A386497 - OEIS

A386497

Number of sets of lists of [n] such that one list is the largest.

1, 1, 2, 12, 60, 440, 3390, 33852, 338072, 4116240, 51776730, 736751180, 11075784852, 183142075272, 3157190863190, 59336602681020, 1164223828582320, 24348331444705952, 533422896546272562, 12365952739192923660, 298208300418298756460, 7570420981014167756760

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Here sets of lists are set partitions of [n] such that the elements within each block are ordered but the blocks themselves are unordered.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..444

FORMULA

E.g.f.: 1 + Sum_{j>0} x^j * exp((x - x^j)/(1 - x)).

EXAMPLE

a(3) = 12 counts: {(1),(2,3)}, {(1),(3,2)}, {(1,2),(3)}, {(1,3),(2)}, {(2),(3,1)}, {(2,1),(3)}, {(1,2,3)}, {(1,3,2)}, {(2,1,3)}, {(2,3,1)}, {(3,1,2)}, {(3,2,1)}.

MAPLE

b:= proc(n, m, t) option remember; `if`(n=0, t, add(b(n-j, max(m, j),

`if`(j>m, 1, `if`(j=m, 0, t)))*(n-1)!*j/(n-j)!, j=1..n))

end:

a:= n-> b(n, 0, 1):

seq(a(n), n=0..21); # Alois P. Heinz, Jul 23 2025

MATHEMATICA

With[{m = 21}, CoefficientList[Series[1 + Sum[x^j*Exp[(x - x^j)/(1 - x)], {j, 1, m}], {x, 0, m}], x] * Range[0, m]!] (* Amiram Eldar, Jul 24 2025 *)

PROG

(PARI) B_x(N) = {my(x='x+O('x^(N+1))); Vec(serlaplace(1+sum(j=1, N, x^j*exp((x-x^j)/(1-x)))))}

CROSSREFS

Cf. A000262, A088009, A088311, A088312, A088313, A097979, A113775, A351884, A386474.