OFFSET
3,2
COMMENTS
The reduced partition lattice on n elements is the lattice of set partitions ordered by refinement, with the minimum and maximum partitions removed. A chain in a lattice is a subset of lattice elements which is totally ordered. The reduced partition lattice on n elements is ranked, with rank n-2, so a maximal chain has n-2 partitions. - Harry Richman, Mar 30 2023
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 3..269
EXAMPLE
From Harry Richman, Mar 30 2023: (Start)
For n = 4, a chain of 1 partition is just a partition in the reduced partition lattice. There are 13 such partitions:
{123|4}
{124|3}
{134|2}
{1|234}
{12|34}
{13|24}
{14|23}
{12|3|4}
{13|2|4}
{14|2|3}
{1|23|4}
{1|24|3}
{1|2|34}
(End)
MAPLE
b:= proc(n) option remember; expand(`if`(n=1, 1,
add(Stirling2(n, j)*b(j)*x, j=0..n-1)))
end:
a:= n-> coeff(b(n), x, n-2):
seq(a(n), n=3..20); # Alois P. Heinz, Mar 31 2023
MATHEMATICA
a[1, _] = 1; a[n_, x_] := a[n, x] = Sum[StirlingS2[n, k]*a[k, x]*x, {k, 0, n-1}]; Table[CoefficientList[a[n, x], x][[-2]], {n, 3, 17}] (* Jean-François Alcover, Nov 28 2013, after Vladeta Jovovic *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 27 2001
EXTENSIONS
More terms from Vladeta Jovovic, Jan 02 2004
Name changed by Harry Richman, Mar 30 2023
STATUS
approved