On the Number of Simple Arrangements of Five Double Pseudolines

Abstract

Abstract. We describe an incremental algorithm to enumerate the isomorphism classes of double pseudoline arrangements. The correction of our algorithm is based on the connectedness under mutations of the spaces of one-extensions of double pseudoline arrangements, proved in this paper. Counting results derived from an implementation of our algorithm are also reported.

LR Characterization of Chirotopes of Finite Planar Families of Pairwise Disjoint Convex bodies

“…A positive answer to that question, providing the key to a practical enumeration algorithm for simple arrangements of at most 5 double pseudolines, is given in [20]. The proof presented in [20] of this connectedness result is based on an enhanced version of the pumping lemma which says that, given a double pseudoline γ of an arrangement Γ with the property that the vertices of the arrangement Γ lying on the curve γ are ordinary, either there are (at least) two fans contained in the crosscap side of the double pseudoline γ with base sides supported by γ or there are no vertices of the arrangement contained in the crosscap side of γ. The enhanced version of the pumping lemma can be easily proved using the geometric representation theorem for arrangements of double pseudolines.…”

“…The enhanced version of the pumping lemma can be easily proved using the geometric representation theorem for arrangements of double pseudolines. It will be interesting to have a direct proof of it since, as explained in [20], the geometric representation theorem for arrangements of double pseudolines can be derived from it.…”

“…13. Thus there are 13 isomorphism classes of arrangements of three double pseudolines and 216 isomorphism classes of indexed arrangements of three oriented double pseudolines on a given set of three indices (and not 214 as indicated by error in [20]). This latter number is computed as the sum…”

“…where g k is the number of arrangements with group of automorphisms of order k. For the number of isomorphism classes of arrangements of four double pseudolines and for the number of isomorphism classes of simple arrangements of five double pseudolines we refer to [20]. Using the exhaustive list of simple arrangements of three double pseudolines we now compute the martagons on three and four double pseudolines.…”

Arrangements of double pseudolines

“…Preliminary counting results, concerning only simple arrangements, are reported in the following table: where an denotes the number of isomorphism classes of simple arrangements of n double pseudolines. For details on the incremental enumeration algorithm and comments on its implementation we refer to [4]. Figure 6 shows representatives of the thirteen isomorphism classes of simple arrangements of three double pseudolines.…”