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A367969 Number of partitions of [n] whose block maxima sum to k, where k is chosen so as to maximize this number.

%I A367969 #21 Dec 16 2023 21:23:38
%S A367969 1,1,1,2,5,11,37,129,431,1921,9544,43844,223512,1407320,8519457,
%T A367969 52422985,373424140,2685768084,20354852852,160370778238,1318493838635,
%U A367969 11239312718146,98700416575613,916309760098349,8735277842452542,84921152781222758,860903677319960583
%N A367969 Number of partitions of [n] whose block maxima sum to k, where k is chosen so as to maximize this number.
%H A367969 Alois P. Heinz, <a href="/A367969/b367969.txt">Table of n, a(n) for n = 0..200</a>
%H A367969 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A367969 a(5) = 11 = A367955(5,12) is the largest value in row 5 of A367955 and counts the partitions of [5] having block maxima sum 12: 123|4|5, 124|3|5, 125|3|4, 13|24|5, 13|25|4, 14|23|5, 15|23|4, 14|25|3, 15|24|3, 1|2|34|5, 1|2|35|4.
%p A367969 b:= proc(n, m) option remember; `if`(n=0, 1,
%p A367969       b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
%p A367969     end:
%p A367969 a:= n-> max(coeffs(b(n, 0))):
%p A367969 seq(a(n), n=0..30);
%p A367969 # second Maple program:
%p A367969 b:= proc(n, i, t) option remember; `if`(i*(i+1)/2<n, 0,
%p A367969       `if`(n=0, t^i, `if`(t=0, 0, t*b(n, i-1, t))+
%p A367969        (t+1)^max(0, 2*i-n-1)*b(n-i, min(n-i, i-1), t+1)))
%p A367969     end:
%p A367969 a:= n-> max(seq(b(k, n, 0), k=n..n*(n+1)/2)):
%p A367969 seq(a(n), n=0..30);
%t A367969 b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m]*m + Expand[x^n*b[n-1, m+1]]];
%t A367969 a[n_] := Max[CoefficientList[b[n, 0], x]];
%t A367969 Table[a[n], {n, 0, 30}]
%t A367969 (* second program: *)
%t A367969 b[n_, i_, t_] := b[n, i, t] = If[i*(i + 1)/2 < n, 0, If[n == 0, t^i, If[t == 0, 0, t*b[n, i - 1, t]] + (t + 1)^Max[0, 2*i - n - 1]*b[n - i, Min[n - i, i - 1], t + 1]]];
%t A367969 a[n_] := If[n == 0, 1, Max[Table[b[k, n, 0], { k, n, n*(n + 1)/2}]]];
%t A367969 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Dec 13 2023, after _Alois P. Heinz_ *)
%Y A367969 Row maxima of A367955.
%Y A367969 Cf. A365441, A365903.
%K A367969 nonn
%O A367969 0,4
%A A367969 _Alois P. Heinz_, Dec 06 2023