A homotopy equivalence for partition posets related to liftings of \(S_{n-1}\)-modules to \(S_n\) (Q5937132)

The partition lattices \(\Pi_n\) of set partitions of the set \(\{1,\dots,n\}\) ordered by refinement, for the natural numbers \(n=1,2, \dots\) is of great (pure) combinatorial interest, with many of its parameters, width, length, etc., important entities of great usefulness elsewhere. Thus, as a class of posets they are also of interest and its properties have been well-studied. A well-trodden path, as is clear from the literature, especially in this case, is to consider the order-complex \(\Delta (\Pi_n)\), i.e., the simplicial complex generated by the maximal chains, and to study the homology of it and that of subposets \(P\) of \(\Pi_n\), usually of other special types defined by certain conditions. This has produced a rich deposit of results added to on a rather regular schedule by several authors, the present one included. In this substantial paper many new instances of an isomorphism \(V_n \downarrow S_{n-1} \cong V_{n-1} \downarrow S_{n-2} \uparrow S_{n-1}\), with \(V_k\) an \(S_k\)-module, \(S_k\) the symmetric group, \(\downarrow\) restriction and \(\uparrow\) induction, are observed beyond the ones already known, leading to new results as well as novel confirmations of recent and older results to demonstrate both technical skill and the availability of nice results in this particular corner of the garden of mathematical delights.