A327673 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 1, 3, 0, 3, 18, 19, 0, 3, 60, 171, 121, 0, 5, 210, 1173, 1996, 1041, 0, 11, 798, 7512, 22784, 27225, 11191, 0, 13, 2462, 39708, 196904, 411115, 382086, 130663, 0, 19, 7891, 204987, 1546042, 4991815, 7843848, 5932843, 1731969
Offset: 0
Examples
T(3,1) = 3: 3aaa, 2aa1a, 1a2aa. T(3,2) = 18: 3aab, 3aba, 3baa, 3abb, 3bab, 3bba, 2aa1b, 2ab1a, 2ba1a, 2ab1b, 2ba1b, 2bb1a, 1a2ab, 1a2ba, 1a2bb, 1b2aa, 1b2ab, 1b2ba. T(3,3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c. Triangle T(n,k) begins: 1; 0, 1; 0, 1, 3; 0, 3, 18, 19; 0, 3, 60, 171, 121; 0, 5, 210, 1173, 1996, 1041; 0, 11, 798, 7512, 22784, 27225, 11191; 0, 13, 2462, 39708, 196904, 411115, 382086, 130663; ...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
Crossrefs
Programs
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Maple
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)* b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i))) end: T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*binomial(k, i), i=0..k): seq(seq(T(n, k), k=0..n), n=0..10); -
Mathematica
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n==0, p!, If[i<1, 0, Sum[Binomial[ k^i, j] b[n - i j, Min[n - i j, i - 1], k, p + j]/j!, {j, 0, n/i}]]]; T[n_, k_] := Sum[b[n, n, i, 0] (-1)^(k - i) Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 30 2020, after Maple *)
Formula
Sum_{k=1..n} k * T(n,k) = A327676(n).