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

0,8

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.

Links

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

Example

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;

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 *)

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).

Sequence in context: A035640 A079327 A340504 * A238342 A123878 A284148

Adjacent sequences: A242444 A242445 A242446 * A242448 A242449 A242450

Keyword

nonn,tabl

Author

Alois P. Heinz, May 15 2014

Status

approved