A059355 Number of chains of n-3 partitions in the reduced partition lattice on n elements.

1, 13, 205, 4245, 114345, 3919860, 167310360, 8719666200, 545594049000, 40394317194000, 3494634235092000, 349446163958892000, 40005208010427660000, 5199553600938496800000, 761551300698921532800000, 124863678342008772566400000, 22782147644564103946550400000

Offset: 3

Author

N. J. A. Sloane, Jan 27 2001

Keywords

• nonn

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

Examples

			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)
		

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..269

Crossrefs

A diagonal of triangle in A008826.

Programs

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 *)

Extensions

More terms from Vladeta Jovovic, Jan 02 2004

Name changed by Harry Richman, Mar 30 2023