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

0,4

Links

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

Example

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;

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 *)

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.

Sequence in context: A161282 A226517 A185214 * A269572 A029198 A029175

Adjacent sequences: A241716 A241717 A241718 * A241720 A241721 A241722

Keyword

nonn,tabf,look

Author

Alois P. Heinz, Apr 27 2014

Status

approved