Let $A$ be a finite dimensional quiver algebra over a field $K$ and let $D(-):=Hom_K(-,K)$ denote the natural duality (assume algebras are connected).

Define $\psi_A:=sup \{ n \geq 1 | D(A)^{\otimes n} \neq 0 \}$ and $W_A:=D(A)^{\otimes \psi_A}$.

Here for an $A$-bimodule $M$, $M^{\otimes n}$ denotes the $n$-fold tensor product of $M$ with itself over $A$.

It seems horribly complicated, but in special cases those definitions might have nice properties. Were they studied before?

Note that Nakayama algebras with a linear quiver and $n$ simple modules are counted by the Catalan nubmers $C_{n-1}$ and thus are in bijection with 321-avoiding permutations on $n-1$-symbols. (Nakayama algebras with a linear quiver are exactly the quotient algebras of the ring of upper triangular matrices over the field $K$ by an admissible ideal)

Here some observations/guesses with the computer for Nakayama algebras with a linear quiver:

$W_A$ is injective.

There are exactly $2^{n-2}$ algebras with $\psi_A = 2$ and all of those algebras seem to have global dimension at most 3.

The generating function of the statistic $A \rightarrow \psi_A$ (http://www.findstat.org/StatisticsDatabase/St001290/) seems to coincide with the generating function on 321-avoiding permutations given by $ \pi \rightarrow f(g(\pi))$, where $g$ is the first fundamental transformation on permutations (http://www.findstat.org/MapsDatabase/Mp00086) and $f$ number of right-to-left minima of a permutation (http://www.findstat.org/StatisticsDatabase/St000991).

Now it looks rather horribly to try to prove those things by direct computations but maybe there is a trick or a nice interpretation of $\psi_A$ and $W_A$ for Nakayama algebras and maybe even more general algebras (maybe QF-3 algebras)?

edit: I found a good reason for 1. and 2. by direct computations with the Nakayama functor applied to the minimal faithful projective-injective module. But maybe there is an argument for 1. that avoids any calculations? 3. remains mysterious.