OFFSET
0,5
COMMENTS
A refinement of A227682.
A colored composition of n is defined as a composition of n where each part p comes in one of p colors (denoted by an integer from 1 to p) and the color numbers are nondecreasing through the composition. The color numbers thus form a partition, called the color partition, of some integer.
For example, the composition 1 + 3 + 2 of 6 gives rise to three colored compositions of 6, namely, 1(c1) + 3(c1) + 2(c1), 1(c1) + 3(c1) + 2(c2) and 1(c1) + 3(c2) + 2(c2), where the color number of a part is shown after the part prefaced by the letter c.
T(n,k) equals the number of colored compositions of n into k parts.
See A253830 for the enumeration of colored compositions having parts with distinct colors.
LINKS
FORMULA
G.f.: G(x,z) := Product_{n >= 1} (1 - z)/(1 - z - x*z^n) = exp( Sum_{n >= 1} (x*z)^n/(n*(1 - z)^n*(1 - z^n)) ) =
1 + Sum_{n >= 1} (x*z/(1 - z))^n/( Product_{i = 1..n} 1 - z^i ) = 1 + x*z + (2*x + x^2)*z^2 + (3*x + 3*x^2 + x^3)*z^3 + ....
Note, G(x*(1 - z),z) is the generating function of A008284.
T(n,k) = Sum_{i = k..n} binomial(i-1,k-1)*A008284(n+k-i,k).
Recurrence equation: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-k,k) - T(n-k-1,k) with boundary conditions T(n,n) = 1, T(n,0) = 0 for n >= 1 and T(n,k) = 0 for n < k.
Row sums are A227682.
EXAMPLE
Triangle begins
n\k| 0 1 2 3 4 5 6 7
= = = = = = = = = = = = = = = = =
0 | 1
1 | 0 1
2 | 0 2 1
3 | 0 3 3 1
4 | 0 4 7 4 1
5 | 0 5 13 11 5 1
6 | 0 6 22 25 16 6 1
7 | 0 7 34 50 41 22 7 1
...
T(4,2) = 7: The compositions of 4 into two parts are 2 + 2, 1 + 3 and 3 + 1. Coloring the parts as described above produces seven colored compositions of 4 into two parts:
2(c1) + 2(c1), 2(c1) + 2(c2), 2(c2) + 2(c2),
1(c1) + 3(c1), 1(c1) + 3(c2), 1(c1) + 3(c3),
3(c1) + 1(c1).
MAPLE
G := 1/(product(1-x*z^j/(1-z), j = 1 .. 12)): Gser := simplify(series(G, z = 0, 14)): for n to 12 do P[n] := coeff(Gser, z^n) end do: for n to 12 do seq(coeff(P[n], x^j), j = 1 .. n) end do;
CROSSREFS
KEYWORD
AUTHOR
Peter Bala, Jan 19 2015
STATUS
approved