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A059355
Number of chains of n-3 partitions in the reduced partition lattice on n elements.
2
1, 13, 205, 4245, 114345, 3919860, 167310360, 8719666200, 545594049000, 40394317194000, 3494634235092000, 349446163958892000, 40005208010427660000, 5199553600938496800000, 761551300698921532800000, 124863678342008772566400000, 22782147644564103946550400000
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
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
A diagonal of triangle in A008826.
Sequence in context: A215621 A299960 A194727 * A243783 A206348 A240393
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