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Combinatorics of Complete Non-Ambiguous Trees Thomas Selig joint work with Mark Dukes, Jason P. Smith and Einar Steingr msson ICE-TCS Seminar, University of Reykjavik 20 November, 2018 Thomas Selig Combinatorics of CNATs Permutations

Combinatorics of Complete Non-Ambiguous Trees Thomas Selig joint work with Mark Dukes, Jason P. Smith and Einar Steingr´ ımsson ICE-TCS Seminar, University of Reykjavik 20 November, 2018 Thomas Selig Combinatorics of CNATs

Permutations Permutation = Bijection { 1 , . . . , n } → { 1 , . . . , n } . Thomas Selig Combinatorics of CNATs

Permutations Permutation = Bijection { 1 , . . . , n } → { 1 , . . . , n } . One-line notation π = 561243 ∈ S 6 . Thomas Selig Combinatorics of CNATs

Permutations Permutation = Bijection { 1 , . . . , n } → { 1 , . . . , n } . One-line notation π = 561243 ∈ S 6 . Diagram representation: 6 5 4 3 2 1 1 2 3 4 5 6 Thomas Selig Combinatorics of CNATs

Inversions Inversions are pairs ( π i , π j ) s.t. i < j and π i > π j . Thomas Selig Combinatorics of CNATs

Inversions Inversions are pairs ( π i , π j ) s.t. i < j and π i > π j . π = 561243 Inv ( π ) = { (5 , 1) , (5 , 2) , (5 , 4) , (5 , 3) , (6 , 1) , (6 , 2) , (6 , 4) , (6 , 3) , (4 , 3) } . Thomas Selig Combinatorics of CNATs

Inversions Inversions are pairs ( π i , π j ) s.t. i < j and π i > π j . π = 561243 Inv ( π ) = { (5 , 1) , (5 , 2) , (5 , 4) , (5 , 3) , (6 , 1) , (6 , 2) , (6 , 4) , (6 , 3) , (4 , 3) } . 6 5 4 3 2 1 1 2 3 4 5 6

Inversions Inversions are pairs ( π i , π j ) s.t. i < j and π i > π j . π = 561243 Inv ( π ) = { (5 , 1) , (5 , 2) , (5 , 4) , (5 , 3) , (6 , 1) , (6 , 2) , (6 , 4) , (6 , 3) , (4 , 3) } . 6 5 4 3 2 1 1 2 3 4 5 6 Thomas Selig Combinatorics of CNATs

Permutation graphs Permutation graph G π for π ∈ S n : Vertex set [ n ] := { 1 , . . . , n } ; Edge set Inv ( π ). Thomas Selig Combinatorics of CNATs

Permutation graphs Permutation graph G π for π ∈ S n : Vertex set [ n ] := { 1 , . . . , n } ; Edge set Inv ( π ). 6 5 4 3 2 1 1 2 3 4 5 6

Permutation graphs Permutation graph G π for π ∈ S n : Vertex set [ n ] := { 1 , . . . , n } ; Edge set Inv ( π ). 6 5 4 3 2 1 1 2 3 4 5 6 Thomas Selig Combinatorics of CNATs

Example π = 561243 ∈ S 6 . 6 3 4 1 2 5 Thomas Selig Combinatorics of CNATs

Connectivity π ∈ S n is decomposable if ∃ k < n , { π 1 , . . . , π k } = { 1 , . . . , k } , indecomposable otherwise. Thomas Selig Combinatorics of CNATs

Connectivity π ∈ S n is decomposable if ∃ k < n , { π 1 , . . . , π k } = { 1 , . . . , k } , indecomposable otherwise. Theorem A permutation graph G π is connected if, and only if, π is indecomposable. Thomas Selig Combinatorics of CNATs

Non-ambiguous trees Definition (Aval, Boussicault, Bouvel, Silimbani 2014) A non-ambiguous tree (NAT) is a filling of a rectangular tableau m × n such that: 1 Every row and every column has a dot; 2 Except for the bottom-left cell, every dot has either a dot below it in its column or to its left in its row, but not both. Thomas Selig Combinatorics of CNATs

Underlying binary tree

Underlying binary tree Thomas Selig Combinatorics of CNATs

Underlying binary tree NAT is complete if binary tree is complete, i.e. every dot has either a dot above it and to its right ( internal dot), or neither of these ( leaf dot). Thomas Selig Combinatorics of CNATs

Underlying binary tree NAT is complete if binary tree is complete, i.e. every dot has either a dot above it and to its right ( internal dot), or neither of these ( leaf dot). Above NAT is incomplete. Thomas Selig Combinatorics of CNATs

History NATs introduced as a special case of tree-like tableaux. Links to PASEP, other types of tableaux. NATs counted by certain permutations. Thomas Selig Combinatorics of CNATs

History NATs introduced as a special case of tree-like tableaux. Links to PASEP, other types of tableaux. NATs counted by certain permutations. CNATs a natural subset of NATs. Count them: 1 , 1 , 4 , 33 , 456 , 9460 , . . . . Sequence A002190 in OEIS: e.g.f. = − log(BesselJ(0 , 2 ∗ √ x )) (first combinatorial interpretation of sequence). Thomas Selig Combinatorics of CNATs

Properties of CNATs

Properties of CNATs Permutation? Thomas Selig Combinatorics of CNATs

Properties of CNATs Lemma (Dukes, S., Smith, Steingr´ ımsson 18) Let N be a CNAT. Then | rows( N ) | = | columns( N ) | . Moreover, the leaf dots of N form the diagram of an indecomposable permutation, denoted Perm( N ). Thomas Selig Combinatorics of CNATs

Counting CNATs Definition The Tutte polynomial of a connected graph G = ( V , E ) is defined by � ( x − 1) cc ( S ) − 1 ( y − 1) cc ( S )+ | S |−| V | , T G ( x , y ) := S ⊆ E where cc ( S ) = number of connected components of ( V , S ). Theorem (Dukes, S., Smith, Steingr´ ımsson 18) Permutation π indecomposable. Then � � � { N ∈ CNAT; Perm( N ) = π } � = T G π (1 , 0) . � � Thomas Selig Combinatorics of CNATs

External activity Definition G = ( V , E ) a graph, T ⊆ E a spanning tree of G , < E a total order on E . An edge e / ∈ T is externally active if it is minimal for < E in the unique cycle of T ∪ { e } . External activity of T = ext( T ) = number of externally active edges. Proposition For a connected graph G = ( V , E ) and a total order < E , we have � � T G (1 , 0) = � { T spanning tree of G ; ext( T ) = 0 } � . Thomas Selig Combinatorics of CNATs

Edges of permutation graph 6 c ′ 5 c 4 3 2 1 1 2 3 4 5 6 Define f ( i , j ) = ( i , π j ). Thomas Selig Combinatorics of CNATs

Edges of permutation graph 6 c ′ 5 c 4 3 2 1 1 2 3 4 5 6 Define f ( i , j ) = ( i , π j ). f ( c ) = (4 , 6), f ( c ′ ) = (5 , 2). Thomas Selig Combinatorics of CNATs

Edges of permutation graph 6 c ′ 5 c 4 3 2 1 1 2 3 4 5 6 Define f ( i , j ) = ( i , π j ). f ( c ) = (4 , 6), f ( c ′ ) = (5 , 2). Lemma: c has a leaf dot above and a leaf dot to its right iff f ( c ) ∈ E ( G π ). Thomas Selig Combinatorics of CNATs

Proof of theorem 6 6 5 4 3 4 1 2 3 2 1 5 1 2 3 4 5 6

Proof of theorem 6 6 6 5 4 ˆ f 3 3 4 4 1 1 2 2 3 2 1 5 5 1 2 3 4 5 6 Thomas Selig Combinatorics of CNATs

Proof of theorem 6 6 6 5 4 ˆ f 3 3 4 4 1 1 2 2 3 2 1 5 5 1 2 3 4 5 6 Theorem (Dukes, S., Smith, Steingr´ ımsson 18) Order edges of G π lexicographically according to corresponding cells in tableau. Then ˆ f : { N ∈ CNAT; Perm( N ) = π } → { T ∈ ST ( G π ) ; ext( T ) = 0 } is a bijection. Thomas Selig Combinatorics of CNATs

Open questions Other combinatorial interpretations of T G (1 , 0) (acyclic orientations, Abelian sandpile model). Other interpretations of CNATs? Thomas Selig Combinatorics of CNATs

Open questions Other combinatorial interpretations of T G (1 , 0) (acyclic orientations, Abelian sandpile model). Other interpretations of CNATs? All spanning trees: a “multi-rooted” generalisation of CNATs? Thomas Selig Combinatorics of CNATs

The decreasing case A CNAT N is decreasing if Perm( N ) = ( n + 1) n · · · 1. Thomas Selig Combinatorics of CNATs

The decreasing case A CNAT N is decreasing if Perm( N ) = ( n + 1) n · · · 1. 1 2 3 4 5 6 Thomas Selig Combinatorics of CNATs

The decreasing case A CNAT N is decreasing if Perm( N ) = ( n + 1) n · · · 1. 1 2 3 4 5 6 G = K n +1 and T G (1 , 0) = n !. Thomas Selig Combinatorics of CNATs

The decreasing case A CNAT N is decreasing if Perm( N ) = ( n + 1) n · · · 1. 1 2 3 4 5 6 G = K n +1 and T G (1 , 0) = n !. Bijective proof? Thomas Selig Combinatorics of CNATs

Bottom row decomposition = + 1 2 3 4 5 6 7 1 2 5 7 3 4 6 7 Thomas Selig Combinatorics of CNATs

The bijection Defined recursively: Thomas Selig Combinatorics of CNATs

The bijection Defined recursively: Ψ( ) = Ψ( ) · Ψ( ) 1 2 3 4 5 6 7 3 4 6 7 1 2 5 7 Thomas Selig Combinatorics of CNATs

The bijection Defined recursively: Ψ( ) = Ψ( ) · Ψ( ) 1 2 3 4 5 6 7 3 4 6 7 1 2 5 7 Ψ( ) = 3 · Ψ( ) 4 6 7 3 4 6 7 Thomas Selig Combinatorics of CNATs