A113896 Number of maximal (and largest) superset towers with n base elements.

1, 4, 864, 2579890176

Offset

0,2

Comments

On 3 elements ABC, some tower (Halmos, "Naive Set Theory" among many) that begins with the empty set can be written without loss of generality as {0, A, AB, ABC}. But we need to have sets B, C, BC, AC included somewhere too so that the thing is "largest", i.e., includes every subset of {A,B,C}. For ABCD, there are 3^3 ways to include B,C,D into AB,AC,AD,BC,BD,CD and 2^5 ways to include AC,AD,BC,BD,CD into ABC,ABD,ACD,BCD. So a(4) = 3^3*2^5.

Links

Table of n, a(n) for n=0..3.

Formula

(n-1)^(n-1) * (n-2)^(nC2 - 1) * (n-3)^(nC3 - 1) *...* 2^(nC(n-2) - 1) * 1^(n-1)

Example

a(2) = 4 because:
(1) 0->A A->AB B->AB C->AB AB->ABC AC->ABC BC->ABC ABC->ABC maximal
(2) 0->A A->AB B->AB C->AC AB->ABC AC->ABC BC->ABC ABC->ABC maximal
(3) 0->A A->AB B->AC C->AB AB->ABC AC->ABC BC->ABC ABC->ABC maximal
(4) 0->A A->AB B->AC C->AC AB->ABC AC->ABC BC->ABC ABC->ABC maximal

Crossrefs

Sequence in context: A221232 A272167 A255269 * A159706 A349077 A188978

Adjacent sequences: A113893 A113894 A113895 * A113897 A113898 A113899

Keyword

nonn

Author

Lee Corbin (lcorbin(AT)tsoft.com), Jan 28 2006

Status

approved