A241719 Number T(n,k) of compositions of n into distinct parts with exactly k descents; triangle T(n,k), n>=0, 0<=k<=max(floor((sqrt(1+8*n)-3)/2),0), read by rows.

1, 1, 1, 2, 1, 2, 1, 3, 2, 4, 6, 1, 5, 7, 1, 6, 11, 2, 8, 16, 3, 10, 31, 15, 1, 12, 36, 16, 1, 15, 55, 29, 2, 18, 71, 41, 3, 22, 101, 65, 5, 27, 147, 144, 32, 1, 32, 188, 179, 35, 1, 38, 245, 269, 63, 2, 46, 327, 382, 93, 3, 54, 421, 549, 148, 5, 64, 540, 739, 205, 7

Offset: 0

Author

Alois P. Heinz, Apr 27 2014

Keywords

• nonn
• tabf
• look

Examples

			T(6,0) = 4: [6], [2,4], [1,5], [1,2,3].
T(6,1) = 6: [5,1], [4,2], [3,1,2], [1,3,2], [2,1,3], [2,3,1].
T(6,2) = 1: [3,2,1].
T(7,0) = 5: [7], [3,4], [2,5], [1,6], [1,2,4].
T(7,1) = 7: [6,1], [4,3], [5,2], [2,1,4], [1,4,2], [2,4,1], [4,1,2].
T(7,2) = 1: [4,2,1].
Triangle T(n,k) begins:
00:   1;
01:   1;
02:   1;
03:   2,   1;
04:   2,   1;
05:   3,   2;
06:   4,   6,   1;
07:   5,   7,   1;
08:   6,  11,   2;
09:   8,  16,   3;
10:  10,  31,  15,  1;
11:  12,  36,  16,  1;
12:  15,  55,  29,  2;
13:  18,  71,  41,  3;
14:  22, 101,  65,  5;
15:  27, 147, 144, 32, 1;
		

Links

• Alois P. Heinz, Rows n = 0..500, flattened

Crossrefs

Columns k=0-10 give: A000009, A241720, A241721, A241722, A241723, A241724, A241725, A241726, A241727, A241728, A241729.

Row sums give A032020.

T(A000217(k+1)-1,k-1) = A000041(k) for k>0.

Cf. A052146.

Programs

Maple

g:= proc(u, o) option remember; `if`(u+o=0, 1, expand(
      add(g(u+j-1, o-j)  , j=1..o)+
      add(g(u-j, o+j-1)*x, j=1..u)))
    end:
b:= proc(n, i) option remember; local m; m:= i*(i+1)/2;
      `if`(n>m, 0, `if`(n=m, x^i,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
T:= n-> (p-> (q-> seq(coeff(q, x, i), i=0..degree(q)))(add(
         coeff(p, x, k)*g(0, k), k=0..degree(p))))(b(n$2)):
seq(T(n), n=0..20);

Mathematica

g[u_, o_] := g[u, o] = If[u+o == 0, 1, Expand[Sum[g[u+j-1, o-j], {j, 1, o}] + Sum[g[u-j, o+j-1]*x, {j, 1, u}]]]; b[n_, i_] := b[n, i] = Module[{m}, m = i*(i+1)/2; If[n>m, 0, If[n == m, x^i, Expand[b[n, i-1] + If[i>n, 0, x*b[n-i, i-1]]]]]]; T[n_] := Function [p, Function[q, Table[Coefficient[q, x, i], {i, 0, Exponent[q, x]}]][Sum[Coefficient[p, x, k]*g[0, k], {k, 0, Exponent[p, x]}]]][b[n, n]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Apr 28 2014, after Alois P. Heinz *)