|
%I #22 Mar 31 2023 14:55:49
%S 1,13,205,4245,114345,3919860,167310360,8719666200,545594049000,
%T 40394317194000,3494634235092000,349446163958892000,
%U 40005208010427660000,5199553600938496800000,761551300698921532800000,124863678342008772566400000,22782147644564103946550400000
%N Number of chains of n-3 partitions in the reduced partition lattice on n elements.
%C 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
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 148.
%H Alois P. Heinz, <a href="/A059355/b059355.txt">Table of n, a(n) for n = 3..269</a>
%e From _Harry Richman_, Mar 30 2023: (Start)
%e For n = 4, a chain of 1 partition is just a partition in the reduced partition lattice. There are 13 such partitions:
%e {123|4}
%e {124|3}
%e {134|2}
%e {1|234}
%e {12|34}
%e {13|24}
%e {14|23}
%e {12|3|4}
%e {13|2|4}
%e {14|2|3}
%e {1|23|4}
%e {1|24|3}
%e {1|2|34}
%e (End)
%p b:= proc(n) option remember; expand(`if`(n=1, 1,
%p add(Stirling2(n, j)*b(j)*x, j=0..n-1)))
%p end:
%p a:= n-> coeff(b(n), x, n-2):
%p seq(a(n), n=3..20); # _Alois P. Heinz_, Mar 31 2023
%t 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_ *)
%Y A diagonal of triangle in A008826.
%K nonn
%O 3,2
%A _N. J. A. Sloane_, Jan 27 2001
%E More terms from _Vladeta Jovovic_, Jan 02 2004
%E Name changed by _Harry Richman_, Mar 30 2023
|
