The number of matchings on [2n] avoiding both patterns 12312 and 121323 with m crossings is equal towhich is a refinement of the super-Catalan numbers.

• 通过运用Stanley的振荡杨表与避免12312模式的匹配之间的一一对应，我们刻划了对应于避免12312模式的匹配的振荡杨表。