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

1, 1, 1, 2, 5, 11, 37, 129, 431, 1921, 9544, 43844, 223512, 1407320, 8519457, 52422985, 373424140, 2685768084, 20354852852, 160370778238, 1318493838635, 11239312718146, 98700416575613, 916309760098349, 8735277842452542, 84921152781222758, 860903677319960583

Offset: 0

Author

Alois P. Heinz, Dec 06 2023

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Wikipedia, Partition of a set

Crossrefs

Row maxima of A367955.

Cf. A365441, A365903.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0, 1,
      b(n-1, m)*m + expand(x^n*b(n-1, m+1)))
    end:
a:= n-> max(coeffs(b(n, 0))):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 max(seq(b(k, n, 0), k=n..n*(n+1)/2)):
seq(a(n), n=0..30);

Mathematica

b[n_, m_] := b[n, m] = If[n == 0, 1, b[n-1, m]*m + Expand[x^n*b[n-1, m+1]]];
a[n_] := Max[CoefficientList[b[n, 0], x]];
Table[a[n], {n, 0, 30}]
(* second program: *)
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]]];
a[n_] := If[n == 0, 1, Max[Table[b[k, n, 0], { k, n, n*(n + 1)/2}]]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 13 2023, after Alois P. Heinz *)