In this paper, we study arithmetic properties of weighted Catalan numbers. Previously, Postnikov and Sagan found conditions under which the 2-adic valuations of the weighted Catalan numbers are equal to the 2-adic valuations of the Catalan numbers. We obtain the same result under weaker conditions by considering a map from a class of functions to 2-adic integers. These methods are also extended to q-weighted Catalan numbers, strengthening a previous result by Konvalinka. Finally, we prove some results on the periodicity of weighted Catalan numbers modulo an integer and apply them to the specific case of the number of combinatorial types of Morse links. Many open questions are mentioned.

在本文中，我们研究加权加泰罗尼亚数的算术性质。以前，Postnikov和Sagan发现了加权加泰罗尼亚语数字的2 adic估值等于加泰罗尼亚语数字的2 adic估值的条件。通过考虑从一类函数到2-adic整数的映射，我们可以在较弱的条件下获得相同的结果。这些方法还扩展到q加权加泰罗尼亚数，从而增强了Konvalinka的先前结果。最后，我们证明了加权加泰罗尼亚数以整数为模的周期性的一些结果，并将它们应用于摩尔斯链接的组合类型数的特定情况。提到了许多悬而未决的问题。