[seqfan] item A213058 of O.E.I.S.

Bhaskar Bagchi bhaskarbagchi53 at gmail.com
Fri Nov 20 14:53:03 CET 2020

I wish to post in O.E.I.S. the following comment on this sequence. In case
this is not the correct format for a post (for instance, if you require a
p.d.f.), please let me know.

A gracefully labelled tree of order n is a connected graph T with vertex
set {1,2, ...,n} with the property that , for every element k of
{1,2,...,n-1}, there is a unique edge {i,j} of T such that |i-j|=k. We do
not identify isomorphic trees.  I want to conjecture :
for every n, the total number of gracefully labelled trees of order n is
the nth term of the sequence A213058.

As far as I can see, no such combinatorial description of this sequence is
presently known; it is described by a functional equation for its
exponential generating function (which is tantamount to a recurrence
relation). An affirmative resolution of my conjecture is likely to yield an
extremely efficient algorithm for recursively enumerating the gracefully
labelled trees. This in turn should be a great step towards proving the
well known conjecture that all trees can be gracefully labelled.

With best regards,
Bhaskar Bagchi

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