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arXiv · 2212.14666

On the lattice of weighted partitions

Keiichi Shigechi
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Abstract

We introduce and study the lattice of generalized partitions, called weighted partitions. This lattice possesses similar properties of the lattice of partitions. By use of the pictorial representation of a weighted partition, the total number is given by the successive Stirling transforms of the Stirling number of the second kind. We construct an explicit $EL$-labeling on the lattice, which implies this lattice is $EL$-shellable and hence shellable. We compute the M\"obius function and the characteristic polynomial by use of a pictorial representation of a maximal decreasing chain. Further, a maximal decreasing chain is shown to be bijective to a labeled rooted complete binary tree.

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Keiichi Shigechi. 2022-12-30. On the lattice of weighted partitions. https://arxiv.org/abs/2212.14666

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