平方分拆与Catalan数
Square Partitions and Catalan Numbers
摘要：
对于每个整数$k\ge 1$，我们定义了一个算法，该算法将所有分区的某个子集与其最大值至多为$k$的分区相关联。如果我们从一个正方形的分区$λ$开始，即$λ=λ_1\ge...\geλ_k>0$和$λ_1=k,λ_k=1$，那么应用$\ell$ times算法会产生一个集，其基数是加泰罗尼亚数$c_{\ell-k+1}$（自对偶情况）或加泰罗尼亚数的两倍。该算法定义了一棵树，并研究了该树的传播，该树不属于通常加泰罗尼亚树的同构类。该算法也可以修改，以产生一个双参数的家庭的集合和所产生的基数集的选票号码。最后，我们给出了关于$2\ell+m$变量对称函数环的一个特殊模的秩的猜想。
英文摘要：
For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $λ$ which is square, i.e $λ=λ_1\ge...\geλ_k>0$, and $λ_1=k,λ_k=1$, then applying the algorithm $\ell$ times gives rise to a set whose cardinality is either the Catalan number $c_{\ell-k+1}$ (the self dual case) or twice the Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two--parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in $2\ell+m$ variables.
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