A037306 Triangle T(n,k) read by rows: the number of compositions of n into k parts, modulo cyclic shifts.

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, 1, 6, 19, 43, 66, 80, 66, 43, 19, 6, 1, 1, 1, 6, 22, 55, 99, 132, 132, 99, 55, 22, 6, 1

Offset: 1

Author

Jens Voß, Jun 30 2001

Keywords

• easy
• nonn
• tabl
• nice

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

Examples

			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).
		

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]

Crossrefs

A047996 and A241926 are essentially identical to this entry.

Cf. A008965 (row sums), A000010, A007318, A027750, A215251, A004526 (column 2), A007997 (column 3), A008610 (column 4), A008646 (column 5), A032191 (column 6).

See A245558, A245559 for a closely related array.

See A052307 for compositions modulo cyclic shifts and reversal.

Programs

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

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 *)

PARI

T(n, k) = sumdiv(gcd(n,k), d, eulerphi(d)*binomial(n/d, k/d))/n; \\ Michel Marcus, Feb 10 2016

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

Extensions

More terms from David Wasserman, Mar 11 2002

Comments, reference, example from Paul Weisenhorn, Dec 18 2010