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A372722
Number T(n,k) of partitions of [n] having exactly k blocks of maximal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
5
1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 3, 0, 1, 0, 36, 15, 0, 0, 1, 0, 132, 55, 15, 0, 0, 1, 0, 596, 175, 105, 0, 0, 0, 1, 0, 2809, 805, 420, 105, 0, 0, 0, 1, 0, 14608, 4053, 1540, 945, 0, 0, 0, 0, 1, 0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1, 0, 461748, 151371, 37730, 17325, 10395, 0, 0, 0, 0, 0, 1
OFFSET
0,8
LINKS
FORMULA
Sum_{k=0..n} k * T(n,k) = A372649(n).
EXAMPLE
T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
T(4,2) = 3: 12|34, 13|24, 14|23.
T(4,3) = 0.
T(4,4) = 1: 1|2|3|4.
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 4, 0, 1;
0, 11, 3, 0, 1;
0, 36, 15, 0, 0, 1;
0, 132, 55, 15, 0, 0, 1;
0, 596, 175, 105, 0, 0, 0, 1;
0, 2809, 805, 420, 105, 0, 0, 0, 1;
0, 14608, 4053, 1540, 945, 0, 0, 0, 0, 1;
0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1;
...
MAPLE
b:= proc(n, m, t) option remember; `if`(n=0, x^t,
add(binomial(n-1, j-1)*b(n-j, max(j, m),
`if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0$2)):
seq(T(n), n=0..12);
CROSSREFS
Columns k=0-1 give: A000007, A372721.
Row sums give A000110.
T(2n,n) gives A001147.
T(3n,n) gives A271715.
Sequence in context: A363973 A046781 A244530 * A271424 A117435 A282252
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 11 2024
STATUS
approved