Notation¶
The symbols used on this site, fixed once. Where the literature has a standard, it follows Grünbaum's Arrangements and Spreads.1
Arrangements¶
| symbol | meaning |
|---|---|
| \(\mathbb{RP}^2\) | the real projective plane; points \((x:y:z)\), lines \(\{ax+by+cz=0\}\) written \([a:b:c]\) |
| \(\mathcal A\) | an arrangement: a set of \(n\) distinct lines in \(\mathbb{RP}^2\) (no "general position" assumed) |
| \(n = \lvert\mathcal A\rvert\) | the number of lines |
| \(\mu(P)\) | the multiplicity of a point: the number of lines of \(\mathcal A\) through \(P\) |
| pencil | the set of lines through an intersection point; its size is \(\mu(P)\) |
| \(t_k(\mathcal A)\) | the number of points with \(\mu(P) = k\), \(k \ge 2\) (Grünbaum's \(t_k\)); \(t_2\) counts the ordinary points |
| \(f_0(\mathcal A)\) | \(\sum_{k\ge2} t_k\), the number of intersection points (vertices) |
| \(m(\mathcal A)\) | \(\max\{k : t_k > 0\}\), the size of the largest pencil |
| \(\ell(\mathcal A)\) | the loss \(\binom n2 - f_0 = \sum_{k \ge 3}(\binom k2 - 1)\,t_k\) |
| pencil type | the multiset \(\{k^{t_k} : k \ge 2\}\); written without the \(2\)s, e.g. \(\{6, 4, 3^{12}\}\) |
Synonyms found in the literature for the pencil type: t-vector \((t_2, t_3, \dots)\), weak combinatorics, point-multiplicity sequence. We prefer "type" and write it as a multiset because the sequences count multisets.
Pencils of size two are always omitted from the written type; their number is \(t_2 = \binom n2 - \sum_{k\ge3}\binom k2 t_k\). When an explicit \(t_2\) helps, we write \(\{6, 4, 3^{12}, 2^9\}\).
Combinatorial structures¶
| symbol | meaning |
|---|---|
| linear space | a set of \(n\) elements with a family of blocks (subsets of size \(\ge 2\)) such that every pair of elements lies in exactly one block; for an arrangement, the elements are the lines and the blocks are the pencils |
| blocks of a type | for a type \(\{k_1, \dots, k_r\}\): blocks of sizes \(k_1, \dots, k_r\), pairwise sharing at most one point; the uncovered pairs are the 2-blocks |
| chirotope \(\chi\) | an alternating map \(\binom{[n]}{3} \to \{+, 0, -\}\) satisfying the three-term Grassmann–Plücker relations; \(\chi = 0\) exactly on the collinear triples |
| orientable | a linear space is orientable if some chirotope has exactly its blocks as zero set; equivalently it is the underlying matroid of a rank-3 oriented matroid, equivalently (Folkman–Lawrence) it is the incidence structure of an arrangement of pseudolines |
In the dual (point) picture a pencil of \(k\) lines is a \(k\)-point line, so a linear space on \(n\) points with blocks of sizes \(k_i\) is exactly the combinatorial skeleton of a set of \(n\) points with lines of those sizes.
The sequences¶
| symbol | definition | first terms |
|---|---|---|
| \(a(n)\) | number of arithmetically admissible types for \(n\) lines | 1, 1, 2, 4, 7, 15, 32, 66, 141, 295, 619, 1286 |
| \(b(n)\) | number of types admitting a linear space | 1, 1, 2, 3, 5, 9, 17, 29, 56, 106, 202, 384 |
| \(c(n)\) | number of types of arrangements of \(n\) real lines | 1, 1, 2, 3, 5, 9, 16, 28, 53, 98, 180, 336 |
| \(d(n)\) | number of distinct values of \(f_0\) over arrangements of \(n\) lines | 1, 1, 2, 3, 5, 9, 13, 18, 24, 30, 38, 46, 56 |
| \(V(n)\) | the set of values of \(f_0\); \(d(n) = \lvert V(n) \rvert\) | |
| \(A_n\) | the set of arithmetically admissible types; \(a(n) = \lvert A_n\rvert\) |
All sequences are indexed by \(n \ge 1\), the number of lines; \(n = 1\) gives the single line with no intersection points, counted as one (empty) type and one value \(f_0 = 0\).
Status labels¶
Results on this site are labelled by how they are established.
- proved — a complete proof is given or a precise reference is cited;
- computer-verified — established by a program whose algorithm is described on the algorithms page, with the exact command, log and (for examples) an exact certificate checked by the verifier;
- bound — only an interval is known; both ends are justified;
- conjectured — supported by data, not proved.
Numbering¶
Theorems, lemmas and propositions are numbered consecutively within each page. Proofs end with \(\square\). A "Theorem" whose proof relies on a computation says so explicitly and points to the data.
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B. Grünbaum, Arrangements and Spreads, CBMS Regional Conference Series in Mathematics 10, American Mathematical Society, Providence, 1972. ↩