Equivalence classes (under reflection, cycling) of compositions of n

[Neil Fernandez]

I've been looking at equivalence classes of compositions of n where the
equivalence relation is different from the 'order insignificant' one
that defines partitions.

Equivalence under reflection, cycling, or a combination of both, gives
the following nos. of equivalence classes for n = 1, 2, 3, ... :

1, 2, 3, 5, 7, 12, 17, 29, 45, ...

7 has 15 partitions and 64 compositions. Call the former p-classes of
the latter. The lexicographically first compositions in the p-classes
are:

7
1,6
2,5
3,4
1,1,5
1,2,4
1,3,3
2,2,3
1,1,1,4
1,1,2,3
1,2,2,2
1,1,1,1,3
1,1,1,2,2
1,1,1,1,1,2
1,1,1,1,1,1,1

In most cases, the given composition can by reflection, cycling, or a
combination of both, yield all of the other compositions in the
respective p-class. The exceptions are {1,1,2,3}, which cannot yield
{1,2,1,3} (and equivalents), and {1,1,1,2,2}, which cannot yield
{1,1,2,1,2} (and equivalents).

So a(7)= 15 + 2 = 17

Neil

-- 
Neil Fernandez