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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Wikipedia, Partition of a set

Maple

b:= proc(n, i, t, m) option remember; `if`(n=0, t^(m-i+1),
     `if`((i+m)*(m+1-i)/2<n or i>n, 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 *)

Crossrefs

Row maxima of A124327.

Cf. A367969.

Sequence in context: A261041 A047051 A271077 * A126349 A076315 A304124

Adjacent sequences: A365900 A365901 A365902 * A365904 A365905 A365906

Keyword

nonn

Author

Alois P. Heinz, Dec 14 2023

Status

approved