A242447 Number T(n,k) of compositions of n in which the maximal multiplicity of parts equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 3, 4, 0, 1, 0, 5, 6, 4, 0, 1, 0, 11, 10, 5, 5, 0, 1, 0, 13, 21, 18, 5, 6, 0, 1, 0, 19, 40, 34, 21, 6, 7, 0, 1, 0, 27, 87, 59, 40, 27, 7, 8, 0, 1, 0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1, 0, 65, 219, 272, 210, 131, 63, 44, 9, 10, 0, 1

Offset: 0

Author

Alois P. Heinz, May 15 2014

Keywords

• nonn
• tabl

Comments

T(0,0) = 1 by convention. T(n,k) counts the compositions of n in which at least one part has multiplicity k and no part has a multiplicity larger than k.

Examples

			T(6,1) = 11: [1,2,3], [1,3,2], [2,1,3], [2,3,1], [3,1,2], [3,2,1], [2,4], [4,2], [1,5], [5,1], [6].
T(6,2) = 10: [1,1,2,2], [1,2,1,2], [1,2,2,1], [2,1,1,2], [2,1,2,1], [2,2,1,1], [3,3], [1,1,4], [1,4,1], [4,1,1].
T(6,3) = 5: [2,2,2], [1,1,1,3], [1,1,3,1], [1,3,1,1], [3,1,1,1].
T(6,4) = 5: [1,1,1,1,2], [1,1,1,2,1], [1,1,2,1,1], [1,2,1,1,1], [2,1,1,1,1].
T(6,6) = 1: [1,1,1,1,1,1].
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  3,   0,   1;
  0,  3,   4,   0,   1;
  0,  5,   6,   4,   0,  1;
  0, 11,  10,   5,   5,  0,  1;
  0, 13,  21,  18,   5,  6,  0, 1;
  0, 19,  40,  34,  21,  6,  7, 0, 1;
  0, 27,  87,  59,  40, 27,  7, 8, 0, 1;
  0, 57, 121, 132, 100, 49, 35, 8, 9, 0, 1;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A032020 (for n>0), A243119, A243120, A243121, A243122, A243123, A243124, A243125, A243126, A243127.

T(2n,n) = A232665(n).

Row sums give A011782.

Cf. A242451 (the same for minimal multiplicity).

Programs

Maple

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
      add(b(n-i*j, i-1, p+j, k)/j!, j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n$2, 0, k) -`if`(k=0, 0, b(n$2, 0, k-1)):
seq(seq(T(n, k), k=0..n), n=0..14);

Mathematica

b[n_, i_, p_, k_] := b[n, i, p, k] = If[n == 0, p!, If[i<1, 0, Sum[b[n - i*j, i-1, p + j, k]/j!, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, 0, k] - If[k == 0, 0, b[n, n, 0, k-1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 22 2015, after Alois P. Heinz *)