Skeletal posets of the Tamari lattice and beyond
Time: 14:00 -- Location: bat 650, 445
summary: Given a lattice \(L\), the subposet \(\mathrm{Spine}(L)\) of \(L\) is the union of the longest maximal chains in \(L\). Dually, the subposet \(\mathrm{Spine}'(L)\) of \(L\) is the union of the shortest maximal chains in \(L\). For certain lattices, these subposets are particularly well-behaved; an example being trim lattices. Using \(\mathrm{Spine}(L)\) and \(\mathrm{Spine}'(L)\), we introduce two posets for any ungraded lattice called the long and wide skeletal posets. When \(L\) is the Tamari lattice, we show that the intervals of its skeletal posets have nice enumeration formulas. Specifically, the number of intervals of the long skeletal poset is the number of ternary trees \(\mathrm{Cat}_n^2 = \frac{1}{2n+1}\binom{3n}{n}\), and the number of intervals of the wide skeletal poset is the number of hex trees \(\mathrm{Hex}_n = 3\mathrm{Hex}_{n-1} + \sum_{k=1}^{n-2} Hex_{k}Hex_{n-k-1}\). In addition, we present some additional properties/enumerations and a relationship between the long and wide skeletal posets of the Tamari lattice. We list many enumerative conjectures regarding subfamilies of intervals of the long skeletal poset of the Tamari lattice, and suggest several directions of research regarding skeletal posets of ungraded lattices.