Counting posets

[Michael Somos]

Recently, Thomas Ward posted an article with Subject "Counting posets"
on 14 Jul 1999 to sci.math asking about the sequence A001035 which
enumerates posets on an n-set. He noticed that:

\sum_{d | n} \mu(n/d) * t(d)  =  0  mod n.

and asked "Why?".  It would be amazing if the congruence were true for all n.
For more background on why he is interested in this, try looking at his:

==============================================================================
E-print math.DS/9907003
   
   Title: Arithmetic and growth of orbits: what is possible?
   Authors: Y. Puri, T. Ward
   Categories: DS Dynamical Systems (NT Number Theory)
    
   Abstract: We give necessary and sufficient conditions for a sequence
   to be exactly realizable as the sequence of numbers of periodic points
   in a dynamical system. Using these conditions, we show that no
   non-constant polynomial is realizable, and give some conditions on
   realizable binary recurrence sequences. Realization in rate is always
   possible for sufficiently rapidly-growing sequences, and is never
   possible for slowly-growing sequences. Finally, we discuss the
   relationship between the growth rate of periodic points and the growth
   rate of points with specified least period.

   From: Thomas Ward <t.ward at uea.ac.uk>
   Date: Thu 1 Jul 1999 12:02:39 GMT
==============================================================================

Shalom, Michael