A242451 Number T(n,k) of compositions of n in which the minimal 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, 6, 1, 0, 1, 0, 15, 0, 0, 0, 1, 0, 23, 7, 1, 0, 0, 1, 0, 53, 10, 0, 0, 0, 0, 1, 0, 94, 32, 0, 1, 0, 0, 0, 1, 0, 203, 31, 21, 0, 0, 0, 0, 0, 1, 0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1, 0, 855, 77, 91, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1648, 222, 105, 71, 0, 1, 0, 0, 0, 0, 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 smaller than k.

T(n,n) = T(2n,n) = 1.

T(3n,n) = A244174(n).

Links

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

Example

T(5,1) = 15: [1,1,1,2], [1,1,2,1], [1,2,1,1], [2,1,1,1], [1,2,2], [2,1,2], [2,2,1], [1,1,3], [1,3,1], [3,1,1], [2,3], [3,2], [1,4], [4,1], [5].
T(6,2) = 7: [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].
T(6,3) = 1: [2,2,2].
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,  1;
  0,   3,  0,  1;
  0,   6,  1,  0, 1;
  0,  15,  0,  0, 0, 1;
  0,  23,  7,  1, 0, 0, 1;
  0,  53, 10,  0, 0, 0, 0, 1;
  0,  94, 32,  0, 1, 0, 0, 0, 1;
  0, 203, 31, 21, 0, 0, 0, 0, 0, 1;
  0, 404, 71, 35, 0, 1, 0, 0, 0, 0, 1;

Maple

b:= proc(n, i, p, k) option remember; `if`(n=0, p!, `if`(i<1, 0,
       b(n, i-1, p, k) +add(b(n-i*j, i-1, p+j, k)/j!,
       j=max(1, k)..floor(n/i))))
    end:
T:= (n, k)-> b(n$2, 0, k) -`if`(n=0 and 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, b[n, i - 1, p, k] + Sum[b[n - i*j, i - 1, p + j, k]/j!, {j, Max[1, k], Floor[n/i]}]]]; T[n_, k_] := b[n, n, 0, k] - If[n == 0 && 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 27 2015, after Alois P. Heinz *)

Crossrefs

Columns k=0-10 give: A000007, A244164, A244165, A244166, A244167, A244168, A244169, A244170, A244171, A244172, A244173.

Row sums give A011782.

Cf. A242447 (the same for maximal multiplicity), A243978 (the same for partitions).

Sequence in context: A187253 A022904 A238341 * A363978 A262964 A135481

Adjacent sequences: A242448 A242449 A242450 * A242452 A242453 A242454

Keyword

nonn,tabl,look

Author

Alois P. Heinz, May 15 2014

Status

approved