1. Count Triangulations Count, sample, enumerate [ V. Alvarez, R. Seidel, 2013 ] triangulation → sequence of x -monotone ropes ? ? ? ? Not one-to-one! Always choose the LEFTmost triangle! MARK the position of change. (Ex. 1) → path in a DAG with 2 n − 2 nodes G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
1. Count Triangulations Count, sample, enumerate [ V. Alvarez, R. Seidel, 2013 ] triangulation → sequence of x -monotone ropes forbidden ? ? ? ? Not one-to-one! Always choose the LEFTmost triangle! MARK the position of change. (Ex. 1) → path in a DAG with 2 n − 2 nodes G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
1. Count Triangulations Count, sample, enumerate [ V. Alvarez, R. Seidel, 2013 ] triangulation → ↔ sequence of x -monotone ropes MARKED ropes Not one-to-one! Always choose the LEFTmost triangle! MARK the position of change. (Ex. 1) O ( n 2 n ) → path in a DAG with 2 n − 2 nodes G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v Compute N ( v ) from source to sink. G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 28 Compute N ( v ) from source to sink. 14 11 6 3 8 2 2 4 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 28 Compute N ( v ) from source to sink. 14 11 How to SAMPLE a random path: 6 3 8 Find a random number between 1 and 28. . . . 2 2 4 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 14 11 How to SAMPLE a random path: 6 3 8 Find a random number between 1 and 28. . . . 2 2 4 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 7 / 14 14 11 because 21 = 3 + 11 + 7 How to SAMPLE a random path: 6 3 8 Find a random number between 1 and 28. . . . 2 2 4 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 7 / 14 14 11 How to SAMPLE a random path: 6 3 7 / 8 8 Find a random number between 1 and 28. . . . 2 2 4 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 7 / 14 14 11 How to SAMPLE a random path: 6 3 7 / 8 8 Find a random number between 1 and 28. . . . because 7 = 2 + 4 + 1 2 1 / 2 2 4 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 7 / 14 14 11 How to SAMPLE a random path: 6 3 7 / 8 8 Find a random number between 1 and 28. . . . 2 1 / 2 2 4 1 / 1 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 7 / 14 14 11 How to SAMPLE a random path: 6 3 7 / 8 8 Find a random number between 1 and 28. . . . 2 1 / 2 2 4 1 / 1 1 1 1 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Counting source-sink paths in a DAG N ( v ) := # paths from source to v 21 / 28 28 Compute N ( v ) from source to sink. 7 / 14 14 11 How to SAMPLE a random path: 6 3 7 / 8 8 Find a random number between 1 and 28. . . . 2 1 / 2 2 4 1 / 1 1 1 1 Essentially, this is UNRANKING: Compute a function { 1 , . . . , N } → path 1 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Summary The number of triangulations can be found in O ( n 2 2 n ) time and O ( n 2 n ) space. With this much preprocessing and space: • The triangulations can be enumerated with O ( n ) delay. • A random triangulation can be determined in O ( n log n ) steps. WARNING: Have to deal with large numbers. Counting algorithm can use modular arithmetic (Chinese remainder theorem). Can be applied to other structures (e.g. matchings, Ex. 6) Can be used for optimizing decomposable objective functions. (Nonuniqueness is not an issue.) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Other algorithms for counting There are many other approaches (divide-and-conquer, sweep, dynamic programming). The theoretically fastest algorithm for counting triangulations uses divide-and-conquer, based on balanced separators of size O ( √ n ) and has supexponential runtime: n O ( √ n ) Also for counting other structures. [“cactus layers”, Marx and Miltzow, 2016 ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Overview 0. Introduction 1. Count triangulations [ Alvarez and Seidel, 2013 ] • and perfect matchings [ Wettstein 2014 ] • Optimal triangulations 2. Coordinated primal-dual sweep [ Biedl, Chambers, Kostitsyna, Rote, Felsner 2020 ] 3. Count perfect matchings of structured point sets [ Asinowski and Rote, 2018 ] 4. Production matrices [ Huemer, Pilz, Silveira 2016 ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
2. Bipolar orientations ( s - t -planar graphs) • plane directed acyclic graph t • a single source s and a single sink t s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
2. Bipolar orientations ( s - t -planar graphs) • plane directed acyclic graph t • a single source s and a single sink t • split the outer face: → dual graph with a left outer vertex s ′ and a right vertex t ′ s ′ t ′ s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
2. Bipolar orientations ( s - t -planar graphs) • plane directed acyclic graph t • a single source s and a single sink t • split the outer face: → dual graph with a left outer vertex s ′ and a right vertex t ′ s ′ t ′ s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
2. Bipolar orientations ( s - t -planar graphs) • plane directed acyclic graph t • a single source s and a single sink t • split the outer face: → dual graph with a left outer vertex s ′ and a right vertex t ′ s ′ t ′ • The dual graph is also a bipolar orientation. (may be a multigraph) s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
2. Bipolar orientations ( s - t -planar graphs) • plane directed acyclic graph t • a single source s and a single sink t • split the outer face: → dual graph with a left outer vertex s ′ and a right vertex t ′ s ′ t ′ • The dual graph is also a bipolar orientation. (may be a multigraph) • All faces in the overlay of the two graphs are quadrilaterals: s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face s ′ t ′ G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face s ′ t ′ G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face s ′ t ′ G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face s ′ t ′ G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face Sweep is always possible! (Ex. 3) s ′ t ′ G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face Sweep is always possible! (Ex. 3) s ′ t ′ General form of a face (Ex. 2b) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face Sweep is always possible! (Ex. 3) s ′ t ′ General form of a face (Ex. 2b) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep • sweep the dual graph with an s ′ – t ′ rope from bottom to top sweep over the leftmost possible face Sweep is always possible! (Ex. 3) s ′ t ′ General form of a face (Ex. 2b) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep t • sweep the primal graph with an s – t rope from left to right s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep t • sweep the primal graph with an s – t rope from left to right s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep t • sweep the primal graph with an s – t rope from left to right s G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep t • sweep the primal graph with an s – t rope from left to right s sweep over the lowest possible face G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated primal-dual sweep t • sweep the primal graph with an s – t rope from left to right s sweep over the lowest possible face G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Animation t s ′ t ′ s dual rope in the dual (multi-)graph dual rope in the dual (multi-)graph primal rope (The primal graph is not shown.) primal rope (The primal graph is not shown.) page.mi.fu-berlin.de/rote/Papers/slides/Wuerzburg-2020-Simultaneous-sweep-Animation.pdf G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep There is a (unique) coordinated primal-dual sweep with the following properties: • The primal rope always crosses the dual rope exactly once. • The primal and the dual rope stay “close” to each other. • Exactly one rope can advance, depending on the situation at the crossing. • Every primal-dual edge pair is visited exactly once. • Each individual sweep is a leftmost/bottommost sweep. [ Biedl, Chambers, Kostitsyna, Rote, Felsner 2020 ] in connection with sweeping over a pseudoline arrangement, see Ex. 4. G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: ? ? G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: The other case is symmetric. G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: The other case is symmetric. G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: The other case is symmetric. G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Coordinated sweep general situation: The other case is symmetric. G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Overview 0. Introduction 1. Count triangulations [ Alvarez and Seidel, 2013 ] • and perfect matchings [ Wettstein 2014 ] • Optimal triangulations 2. Coordinated primal-dual sweep [ Biedl, Chambers, Kostitsyna, Rote, Felsner 2020 ] 3. Count perfect matchings of structured point sets [ Asinowski and Rote, 2018 ] 4. Production matrices [ Huemer, Pilz, Seara, Silveira 2016 ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
3. Perf. matchings in structured points P “flat”! Q convex position double-chain Θ ∗ (3 n ) smallest possible number of perfect matchings: Θ ∗ (2 n ) [Garc´ ıa, Noy, Tejel 2000] Upper bound: O ∗ (10 . 06 n ) [Sharir, Welzl 2006] ∗ = up to a polynomial factor G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
3. Perf. matchings in structured points P noncrossing! Q convex position double-chain Θ ∗ (3 n ) smallest possible number of perfect matchings: Θ ∗ (2 n ) [Garc´ ıa, Noy, Tejel 2000] Upper bound: O ∗ (10 . 06 n ) [Sharir, Welzl 2006] ∗ = up to a polynomial factor G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
3. Perf. matchings in structured points Current lower bound record: The generalized double-zigzag chain | P | = rn + 1 r = 5 n 1 2 3 r = 8 : Θ ∗ (3 . 0930 n ) [ Asinowski and Rote 2018 ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
3. Perf. matchings in structured points Current lower bound record: The generalized double-zigzag chain | P | = rn + 1 r = 5 n 1 2 3 r = 8 : Θ ∗ (3 . 0930 n ) [ Asinowski and Rote 2018 ] Here: r = 3 without corners : Θ ∗ (3 . 037 n ) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
3. Perf. matchings in structured points Current lower bound record: The generalized double-zigzag chain | P | = rn + 1 | P | = 3 n n 1 2 3 r = 8 : Θ ∗ (3 . 0930 n ) [ Asinowski and Rote 2018 ] Here: r = 3 without corners : Θ ∗ (3 . 037 n ) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Perfect matchings in double-X P Q | P | = | Q | = n G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Perfect matchings in double-X matching with k edges P P Q Q matching with k edges | P | = | Q | = n G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Perfect matchings in double-X matching with k edges P P n − 2 k unique edges P — Q Q Q matching with k edges | P | = | Q | = n G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Perfect matchings in double-X matching with k edges P P n − 2 k unique edges P — Q Q Q matching with k edges | P | = | Q | = n n/ 2 n/ 2 � � M k ( X ) 2 PM(double- X ) = M ( X ) = M k ( X ) k =0 k =0 M ( X ) , M k ( X ) = # matchings of X (with k edges) ⇒ M ( X ) 2 / n X ) ≤ M ( X ) 2 2 ≤ PM(double- = ⇒ Θ ∗ (3 2 n ) (Ex. 5) = G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
More general “flat” X P G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
More general “flat” X P G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
More general “flat” X P G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
More general “flat” X P Must count only down-free matchings of P : The unmatched points must be visible from below! G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Dynamic Programming Recursion A dangling edges B dangling edges ℓ n ℓ n +1 X n +1 X n A B X n A = # possibilities after n arcs with A dangling edges G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Dynamic Programming Recursion X n +1 = X n 2 + 3 X n 3 + 7 X n 4 + 6 X n 5 + 7 X n 6 + 3 X n 7 + X n 5 8 X n +1 X n 3 6 3 1 0 0 0 0 · · · 0 0 X n +1 X n 6 6 7 3 1 0 0 0 · · · 1 1 X n +1 X n · · · 3 7 6 7 3 1 0 0 2 2 X n +1 X n · · · 1 3 7 6 7 3 1 0 3 3 X n +1 X n 0 1 3 7 6 7 3 1 · · · = 5 5 X n +1 X n · · · 0 0 1 3 7 6 7 3 6 6 X n X n +1 · · · 0 0 0 1 3 7 6 7 7 7 X n X n +1 0 0 0 0 1 3 7 6 · · · 8 8 . total #points . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . ⇒ vectors grow like 28 n /n 3 / 2 = ⇒ Θ ∗ (3 . 037 N ) row sum 28 = [ Banderier and Flajolet, 2002 ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Weighted lattice paths A 1 3 X n A 7 6 7 3 1 n G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Overview 0. Introduction 1. Count triangulations [ Alvarez and Seidel, 2013 ] • and perfect matchings [ Wettstein 2014 ] • Optimal triangulations 2. Coordinated primal-dual sweep [ Biedl, Chambers, Kostitsyna, Rote, Felsner 2020 ] 3. Count perfect matchings of structured point sets [ Asinowski and Rote, 2018 ] 4. Production matrices [ Huemer, Pilz, Seara, Silveira 2016 ] G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon d n = 5 d n +1 = 4 n n n + 1 1 1 2 2 4 4 3 3 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon d n = 5 d ′ n = d n + d n +1 − 3 d n +1 = 4 n n n + 1 1 1 2 2 4 4 3 3 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon d n = 5 d ′ n = d n + d n +1 − 3 d n +1 = 4 n n n + 1 1 1 2 2 4 4 3 3 d n = d n d n +1 = 2 , 3 , . . . , d + 1 n n + 1 1 1 2 2 4 4 3 3 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon d n = 5 d ′ n = d n + d n +1 − 3 d n +1 = 4 n n n + 1 1 1 2 2 4 4 3 3 d n = d n d n +1 = 2 , 3 , . . . , d + 1 n n + 1 1 1 2 2 4 4 3 3 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon Triangulation of n -gon with last vertex of degree d n = d → Triangulation of ( n + 1 )-gon with last vertex of degree d n +1 = 2 or 3 or 4 or . . . or d, or d + 1 [ Hurtado & Noy 1999 ] “tree of triangulations” d n = d n d n +1 = 2 , 3 , . . . , d + 1 n n + 1 1 1 2 2 4 4 3 3 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon Triangulation of n -gon with last vertex of degree d n = d → Triangulation of ( n + 1 )-gon with last vertex of degree d n +1 = 2 or 3 or 4 or . . . or d, or d + 1 [ Hurtado & Noy 1999 ] “tree of triangulations” G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
4. Triangulations of a convex n -gon Triangulation of n -gon with last vertex of degree d n = d → Triangulation of ( n + 1 )-gon with last vertex of degree d n +1 = 2 or 3 or 4 or . . . or d, or d + 1 [ Hurtado & Noy 1999 ] d “tree of triangulations” triangulation � 4 lattice path 3 n 2 G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Production matrices count paths in a layered graph n n The answer is 1 1 1 1 . . . 1 1 1 1 . . . 1 0 1 1 1 . . . 0 � 1 . . . � 0 0 0 0 1 1 . . . 0 . 0 0 0 1 . . . . . . . . . ... . . . . . . . . � �� � the “production matrix” P G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Production matrices for enumeration were introduced by Emeric Deutsch, Luca Ferrari, and Simone Rinaldi (2005). were used for counting noncrossing graphs for points in convex position: Huemer, Seara, Silveira, and Pilz (2016) Huemer, Pilz, Seara, and Silveira (2017) 0 1 1 1 . . . 2 3 4 5 . . . 1 1 1 1 . . . 1 0 1 1 . . . 1 2 3 4 . . . 1 3 4 5 . . . 0 1 0 1 . . . 0 1 2 3 . . . 0 1 3 4 . . . 0 0 1 0 . . . 0 0 1 2 . . . 0 0 1 3 . . . 0 0 0 1 . . . 0 0 0 1 . . . 0 0 0 1 . . . . . . . . . . . . . . . ... ... ... . . . . . . . . . . . . . . . . . . . . . . . . spanning trees matchings forests G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Making the degree finite use vertical edges for partial summation n Number of paths is preserved. n G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Making the degree finite Shearing → Dyck paths → Catalan numbers n Number of paths is preserved. n G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
Other examples: graphs, paths 1 1 1 1 1 1 . . . 1 1 1 0 0 0 . . . 1 3 3 3 3 3 . . . 1 0 0 0 0 0 . . . 0 2 4 4 4 4 . . . 0 1 0 0 0 0 . . . 0 0 2 4 4 4 . . . 1 1 1 1 1 1 . . . 0 0 0 2 4 4 . . . 0 1 1 0 1 1 . . . 0 0 0 0 2 4 . . . 0 0 1 0 0 1 . . . . . . . . . . . . . ... ... . . . . . . . . . . . . . . . . . . . . geometric graphs paths Huemer, Seara, Silveira, and Pilz (2016) Huemer, Pilz, Seara, and Silveira (2017) G¨ unter Rote, Freie Universit¨ at Berlin Counting and enumeration in geometry PhD School on Computational Geometry, W¨ urzburg, March 20, 2020
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