A partition \(\alpha \) is said to contain another partition (or pattern) \(\mu \) if the Ferrers board for \(\mu \) is attainable from \(\alpha \) under removal of rows and columns. We say \(\alpha \) avoids \(\mu \) if it does not contain \(\mu \). In this paper we count the number of partitions of n avoiding a fixed pattern \(\mu \), in terms of generating functions and their asymptotic growth rates. We find that the generating function for this count is rational whenever \(\mu \) is (rook equivalent to) a partition in which any two part sizes differ by at least two. In doing so, we find a surprising connection to metacyclic p-groups. We further obtain asymptotics for the number of partitions of n avoiding a pattern \(\mu \). Using these asymptotics we conclude that the generating function for \(\mu \) is not algebraic whenever \(\mu \) is rook equivalent to a partition with distinct parts whose first two parts are positive and differ by 1.

如果可以在删除行和列的情况下从\（\ alpha \）获得\（\ mu \）的Ferrers板，则说分区\（\ alpha \）包含另一个分区（或模式）\（\ mu \）。我们说\（\阿尔法\）避免\（\亩\） ，如果它不包含\（\亩\） 。在本文中，我们根据生成函数及其渐近增长率来计算避免固定模式\（\ mu \）的n个分区的数量。我们发现，每当\（\ mu \）时，此计数的生成函数都是有理数是（相当于）一个分区，其中任何两个零件尺寸至少相差两个。通过这样做，我们发现了与元环p-基团的惊人连接。我们进一步获得n的分区数的渐近性，避免了模式\（\ mu \）。使用这些渐近我们得出结论，为生成函数\（\亩\）不是代数每当\（\亩\）相当于车用不同的部件，其第一两个部分是正的且相差1的分区。