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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.
15
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
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
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 15 2014
STATUS
approved