A367850 Total sum of the block maxima minus the block minima over all partitions of [n].

0, 0, 1, 6, 33, 182, 1034, 6122, 37927, 246030, 1669941, 11844324, 87644672, 675494180, 5413500801, 45040155758, 388441330457, 3467619369538, 31998729152474, 304846692965822, 2994781617653439, 30304301968015582, 315536869771786501, 3377398077726963112

Offset: 0

Author

Alois P. Heinz, Dec 15 2023

Keywords

• nonn

Examples

			a(3) = 6 = 2 + 1 + 2 + 1 + 0: 123, 12|3, 13|2, 1|23, 1|2|3.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..574
• Wikipedia, Partition of a set

Crossrefs

Cf. A000110, A002538 (the same for permutations), A002620, A120325, A124325, A278677, A368338.

Programs

Maple

b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], (p->
      p+[0, p[1]*(n-t)])(b(n-1, m+1, t+1))+m*b(n-1, m, t+1))
    end:
a:= n-> b(n, 0, 1)[2]:
seq(a(n), n=0..23);
# second Maple program:
egf:= (z-2)*exp(2*z+exp(z)-1)+(2*z+1)*exp(z+exp(z)-1)+exp(exp(z)-1):
a:= n-> n!*coeff(series(egf, z, n+1), z, n):
seq(a(n), n=0..23);

Formula

E.g.f.: (z-2)*exp(2*z+exp(z)-1)+(2*z+1)*exp(z+exp(z)-1)+exp(exp(z)-1).

a(n) = A278677(n-1) - A124325(n+1) for n>=1.

a(n) = Bell(n+1)+(n+1)*Bell(n)-Bell(n+2)+Sum_{k=0..n} Stirling2(n+1,k)*(n+1-k).

a(n) = Sum_{k=0..A002620(n)} k * A368338(n,k).

a(n) mod 2 = A120325(n).