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Permutations, EW -tableaux, and the Abelian sandpile model on Ferrers graphs Thomas Selig joint work with Mark Dukes, Jason P. Smith and Einar Steingr´ ımsson University of Strathclyde, Glasgow, UK December 11, 2017 Thomas Selig Permutations and the ASM on Ferrers graphs

Ferrers graphs Definition A Ferrers graph G = { B , T } of size n is a bipartite graph on the vertex set [ n ] := { 0 , 1 , . . . , n } = B ⊔ T , with 0 ∈ B , such that ( b , t ) ∈ E ( G ) iff b < t . Thomas Selig Permutations and the ASM on Ferrers graphs

Ferrers graphs Definition A Ferrers graph G = { B , T } of size n is a bipartite graph on the vertex set [ n ] := { 0 , 1 , . . . , n } = B ⊔ T , with 0 ∈ B , such that ( b , t ) ∈ E ( G ) iff b < t . N.B.: G = { B , T } is connected iff n ∈ T . Assume this is the case. Write G ( n , B ) for B ⊆ [ n − 1] with 0 ∈ B . Thomas Selig Permutations and the ASM on Ferrers graphs

Ferrers graphs Ferrers graphs are in 1-1 correspondence with Ferrers diagrams. 10 8 7 5 4 2 0 1 2 3 5 4 6 8 7 0 1 3 6 9 9 10 G ( F ) F Label rows and columns of the Ferrers diagram 0 , 1 , . . . , n from top-right to bottom-left (SE border). B = { rows } ; T = { columns } ; E ( G ) = { cells } . Thomas Selig Permutations and the ASM on Ferrers graphs

The Abelian sandpile model (ASM) Configuration c = ( c 1 , · · · , c n ) ∈ Z n + . 0 is the sink . c i ≡ number of grains of sand at vertex i . 10 8 7 5 4 2 4 0 1 1 0 0 4 3 1 0 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM Add grains at random. 10 8 7 5 4 2 4 0 1 1 0 0 5 3 1 0 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM Add grains at random. Vertex i is unstable if c i ≥ d i . 10 8 7 5 4 2 4 0 1 1 0 0 5 3 1 1 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM Unstable vertices topple, sending one grain along each incident edge. 10 8 7 5 4 2 4 0 1 1 0 0 5 3 1 1 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM This may cause other vertices to become unstable. 10 8 7 5 4 2 5 0 1 1 0 0 5 3 1 0 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM Unstable vertices topple. The sink absorbs grains. 10 8 7 5 4 2 0 0 1 1 0 0 6 4 2 1 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM Process eventually stabilises. Stabilisation does not depend on order of topplings. σ ( c ) := stabilisation of c . 10 8 7 5 4 2 2 1 2 2 1 1 0 4 2 0 0 1 3 6 9 Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM We define a Markov chain on the set of stable configurations Stab ( G ). At each step: 1 Add a grain at a random vertex. 2 Stabilise (in some cases no topplings necessary). Thomas Selig Permutations and the ASM on Ferrers graphs

The ASM We define a Markov chain on the set of stable configurations Stab ( G ). At each step: 1 Add a grain at a random vertex. 2 Stabilise (in some cases no topplings necessary). Under these dynamics, certain configurations appear infinitely often. These configurations are called recurrent , their set is Rec ( G ). Thomas Selig Permutations and the ASM on Ferrers graphs

Minimal recurrent configurations Define a partial order on Rec ( G ): c � c ′ iff c i ≤ c ′ i for all 1 ≤ i ≤ n . Rec min ( G ) := { c ∈ Rec ( G ); c is minimal for �} . Thomas Selig Permutations and the ASM on Ferrers graphs

Minimal recurrent configurations Define a partial order on Rec ( G ): c � c ′ iff c i ≤ c ′ i for all 1 ≤ i ≤ n . Rec min ( G ) := { c ∈ Rec ( G ); c is minimal for �} . n � Define level ( c ) = c i + d 0 − | E ( G ) | . For c ∈ Rec ( G ), we have i =1 0 ≤ level ( c ) ≤ | E ( G ) | − n . The level polynomial of G is � x level ( c ) . Level G ( x ) = c ∈ Rec ( G ) Thomas Selig Permutations and the ASM on Ferrers graphs

Minimal recurrent configurations Define a partial order on Rec ( G ): c � c ′ iff c i ≤ c ′ i for all 1 ≤ i ≤ n . Rec min ( G ) := { c ∈ Rec ( G ); c is minimal for �} . n � Define level ( c ) = c i + d 0 − | E ( G ) | . For c ∈ Rec ( G ), we have i =1 0 ≤ level ( c ) ≤ | E ( G ) | − n . The level polynomial of G is � x level ( c ) . Level G ( x ) = c ∈ Rec ( G ) Fact(??) We have Rec min ( G ) = { c ∈ Rec ( G ); level ( c ) = 0 } . Thomas Selig Permutations and the ASM on Ferrers graphs

Dhar’s burning criterion Theorem (Dhar 89) c := c + 1 A , where A is the Let G = G ( n , B ), c in Stab ( G ). Set ˜ set of neighbours of the sink, and 1 A i = 1 if i ∈ A and 0 otherwise. c is recurrent iff σ (˜ c ) = c . Moreover, in the stabilisation of ˜ c every (non sink) vertex topples exactly once. N.B.: ˜ c ≡ ( c after toppling the sink). Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K 1 , m 1 ,..., m k with dominating sink (Cori, Poulalhon 02). Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K 1 , m 1 ,..., m k with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs K m , n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes. Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K 1 , m 1 ,..., m k with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs K m , n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes. Graph decompositions (Dukes, S. 16). Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K 1 , m 1 ,..., m k with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs K m , n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes. Graph decompositions (Dukes, S. 16). Count them: Level G ( x ) = T G (1 , x ) where T G =Tutte polynomial (Merino 97, Cori, Le Borgne 03, Bernardi 08). Thomas Selig Permutations and the ASM on Ferrers graphs

Recurrent configurations Question: how to find all recurrent configurations for a given graph or family of graphs? Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K 1 , m 1 ,..., m k with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs K m , n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes. Graph decompositions (Dukes, S. 16). Count them: Level G ( x ) = T G (1 , x ) where T G =Tutte polynomial (Merino 97, Cori, Le Borgne 03, Bernardi 08). This talk: Ferrers graphs. Thomas Selig Permutations and the ASM on Ferrers graphs

EW -tableaux Definition (Ehrenborg, van Willigenburg 04) An EW -tableau (EWT) T is a 0–1 filling of a Ferrers diagram that satisfies the following properties: 1 The top row of T has a 1 in every cell. 2 Every other row has at least one 0. 3 No four cells of T that form the corners of a rectangle have 0s in two diagonally opposite corners and 1s in the other two. The size of an EWT is the size of the underlying Ferrers diagram. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 0 (a) an EWT (b) not an EWT Thomas Selig Permutations and the ASM on Ferrers graphs

EWTs and acyclic orientations G ( F ) F

EWTs and acyclic orientations 1 1 1 1 1 1 1 1 0 0 0 1 0 1 0 0 0 0 1 0 0 G ( F ) F