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

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Wikipedia, Partition of a set

Example

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.

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<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)))
    end:
a:= n-> 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 *)

Crossrefs

Row maxima of A367955.

Cf. A365441, A365903.

Sequence in context: A074497 A131581 A195985 * A056301 A001344 A056302

Adjacent sequences: A367966 A367967 A367968 * A367970 A367971 A367972

Keyword

nonn

Author

Alois P. Heinz, Dec 06 2023

Status

approved