A partition of integer $n$ is a sequence $\lambda = (\lambda_1, \lambda_2, \cdots, \lambda_k, \cdots)$ of non negative integers such that $\lambda_i \ge \lambda_{i+1}$ and $\sum_i \lambda_i = n$. It follows that there are finitely many non-zero $\lambda_i$’s. One can restrict the number of them and the largest value of $\lambda_i$ and observe that the set of such partitions form a poset under a suitable relation. Several natural questions arise in this setting. Some of these questions have been answered by Proctor, Stanley and Kathy O’Hara among others. We take a look at some techniques as given by Stanley and ask if it is possible to extend it to higher dimensions.

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Last updated: 21 Jun 2024