%I #23 Sep 14 2018 03:46:15
%S 1,0,1,0,1,1,0,3,10,7,0,3,15,21,9,0,5,40,96,92,31,0,11,183,832,1562,
%T 1305,403,0,13,266,1539,3908,4955,3090,757,0,19,549,4281,14791,26765,
%U 26523,13671,2873,0,27,1056,10902,50208,124450,178456,148638,66904,12607
%N Number T(n,k) of compositions of n into distinct parts where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order and all k letters occur at least once in the composition; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C Also number of matrices with k rows of nonnegative integer entries and without zero rows or columns such that the sum of all entries is equal to n and the column sums are distinct.
%H Alois P. Heinz, <a href="/A261836/b261836.txt">Rows n = 0..140, flattened</a>
%F T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A261835(n,k-i).
%e T(3,2) = 10: (matrices and corresponding marked compositions are given)
%e [2] [1] [2 0] [0 2] [1 0] [0 1] [1 1] [1 1] [1 0] [0 1]
%e [1] [2] [0 1] [1 0] [0 2] [2 0] [1 0] [0 1] [1 1] [1 1]
%e 3aab, 3abb, 2aa1b, 1b2aa, 1a2bb, 2bb1a, 2ab1a, 1a2ab, 2ab1b, 1b2ab.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 3, 10, 7;
%e 0, 3, 15, 21, 9;
%e 0, 5, 40, 96, 92, 31;
%e 0, 11, 183, 832, 1562, 1305, 403;
%e 0, 13, 266, 1539, 3908, 4955, 3090, 757;
%e 0, 19, 549, 4281, 14791, 26765, 26523, 13671, 2873;
%p b:= proc(n, i, p, k) option remember;
%p `if`(i*(i+1)/2<n, 0, `if`(n=0, p!, b(n, i-1, p, k)+
%p `if`(i>n, 0, b(n-i, i-1, p+1, k)*binomial(i+k-1, k-1))))
%p end:
%p T:= (n, k)-> add(b(n$2, 0, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t b[n_, i_, p_, k_] := b[n, i, p, k] = If[i*(i+1)/2<n, 0, If[n==0, p!, b[n, i -1, p, k] + If[i>n, 0, b[n-i, i-1, p+1, k]*Binomial[i+k-1, k-1]]]]; T[n_, k_] := Sum[b[n, n, 0, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[ Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Feb 21 2016, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A032020 (for n>0), A261853, A261854, A261855, A261856, A261857, A261858, A261859, A261860, A261861.
%Y Main diagonal gives A032011.
%Y Row sums give A261838.
%Y T(2n,n) gives A261828.
%Y Cf. A261781, A261835.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, Sep 02 2015