Characterization for the party algebras (Q2730972)

Let \(D,R\) be sets of \(n\) elements. A seating plan of size \(n\) is a partition of \(D\cup R\) into disjoint non-void subsets \(M_1,\dots, M_m\) such that \(|M_i \cap D|=|M_i \cap R|\). For every seating plan, the author assigns a graph whose connected components, denoted by \(M_i\), are composed of a \(|M_i|\)-vertex and \(|M_i|\)-univalent vertices. Given graphs \(G_1 , G_2 \) corresponding to seating plans \(w_1 , w_2 \), join the vertices \(d_i\) of \(G_1\) and \(r_i\) of \(G_2\). The resulting graph \(G_1 \circ G_2\) corresponds to a seating plan \(w_1w_2\). The author characterizes the resulting algebra of these seating plans by a presentation with \(n\) generators and \(n^2 +n\) relators.