Let $A$ be a finite dimensional algebra with Jacobson radical $J$.
Question 1: In case $A$ is a Nakayama algebra with a linear quiver corresponding to a Dyck path $D$ (via its Auslander-Reiten quiver), then it seems that the number of indecomposable summands of $J$ as a bimodule is equal to the number of touch points of $D$ with the x-axis excluding the origin. (see the statistic http://www.findstat.org/StatisticsDatabase/St000011 ) Is this true and is there a simple explanation for this?
In case this is true, $J$ as a bimodule would be indecomposable if and only if the corresponding Dyck path $D$ is indecomposable. The question has a positive answer for all Nakayama algebras with at most 6 simple modules or equivalently all Dyck paths from (0,0) to (10,0).
A representation-theoretic interpretation for this number is the number of simple non-projective direct summands of $J$ as a one-sided module.
Question 2: Is there a simple criterion when the Jacobson radical $J$ of a general algebra $A$ as a bimodule is indecomposable? Can the number of indecomposable summands of $J$ be counted by a simple formula?
