Otter's method and the homology of homeomorphically irreducible \(k\)-trees (Q1914013)

For a positive integer \(k\) and a non-negative integer \(n\), \(T_n^{(k)}\) denotes the collection of all trees having \(nk + 2\) labelled leaves such that every internal vertex has degree \(mk + 2\) for some \(m > 0\). If \(k = 1\), then the collection \(T_n^{(1)}\) of homeomorphically irreducible trees with \(n + 2\) labelled leaves is already known and known to be interesting. In this paper the ``main theorem'' is a powerful formula determining \(H_{n - 1} (L_n^{(k)})\) as an \(S_{nk + 2}\)-module, where \(L_n^{(k)}\) is \(T_n^{(k)}\) equipped with a partial order \(T_1 \leq T_2\) if \(T_1\) can be obtained from \(T_2\) by contracting a set of internal edges while a maximal element \(\widehat 1\) is added, while \(\widehat 0\), the unique minimal element, is the star with \(nk + 2\) leaves. The computations establishing this formula make excellent us of the proper version of Polya's Cycle Index in the context of Cohen-Macaulay Posets by means of the poset \(\pi_n^{(k)}\) of set partitions of \(\{1, \dots, N\}\) with block sizes congruent to \(1 \pmod k\) and \(\widehat 1\) added if \(N \not \equiv 1 \pmod k\). The bridge needed to produce the required transit lies in the notion of a Lie \(k\)-algebra introduced and developed by the author and M. Wachs, whose properties are exploited in conjunction with a variety of interesting graph theoretical observations in order to construct the proof out of a quite substantial set of results. Certainly an elegant performance and an excellent occasion to observe the interaction of myriad detail in a construction which makes use of a wide spectrum of results out of several distinct areas of interest. This being so, the expectation is then that the ``main theorem'' will prove itself useful in several distinct areas also.