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%I #37 Jan 02 2024 19:37:41
%S 1,1,1,1,1,1,2,1,1,1,2,5,2,3,1,1,1,2,5,10,7,7,11,3,4,1,1,1,2,5,10,23,
%T 15,23,25,37,18,14,19,4,5,1,1,1,2,5,10,23,47,39,49,81,84,129,74,78,70,
%U 87,33,23,29,5,6,1,1,1,2,5,10,23,47,103,81,129,172,261,304,431,299,325,376,317,424,196,183,144,165,52,34,41,6,7,1
%N Number T(n,k) of partitions of [n] whose block maxima sum to k, triangle T(n,k), n>=0, n<=k<=n*(n+1)/2, read by rows.
%C Rows and also columns reversed converge to A365441.
%C T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k < n or k > n*(n+1)/2.
%H Alois P. Heinz, <a href="/A367955/b367955.txt">Rows n = 0..50, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F Sum_{k=n..n*(n+1)/2} k * T(n,k) = A278677(n+1) for n>=1.
%F Sum_{k=n..n*(n+1)/2} (k-n) * T(n,k) = A200660(n) for n>=1.
%F T(n,n) = T(n,n*(n+1)/2) = 1.
%e T(4,7) = 5: 123|4, 124|3, 13|24, 14|23, 1|2|34.
%e T(5,9) = 10: 1234|5, 1235|4, 124|35, 125|34, 134|25, 135|24, 14|235, 15|234, 1|23|45, 1|245|3.
%e T(5,13) = 3: 1|23|4|5, 1|24|3|5, 1|25|3|4.
%e T(5,14) = 4: 12|3|4|5, 13|2|4|5, 14|2|3|5, 15|2|3|4.
%e T(5,15) = 1: 1|2|3|4|5.
%e Triangle T(n,k) begins:
%e 1;
%e . 1;
%e . . 1, 1;
%e . . . 1, 1, 2, 1;
%e . . . . 1, 1, 2, 5, 2, 3, 1;
%e . . . . . 1, 1, 2, 5, 10, 7, 7, 11, 3, 4, 1;
%e . . . . . . 1, 1, 2, 5, 10, 23, 15, 23, 25, 37, 18, 14, 19, 4, 5, 1;
%e ...
%p b:= proc(n, m) option remember; `if`(n=0, 1,
%p b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
%p end:
%p T:= (n, k)-> coeff(b(n, 0), x, k):
%p seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
%p # second Maple program:
%p b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
%p `if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
%p (t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
%p end:
%p T:= (n, k)-> b(k, n, 0):
%p seq(seq(T(n, k), k=n..n*(n+1)/2), n=0..10);
%Y Row sums give A000110.
%Y Column sums give A204856.
%Y Antidiagonal sums give A368102.
%Y T(2n,3n) gives A365441.
%Y T(n,2n) gives A368675.
%Y Row maxima give A367969.
%Y Row n has A000124(n-1) terms (for n>=1).
%Y Cf. A000217, A124327 (the same for block minima), A200660, A278677.
%K nonn,tabf
%O 0,7
%A _Alois P. Heinz_, Dec 05 2023
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