A327673 - OEIS

%I #18 May 30 2020 09:21:09

%S 1,0,1,0,1,3,0,3,18,19,0,3,60,171,121,0,5,210,1173,1996,1041,0,11,798,

%T 7512,22784,27225,11191,0,13,2462,39708,196904,411115,382086,130663,0,

%U 19,7891,204987,1546042,4991815,7843848,5932843,1731969

%N 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.

%H Alois P. Heinz, <a href="/A327673/b327673.txt">Rows n = 0..140, flattened</a>

%F Sum_{k=1..n} k * T(n,k) = A327676(n).

%e T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.

%e T(3,2) = 18: 3aab, 3aba, 3baa, 3abb, 3bab, 3bba, 2aa1b, 2ab1a, 2ba1a, 2ab1b, 2ba1b, 2bb1a, 1a2ab, 1a2ba, 1a2bb, 1b2aa, 1b2ab, 1b2ba.

%e T(3,3) = 19: 3abc, 3acb, 3bac, 3bca, 3cab, 3cba, 2ab1c, 2ac1b, 2ba1c, 2bc1a, 2ca1b, 2cb1a, 1a2bc, 1a2cb, 1b2ac, 1b2ca, 1c2ab, 1c2ba, 1a1b1c.

%e Triangle T(n,k) begins:

%e   1;

%e   0,  1;

%e   0,  1,    3;

%e   0,  3,   18,    19;

%e   0,  3,   60,   171,    121;

%e   0,  5,  210,  1173,   1996,   1041;

%e   0, 11,  798,  7512,  22784,  27225,  11191;

%e   0, 13, 2462, 39708, 196904, 411115, 382086, 130663;

%e   ...

%p b:= proc(n, i, k, p) option remember;

%p      `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*

%p       b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))

%p     end:

%p T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*binomial(k, i), i=0..k):

%p seq(seq(T(n, k), k=0..n), n=0..10);

%t 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 T[n_, k_] := Sum[b[n, n, i, 0] (-1)^(k - i) Binomial[k, i], {i, 0, k}];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 30 2020, after Maple *)

%Y Columns k=0-2 give: A000007, A032020 (for n>0), A327768.

%Y Main diagonal gives A327674.

%Y Row sums give A327675.

%Y T(2n,n) gives A327678.

%Y Cf. A327244, A327676.

%K nonn,tabl

%O 0,6

%A _Alois P. Heinz_, Sep 21 2019