Inverse limits of various posets

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 $[n]={1,2\cdots n}$. 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 (after giving a lattice structure in a natural way). These poset structures can be extended over signed restricted growth functions for standard type B set partitions over $\langle n\rangle={-1,-2,\cdots n,0,1,2\cdots n}$ as well. We investigate properties of the tree and lattice structures of these projective systems. In this scenario we further bring up some other posets like $\mathcal{P}$-Partition posets of snake graph of continued fractions, Ascent lattices on Dyck Paths, certain type of lattice induced by generalisec fibonnaci number and Stanley order, lattices induced by non-crossing set partitions.

💡 Research Summary

This paper presents a comprehensive study on extending the concept of inverse limits and projective systems to various combinatorial posets, moving beyond the traditional setting of permutation-based structures.

The core innovation lies in generalizing the notion of a “P-chain permutational poset” – defined for a subset of permutations P – to a “P-chain-word poset” for subsets of words of length n that are not necessarily permutations. The primary combinatorial objects of focus are Restricted Growth Functions (RGFs), which are in bijection with set partitions of