OFFSET
1,8
COMMENTS
Triangle obtained from A047996 by dropping the first column (k=0), and row (n=0).
T(n, k) = number of different ways the number n can be expressed as ordered sums of k positive integers, counting only once those ordered sums that can be transformed into each other by a cyclic permutation.
These might be described as cyclic compositions, or more loosely as cyclic partitions. - N. J. A. Sloane, Sep 05 2012
REFERENCES
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.
LINKS
Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Ethan Akin and Morton Davis, Bulgarian solitaire, Am. Math. Monthly 92 (4) (1985) 237-250.
R. Baumann, Computer-Knobelei, , LOGIN, 163/164 (2010), 141-142 (in German).
R. Bekes, J. Pedersen and B. Shao, Mad tea party cyclic partitions, College Math. J., 43 (2012), 24-36.
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
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), Article 16.8.2.
Arnold Knopfmacher and Neville Robbins, Some Properties of Cyclic Compositions, Fibonacci Quart. 48 (2010), no. 3, 249-255.
D. M. Y. Sommerville, On Certain Periodic Properties of Cyclic Compositions of Numbers, Proc. London Math. Soc., s2-7, No. 1 (1909), 263-313.
R. Razen, J. Seberry, and K. Wehrhahn, Ordered partitions and codes generated by circulant matrices, J. Combin. Theory Ser. A, 27 (1979), 333-341.
D. Wasserman, Proof of the symmetry [Broken link]
D. Wasserman, Proof of the symmetry [Cached copy]
FORMULA
T(n,k) = (Sum_{d|gcd(n,k)} phi(d)*binomial(n/d,k/d))/n, where phi = A000010 = Euler's totient function. Also T(n,k) = A047996(n,k). - Paul Weisenhorn, Apr 06 2011
EXAMPLE
Triangle begins
1;
1, 1;
1, 1, 1;
1, 2, 1, 1;
1, 2, 2, 1, 1;
1, 3, 4, 3, 1, 1;
1, 3, 5, 5, 3, 1, 1;
1, 4, 7, 10, 7, 4, 1, 1;
1, 4, 10, 14, 14, 10, 4, 1, 1;
1, 5, 12, 22, 26, 22, 12, 5, 1, 1;
1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1;
T(6,3) = 4, since there are 4 essentially different ways 1+1+4, 1+2+3, 1+3+2 and 2+2+2 of expressing 6 as a sum of 3 summands (all others can be obtained by cyclically permuting the summands in one of the above sums).
MAPLE
A037306 := proc(n, k) local a, d; a := 0; for d in numtheory[divisors]( igcd(n, k)) do a := a+numtheory[phi](d)*binomial(n/d, k/d); end do: a/n; end proc:
seq(seq(A037306(n, k), k=1..n), n=1..20); # R. J. Mathar, Jun 11 2011
MATHEMATICA
t[n_, k_] := Total[EulerPhi[#]*Binomial[n/#, k/#] & /@ Divisors[GCD[n, k]]]/n; Flatten[Table[t[n, k], {n, 13}, {k, n}]] (* Jean-François Alcover, Sep 08 2011, after formula *)
nn=15; f[list_]:=Select[list, #>0&]; Map[f, Transpose[Table[Drop[CoefficientList[Series[CycleIndex[CyclicGroup[n], s]/.Table[s[i]->x^i/(1-x^i), {i, 1, n}], {x, 0, nn}], x], 1], {n, 1, nn}]]]//Grid (* Geoffrey Critzer, Oct 30 2012 *)
PROG
(Haskell)
a037306 n k = div (sum $ map f $ a027750_row $ gcd n k) n where
f d = a000010 d * a007318' (div n d) (div k d)
a037306_row n = map (a037306 n) [1..n]
a037306_tabl = map a037306_row [1..]
-- Reinhard Zumkeller, Feb 06 2014
(PARI) T(n, k) = sumdiv(gcd(n, k), d, eulerphi(d)*binomial(n/d, k/d))/n; \\ Michel Marcus, Feb 10 2016
CROSSREFS
KEYWORD
AUTHOR
Jens Voß, Jun 30 2001
EXTENSIONS
More terms from David Wasserman, Mar 11 2002
Comments, reference, example from Paul Weisenhorn, Dec 18 2010
STATUS
approved