A365903 Number of partitions of [n] whose block minima sum to k, where k is chosen so as to maximize this number.

1, 1, 1, 2, 4, 10, 29, 101, 367, 1562, 6891, 37871, 197930, 1121634, 6888085, 46190282, 323250987, 2349020516, 17897285514, 142512956148, 1178963284732, 10248806222398, 91421283039658, 847666112839362, 8100455404172267, 79925567946537362, 814508927747776069

Offset: 0

Author

Alois P. Heinz, Dec 14 2023

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Wikipedia, Partition of a set

Crossrefs

Row maxima of A124327.

Cf. A367969.

Programs

Maple

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2n, 0, `if`(t=0, 0,
      t*b(n, i+1, t, m))+ b(n-i, i+1, t+1, m)))
    end:
a:= n-> max(seq(b(k, 1, 0, n), k=0..n*(n+1)/2)):
seq(a(n), n=0..26);
# second Maple program:
a:= proc(h) option remember; local b; b:=
      proc(n, m) option remember; `if`(n=0, 1,
        b(n-1, m)*m + expand(x^(h-n+1)*b(n-1, m+1)))
      end: forget(b); max(coeffs(b(h, 0)))
    end:
seq(a(n), n=0..26);

Mathematica

Q[1, t_, s_] := t*s;
Q[n_, t_, s_] := Q[n, t, s] = s*D[Q[n-1, t, s], s] + s*t^n*Q[n-1, t, s] // Expand;
P[n_, t_] := Module[{s}, Q[n, t, s] /. s -> 1];
a[n_] := If[n == 0, 1, Module[{t}, CoefficientList[P[n, t], t] // Max]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Oct 03 2024 *)