Catalan numbers and a conjecture on the maximum composition length of a Kac module

Let $f:\mathbb{Z}⟶\{\times \cdot \}$ be a function such that $f(a)=\cdot$ for all except finitely for many $a\in \mathbb{Z}$. We define a set $\mathrm{♭}f$ of non-intersecting arc (or cap) diagrams satisfying certain conditions determined by $f$. Then we give a recursive method for enumeration of $\mathrm{♭}f$ which recalls the Fundamental Recurrence for Catalan numbers. The motivation comes from the problem of enumeration of the composition factors of a Kac module with maximum degree of atypicality for the Lie superalgebra $\mathfrak{g}=\mathfrak{g}\mathfrak{l}(r\mathrm{\mid }r)$. In particular we prove a conjecture that the maximum number of composition factors is a Catalan number.