Categories

[Jonathan Post]

a(n) = 2*A000055(n) - A000081(n/2) = (2* Number of trees with n unlabeled
nodes) - Number of rooted trees with n/2 nodes (or connected functions with
a fixed point)

where a(n) is downwards sloping diagonal sums of main category table, or
what?

It isn't until we look at bicategories that we are forced to look at the
symmetries and structure of the individual categories being enumerated here,
as objects in the higher categories, and have to enumerate the morphisms of
the functions. A (finite, concrete) category whose underlying directed graph
has no symmetry (i.e. automorphism group of order 1) can only map to itself
or an identical copy of itself, point to point, loop to loop, arc to arc.
But any cycle of length >1 in that directed graph allows morphisms to other
directed graphs.  The directed graph C_n, a cycle of length n, can thus map
to the same graph but mapping each point to the next, and each arc to the
next, and so forth. That requires then analysis of cycles in directed
graphs, as by de Bruijn in 1952 and more deeply understood since then. This
is another enumeration problem, unnecessary to completing the answer to
Frank's original question, and more tied to my endofunctions analysis. Sorry
if my interruptions distracted you from the straightforward question.  Good
work, guys!

Funny how Category Theory folks tend to be addicted to theory for its own
sake, and less willing to "get their hands dirty" with anything as old
fashioned as enumeration.  OEIS helps us focus here, by demanding the
integer sequences!

On 11/30/06, franktaw at netscape.net <franktaw at netscape.net> wrote:
>
> A typo there; that should be a(n) = 2*A000055(n) - A000081(n/2).
>
> Franklin T. Adams-Watters
>
>
> -----Original Message-----
> From: bowerc at usa.net
>
> ...
> For n>1 it's A122086
> http://www.research.att.com/~njas/sequences/A122086
> A sequence which could use a nicer description and formula such as
> a(n) = A000055(n) - A000081(n/2) where a sequence evaluated at a
> noninteger is 0.
> http://www.research.att.com/~njas/sequences/A000055
> http://www.research.att.com/~njas/sequences/A000081
>
> Christian
>
>
>
>
>
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