(numtheory);
T:=proc(n, k) local d, s, g, t0;
t0:=0; s:=n+k; g:=gcd(n, k);
for d from 1 to s do
if (g mod d) = 0 then t0:=t0+phi(d)*binomial(s/d, k/d); fi;
od: t0/s; end;
r:=n->[seq(T(n, k), k=1..12)];
[seq(r(n), n=1..12)];
MATHEMATICA
T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#] Binomial[(n+k)/#, n/#]& ]/ (n+k); Table[T[n-k+1, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 02 2015 *)
PROG
(PARI) T(n, k) = sumdiv(gcd(n, k), d, eulerphi(d)*binomial((n+k)\d, n\d))/(n+k)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Franklin T. Adams-Watters, May 02 2014
EXTENSIONS
Edited by N. J. A. Sloane, May 03 2014
Elashvili et al. references supplied by Vladimir Popov, May 17 2014
STATUS
approved