[seqfan] A182080: nice sequence needing more terms

[Neil Sloane]

I just came across this in an old book. It is related to many other
problems.

Please see the entry A182080 for more details.
Neil

A182080

a(n) = maximal depth of an indecomposable exact cover of an n-set.

1,2,3,5,9?,18?,38?


Let U = {1,2,...,n} and let P = collection of all subsets of U.

A block system on U is a function f: P -> {0,1,2,...}; f(S) is the number
of times a subset S occurs as a block in the system.

The sum of two block systems f,g is defined in the obvious way, and z
denotes the zero block system.

f is an exact cover of depth d(f) if for each u in U,

Sum_{ S in P: u in S} f(S) = d(f).

An exact cover is decomposable if f = g+h where g, h are nonzero exact
covers.

Then a(n) is the maximal depth of an indecomposable exact cover of U.


Example showing an indecomposable f of depth 2 for n=3, illustrating a(3) =
2:

.S. | 1 2 3 | f(S)

------------------

..- | 0 0 0 | 0

..1 | 1 0 0 | 0

..2 | 0 1 0 | 0

..3 | 0 0 1 | 0

.12 | 1 1 0 | 1

.13 | 1 0 1 | 1

.23 | 0 1 1 | 1

123 | 1 1 1 | 0