Catalan numbers as discrepancies for a family of substitutions on infinite alphabets

In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit growth behaviour which is impossible for finite alphabets. While both settings the leading term tile counting function exponential (and guided by inflation factor), second-order strikingly different. For setting, it known second also or times polynomial. We large family examples where at least in n divided half-integer powers n, number substitution steps. particular, provide an identity discrepancy terms linear combinations Catalan numbers.