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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.

%I A372722 #21 May 12 2024 14:21:10
%S A372722 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,
%T A372722 175,105,0,0,0,1,0,2809,805,420,105,0,0,0,1,0,14608,4053,1540,945,0,0,
%U A372722 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
%N 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.
%H A372722 Alois P. Heinz, <a href="/A372722/b372722.txt">Rows n = 0..140, flattened</a>
%H A372722 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A372722 Sum_{k=0..n} k * T(n,k) = A372649(n).
%e A372722 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.
%e A372722 T(4,2) = 3: 12|34, 13|24, 14|23.
%e A372722 T(4,3) = 0.
%e A372722 T(4,4) = 1: 1|2|3|4.
%e A372722 Triangle T(n,k) begins:
%e A372722   1;
%e A372722   0,     1;
%e A372722   0,     1,     1;
%e A372722   0,     4,     0,    1;
%e A372722   0,    11,     3,    0,    1;
%e A372722   0,    36,    15,    0,    0,   1;
%e A372722   0,   132,    55,   15,    0,   0, 1;
%e A372722   0,   596,   175,  105,    0,   0, 0, 1;
%e A372722   0,  2809,   805,  420,  105,   0, 0, 0, 1;
%e A372722   0, 14608,  4053, 1540,  945,   0, 0, 0, 0, 1;
%e A372722   0, 79448, 24906, 5950, 4725, 945, 0, 0, 0, 0, 1;
%e A372722   ...
%p A372722 b:= proc(n, m, t) option remember; `if`(n=0, x^t,
%p A372722       add(binomial(n-1, j-1)*b(n-j, max(j, m),
%p A372722      `if`(j>m, 1, `if`(j=m, t+1, t))), j=1..n))
%p A372722     end:
%p A372722 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0$2)):
%p A372722 seq(T(n), n=0..12);
%Y A372722 Columns k=0-1 give: A000007, A372721.
%Y A372722 Row sums give A000110.
%Y A372722 T(2n,n) gives A001147.
%Y A372722 T(3n,n) gives A271715.
%Y A372722 Cf. A372649, A372762.
%K A372722 nonn,tabl
%O A372722 0,8
%A A372722 _Alois P. Heinz_, May 11 2024