# Number of Dyck paths up to stable equivalence

Acyclic (connected) Nakayama algebras can be identified with Dyck paths via their top boundary Auslander-Reiten quivers.

Now two Nakayama algebras $$A$$ and $$B$$ should be stable equivalent in case deleting the projective vertices in their Dyck paths gives the same result up to ordering. (I hope this is correct, since my experience with stable equivalences is very limited and maybe I have a thinking error).

Note that after the deletion we get a disjoint union of Dyck paths.

Dyck paths $$D$$ are always a sequence of irreducible Dyck paths (that do not touch the x-axis except at start and end).

So call two Dyck paths stable equivalent in case their sequence of irreducible Dyck paths is the same as a multiset. In case I have no thinking error this should describe when acyclic Nakayama algebras are stable equivalent using their associated Dyck paths.

For example the Nakayama algebras with Kupisch series [3,2,2,1] (=UDUUDD) and [2,3,2,1] (=UUDDUD) should be stable equivalent since the resulting stable Dyck path is in both cases the disjoint union of the Dyck path UD and the Dyck path consisting only of a point.

Question: Is there an explicit formula for the number of Dyck paths up to stable equivalence?

Here is the generating series:

Generating series of the irreducible Dyck paths is $$B(x)=x \frac{1-\sqrt{1-4x}}{2x}$$ and composing this with the Multiset generating series gives that the number of Dyck paths up to stable equivalence should have the following generating series:

$$e^{\sum\limits_{k=1}^{\infty}{\frac{B(z^k)}{k}}}$$ with $$B(x)=x \frac{1-\sqrt{1-4x}}{2x}$$. I wonder whether one can get an explicit formula for the sequence from the generating series.

The sequence starts with 1,2,4. Is there are good computer program that can calculate more terms for such a complicated generating series?

• This is oeis.org/A052854 – Martin Rubey Jun 8 at 12:10
• Here is FriCAS code. If you plug in 500, the first 500 coefficients are correct. $$\verb|B x == (1-sqrt(1-4*x))/2; z := monomial(1,1)UTS(FRAC INT, 'z, 0); f N == exp(reduce(+, [1/k*B(z^k) for k in 1..N])); s := f 500|$$ – Martin Rubey Jun 8 at 12:36