A class of labeled posets and the Shi arrangement of hyperplanes (Q1369737)

An unlabeled poset is \((C_2+ 1)\)-free if it does not contain the three element poset \(\{p,q,r\}\) where \(p<q\), \(q\circ r\), \(p\circ r\); \(x\circ y\) means that \(x\) and \(y\) are incomparable. It is known that the pogroupoid of such a poset is associative iff it is \((C_2+1)\)-free. The note reviewed here discusses a labeled semi-analog of \((C_2+ 1)\)-free posets. In particular, let \(P_n\) denote the collection of labeled posets on \(\{1,\dots,n\}\) which do not contain \(\{c<b, c\circ a,b\circ a\}\), \(\{b<a, b\circ c, a\circ c\}\), \(\{a< c, a\circ b, c\circ b\}\), as included in subsets. The author shows that \(P_n\) contains \((n+1)^{n-1}\) elements, which is the number of non-isomorphic trees of order \(n+1\) (e.g., on \(\{0,1,\dots, n\}\)) and discerns furthermore that the number of posets of \(P_n\) with \(i\) pairs \((a,b)\) such that \(a<b\) is equal to the number of trees on \(\{0,1,\dots, n\}\) with \({n\choose 2}- i\) inversions by exploiting a bijection between \(P_n\) and the set \(S_n\) of regions produced by hyperplanes in \(\mathbb{R}^n\) of the form \(X_i- X_j= 0\) or \(X_i- X_j= 1\) for \(1\leq i\leq j\leq n\).