A332244 Number of chains of length n in the lattice of set partitions of [2n] ordered by refinement.

1, 2, 45, 8176, 5967927, 12354550875, 58745934381618, 557269710685272585, 9536970947556120868800, 273107814151944184186060560, 12345107536247705318429028256740, 840776466106445249176017830108333910, 83061829400676435859326576547506817501212

Offset: 0

Author

Alois P. Heinz, Feb 07 2020

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..133
• Wikipedia, Lattice (order)
• Wikipedia, Partition of a set

Crossrefs

Cf. A331955.

Programs

Maple

b:= proc(n, k, t) option remember; `if`(k<0, 0, `if`({n, k}={0}, 1,
      add(`if`(k=1, 1, b(v, k-1, 1))*Stirling2(n, v), v=k..n-t)))
    end:
a:= n-> b(2*n, n, 0):
seq(a(n), n=0..14);

Mathematica

b[n_, k_, t_] := b[n, k, t] = If[k < 0, 0, If[Union@{n, k} == {0}, 1, Sum[ If[k == 1, 1, b[v, k - 1, 1]] StirlingS2[n, v], {v, k, n - t}]]];
a[n_] := b[2n, n, 0];
a /@ Range[0, 14] (* Jean-François Alcover, May 08 2020, after Maple *)

Formula

a(n) = A331955(2n,n).

Limit_{n->oo} (a(n)/n!^4)^(1/n) = 4/(w*(2-w)) = 6.17655460948348035823168..., where w = -LambertW(-2*exp(-2)) = -A226775. - Vaclav Kotesovec, Jul 03 2025