0.25
0.5
0.75
1.25
1.5
1.75
2
Computer algebra for lattice path combinatorics
Published on 2019-07-19194 Views
Classifying lattice walks in restricted lattices is an important problem in enumerative combinatorics. Recently, computer algebra has been used to explore and to solve a number of difficult questions
Related categories
Presentation
Computer Algebra for Lattice Path Combinatorics00:00
Computer Algebra for Enumerative Combinatorics00:25
An (innocent looking) combinatorial question01:03
Teasers03:37
Why care about counting walks?04:28
Counting walks is an old topic: the ballot problem - 105:03
Counting walks is an old topic: the ballot problem - 206:39
Counting walks is an old topic: the ballot problem - 308:06
. . . but it is still a very hot topic - 108:34
. . . but it is still a very hot topic - 209:02
Our approach: Experimental Mathematics using Computer Algebra - 109:44
Our approach: Experimental Mathematics using Computer Algebra - 210:19
Lattice walks with small steps in the quarter plane - 110:29
Lattice walks with small steps in the quarter plane - 212:16
Entire books dedicated to small step walks in the quarter plane!12:49
Small-step models of interest13:21
The 79 small steps models of interest15:10
Task: classify their generating functions!15:45
Classification criterion: properties of generating functions - 116:01
Classification criterion: properties of generating functions - 219:07
Classification criterion: properties of generating functions - 319:29
Algebraic reformulation of main task: solving a functional equation - 120:12
Algebraic reformulation of main task: solving a functional equation - 222:41
“Special” models of walks in the quarter plane22:48
Gessel walks (2000) - 123:29
Gessel walks (2000) - 224:11
Gessel walks (2000) - 324:27
Gessel walks (2000) - 425:31
Guess-and-Prove26:10
Guess-and-Prove: a toy example - 126:59
Guess-and-Prove: a toy example - 228:49
Guess-and-Prove: a toy example - 329:01
Guess-and-Prove for Gessel walks29:33
A typical Guess-and-Prove algorithmic proof31:22
Algorithmic classification of models - 136:38
Algorithmic classification of models - 237:34
Algorithmic classification of models - 337:51
Algorithmic classification of models - 438:07
Models 1–19: proofs, explicit expressions and transcendence38:33
Kernel method - 140:21
Kernel method - 241:07
Kernel method - 341:29
Kernel method - 441:50
Kernel method - 542:03
Kernel method - 642:26
Kernel method - 743:18
Kernel method - 843:28
Kernel method - 943:42
Kernel method - 1044:07
Kernel method - 1144:27
Kernel method - 1244:41
Creative Telescoping - 144:51
Creative Telescoping - 246:18
Creative Telescoping - 346:33
Creative Telescoping - 448:11
Creative Telescoping - 548:32
4G Creative Telescoping48:48
A toy transcendence proof50:23
Summary - 150:34
Summary - 251:56
Conclusion52:24