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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.
6
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
0,6
LINKS
FORMULA
Sum_{k=1..n} k * T(n,k) = A327676(n).
EXAMPLE
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;
...
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 *)
CROSSREFS
Columns k=0-2 give: A000007, A032020 (for n>0), A327768.
Main diagonal gives A327674.
Row sums give A327675.
T(2n,n) gives A327678.
Sequence in context: A275080 A128252 A230675 * A279657 A272722 A229694
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 21 2019
STATUS
approved