Projective system of posets from set partitions

It is known when we call a poset \(P\), a \(\mathcal{P}\)-chain permutational poset, given a subset of permutations \(\mathcal{P}\) of the symmetric group \(S_n\). In this work, we use the same idea to study subsets of words of length \(n\), that are not necessarily permutations, for example: especially when they are certain classes of restricted growth functions induced by set partitions in standard form over \([n] = {1, 2, \ldots, n}\). Varying \(n\) only, and also varying \(n\) and \(k\) (the number of blocks of the set partitions) simultaneously, we can show that those posets form a projective system of trees and lattices. These poset structures can be extended over signed restricted growth functions for standard type B set partitions over \(\langle n \rangle = {−1, −2, \ldots, n, 0, 1, 2,\ldots, n}\) as well. We investigate properties of the tree and lattice structures of these projective systems.

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Date published: Wednesday, April 16, 2025