Open arc diagrams and plane walks Mathias Lepoutre Ecole - PowerPoint PPT Presentation

Open arc diagrams and plane walks Mathias Lepoutre ´ Ecole polytechnique Joint work with: • Julien Courtiel • ´ Eric Fusy • Marni Mishna s´ eminaire CALIN, 24 novembre 2017 EUROCOMB 2017, European Journal of Combinatorics

Part I Introduction Part II The Simple case Part III The Hesitant case

Part I Introduction

Domain constraint, marking, ending constraint meander bridge

Domain constraint, marking, ending constraint meander Dyck path with marked steps from 1 to 0 bridge

Domain constraint, marking, ending constraint Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

Walks, Tableaux, Diagrams The Simple case Walk in the ( 2 -dimensional) Walk octant n step of type N, S, E, O ending on the axis at ( i, 0)

Walks, Tableaux, Diagrams The Simple case Diagram Walk matching diagram of length n with i open arcs without 3 -crossing

Walks, Tableaux, Diagrams The Simple case Diagram Walk Respective advantages :

Walks, Tableaux, Diagrams The Simple case Diagram Walk Respective advantages : • well-known objects • easy reccurence relations for generating series • a more natural phrasing of problems

Walks, Tableaux, Diagrams The Simple case Diagram Walk Respective advantages : • well-known objects • new generating trees • easy reccurence relations • easily-removable open for generating series arcs • a more natural phrasing of problems

Walks, Tableaux, Diagrams The Hesitating case Walk in the ( 2 -dimensional) Walk octant n steps of type N, S, E, O, NE, NS, EO, ES enging on the axis at ( i, 0)

Walks, Tableaux, Diagrams The Hesitating case Diagram Walk partition diagram of length n with i open arcs without enhanced 3 -crossing

Domain constraint ↔ ending constraint

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant Excursion in the quarter-plane

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant Excursion in the quarter-plane C n · C n +1 B n +1 Cardinality

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant Excursion in the quarter-plane

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

Domain constraint ↔ ending constraint The simple case The Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

A new approach via open arc diagrams Remove the open arcs in order to get marked excursions in the octant Simple axis-walk in the octant Hesitating axis-walk in the octant

A new approach via open arc diagrams Remove the open arcs in order to get marked excursions in the octant Simple axis-walk in the octant Hesitating axis-walk in the octant Open matcing diagram Open partition diagram without 3 -crossing without enhanced 3 -crossing 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

A new approach via open arc diagrams Remove the open arcs in order to get marked excursions in the octant Simple axis-walk in the octant Hesitating axis-walk in the octant Matching diagram without Partition diagram without enhanced 3 -crossing, with marking 3 -crossing, with marking 2 3 4 6 7 8 1 2 3 4 5 6 7 8 9

A new approach via open arc diagrams Remove the open arcs in order to get marked excursions in the octant Simple excursion in the octant Hesitating excursion in the octant with marking with marking Matching diagram without Partition diagram without enhanced 3 -crossing, with marking 3 -crossing, with marking 2 3 4 6 7 8 1 2 3 4 5 6 7 8 9

Part II The Simple case Bousquet-M´ elou and Mishna’s question

Bousquet-M´ elou and Mishna’s question ? Bousquet-M´ elou Mishna 2010

Bousquet-M´ elou and Mishna’s question ? Bousquet-M´ elou Mishna 2010 Gouyou-Beauchamps 1985 Elizalde 2014 Cori et al. 1986 Bernardi 2007

Bousquet-M´ elou and Mishna’s question ? Bousquet-M´ elou Mishna 2010 Open arc Elizalde 2014 diagrams Cori et al. 1986 Bernardi 2007

The missing part Reminder : Gouyou-Beauchamps 1985 (non-bijective) : Simple axis-walks in the octant are counted by C ⌊ n +1 ⌉ , ⌋ · C ⌈ n +1 2 2 where C n is the n -th Catalan number.

The missing part Reminder : Gouyou-Beauchamps 1985 (non-bijective) : Simple axis-walks in the octant are counted by C ⌊ n +1 ⌉ , ⌋ · C ⌈ n +1 2 2 where C n is the n -th Catalan number. Objective : • Build a bijection between Simple axis-walks in the octant of length 2 n and pairs of Dyck paths of half-lengths n and n + 1 .

The even case Matching diagram Open matching diagram Simple axis-walks in the without 3 -crossing with with no 3 -crossing of octant of length 2 n weights on open intervals, length 2 n of size 2 n - - 2 0 0 1 0 1 0 size = length + weight 2 0 1 0 0 1 0 Simple excursion in the octant, with weights on the axis, of size 2 n

Domain constraint ↔ Ending constraint Simple case Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

Domain constraint ↔ Ending constraint Simple case Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

The even case Matching diagram Open matching diagram Simple axis-walks in the without 3 -crossing with with no 3 -crossing of octant of length 2 n weights on open intervals, length 2 n of size 2 n - - 2 0 0 1 0 1 0 2 0 1 0 0 1 0 Pair of positive paths of Simple inter-diagonals Simple excursion in the length 2 n going from excursion of length 2 n octant, with weights on (1 , 0) to (1 , 0) the axis, of size 2 n

The even case Matching diagram Open matching diagram Simple axis-walks in the without 3 -crossing with with no 3 -crossing of octant of length 2 n weights on open intervals, length 2 n of size 2 n - - 2 0 0 1 0 1 0 2 0 1 0 0 1 0 Pair of Dyck paths of Simple inter-diagonals Simple excursion in the half-lengths ( n, n + 1) excursion of length 2 n octant, with weights on the axis, of size 2 n

The odd case Matching diagram Open matching diagram Simple axis-walks in the without 3 -crossing with with no 3 -crossing of octant of length 2 n + 1 weights on open intervals, length 2 n + 1 of size 2 n + 1 - - 2 0 0 1 0 0 0 2 0 1 0 0 0 0 Pair of Dyck paths of Simple inter-diagonals Simple excursion in the half-lengths n + 1 walk of length 2 n + 1 octant, with weights on the axis, of size 2 n + 1

Answering Bousquet-M´ elou et Mishna’s question : three new bijections ? Bousquet-M´ elou Mishna 2009 Elizalde 2014 Gouyou-Beauchamps 1985 Cori et al. 1986 Bernardi 2007

Answering Bousquet-M´ elou et Mishna’s question : three new bijections ? Bousquet-M´ elou Mishna 2009 Elizalde 2014 Open arc diagrams Cori et al. 1986 Bernardi 2007

Answering Bousquet-M´ elou et Mishna’s question : three new bijections ? Bousquet-M´ elou Mishna 2009 Elizalde 2014 Open arc Schnyder woods with diagrams marked edges meanders and alternating Baxter permutations Cori et al. 1986 Bernardi 2007

Part III The hesitating case Burrill et al.’s Question

Symmetric Baxter families Plane bipolar orientations Rectangulations of the square Baxter permutations ( k − 1 )( n +1 n +1 k )( n +1 k +1 ) B n = � n k =1 ( n +1 1 )( n +1 2 ) Gessel-Viennot Hesitating excursions Non-crossing in the quarter-plane triples of paths

Asymmetric Baxter families Hesitating axis-walks in the octant Hesitating tableaux of height at most 2 with a line shape Open partition diagrams with no Open partition diagrams with no enhanced 3 -crossings enhanced 3 -nestings

Baxter families Symmetric families Xin and Zhang 2008 (non bijective) Asymmetric families

Baxter families Symmetric families Explicit bijection Asymmetric families

Strategy: Go through marked excursions in the octant

Domain contraint ↔ Ending constraint Simple case Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane

Strategy: Go through marked excursions in the octant

Strategy: Go through marked excursions in the octant

Strategy: Go through marked excursions in the octant Open partition diagrams of length n without enhanced 3 -crossings are in bijection with Simple excursions in the octant of length n with marked peaks and marked W-steps on the axis .

Domain contraint ↔ Ending constraint Simple case Hesitating case Axis-walk in the octant Excursion in the octant with marking Excursion in the quarter-plane