A368675 Number of partitions of [n] whose block maxima sum to 2n.

1, 0, 0, 1, 2, 7, 15, 39, 81, 193, 396, 885, 1816, 3915, 7973, 16860, 34165, 71092, 143804, 295963, 596872, 1219950, 2455139, 4989265, 10028841, 20296288, 40745616, 82225558, 164916967, 332045545, 665566046, 1337794545, 2680049287, 5380396625, 10774301183

Offset: 0

Author

Alois P. Heinz, Jan 02 2024

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(3) = 1: 1|2|3.
a(4) = 2: 1|23|4, 1|24|3.
a(5) = 7: 12|3|45, 13|2|45, 1|234|5, 1|235|4, 145|2|3, 1|24|35, 1|25|34.
a(6) = 15: 12|34|56, 12|356|4, 134|2|56, 1356|2|4, 1|2345|6, 1|2346|5, 1|235|46, 1|236|45, 14|2|356, 1|245|36, 1|246|35, 156|2|34, 1|25|346, 1|26|345, 1|2|3|456.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3322
• Wikipedia, Partition of a set

Crossrefs

Cf. A367955, A365441, A368678.

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-> coeff(b(n, 0), x, 2*n):
seq(a(n), n=0..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 b(2*n, n, 0):
seq(a(n), n=0..42);

Mathematica

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, b[2n, n, 0]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Oct 03 2024, after Alois P. Heinz *)

Formula

a(n) = A367955(n,2n).

a(n) ~ c * 2^n, where c = 0.636808431228827742738441592748953932083264824206324529619378074873607293... - Vaclav Kotesovec, Jan 13 2024