论文标题
加泰罗尼亚数字是无限字母替代家族的差异
Catalan numbers as discrepancies for a family of substitutions on infinite alphabets
论文作者
论文摘要
在这项工作中，我们考虑了无限字母上的一类替换，并表明它们表现出生长行为，这对于有限字母的替换是不可能的。尽管对于这两种设置，瓷砖计数函数的前项是指数级（并在通货膨胀因素的指导下），但二阶项的行为截然不同。对于有限设置，众所周知，第二项也是指数级或指数时间为多项式。我们展示了一个大型示例，其中第二个学期至少在$ n $中是$ n $的一半势力，其中$ n $是替换步骤的数量。特别是，我们就加泰罗尼亚数字的线性组合提供了这种差异的身份。
In this work, we consider a class of substitutions on infinite alphabets and show that they exhibit a growth behaviour which is impossible for substitutions on finite alphabets. While for both settings the leading term of the tile counting function is exponential (and guided by the inflation factor), the behaviour of the second-order term is strikingly different. For the finite setting, it is known that the second term is also exponential or exponential times a polynomial. We exhibit a large family of examples where the second term is at least exponential in $n$ divided by half-integer powers of $n$, where $n$ is the number of substitution steps. In particular, we provide an identity for this discrepancy in terms of linear combinations of Catalan numbers.
