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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 msson University of Strathclyde, Glasgow, UK December 11, 2017 Thomas Selig

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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

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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

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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

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Ferrers graphs

Ferrers graphs are in 1-1 correspondence with Ferrers diagrams.

9 6 3 1 8 7 5 4 2 10

0 1 3 6 9 10 8 7 5 4 2

F G(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

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The Abelian sandpile model (ASM)

Configuration c = (c1, · · · , cn) ∈ Zn

+. 0 is the sink.

ci ≡ number of grains of sand at vertex i.

1 3 6 9 10 8 7 5 4 2

4 3 1 4 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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The ASM

Add grains at random.

1 3 6 9 10 8 7 5 4 2

5 3 1 4 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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The ASM

Add grains at random. Vertex i is unstable if ci ≥ di.

1 3 6 9 10 8 7 5 4 2

5 3 1 1 4 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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The ASM

Unstable vertices topple, sending one grain along each incident edge.

1 3 6 9 10 8 7 5 4 2

5 3 1 1 4 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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The ASM

This may cause other vertices to become unstable.

1 3 6 9 10 8 7 5 4 2

5 3 1 5 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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The ASM

Unstable vertices topple. The sink absorbs grains.

1 3 6 9 10 8 7 5 4 2

6 4 2 1 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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The ASM

Process eventually stabilises. Stabilisation does not depend on

  • rder of topplings. σ(c) := stabilisation of c.

1 3 6 9 10 8 7 5 4 2

4 2 2 1 2 2 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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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

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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

  • ften. These configurations are called recurrent, their set is

Rec(G).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Minimal recurrent configurations

Define a partial order on Rec(G): c c′ iff ci ≤ c′

i for all

1 ≤ i ≤ n. Recmin(G) := {c ∈ Rec(G); c is minimal for }.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Minimal recurrent configurations

Define a partial order on Rec(G): c c′ iff ci ≤ c′

i for all

1 ≤ i ≤ n. Recmin(G) := {c ∈ Rec(G); c is minimal for }. Define level(c) =

n

  • i=1

ci + d0 − |E(G)|. For c ∈ Rec(G), we have 0 ≤ level(c) ≤ |E(G)| − n. The level polynomial of G is LevelG(x) =

  • c∈Rec(G)

xlevel(c).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Minimal recurrent configurations

Define a partial order on Rec(G): c c′ iff ci ≤ c′

i for all

1 ≤ i ≤ n. Recmin(G) := {c ∈ Rec(G); c is minimal for }. Define level(c) =

n

  • i=1

ci + d0 − |E(G)|. For c ∈ Rec(G), we have 0 ≤ level(c) ≤ |E(G)| − n. The level polynomial of G is LevelG(x) =

  • c∈Rec(G)

xlevel(c). Fact(??) We have Recmin(G) = {c ∈ Rec(G); level(c) = 0}.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Dhar’s burning criterion

Theorem (Dhar 89) Let G = G(n, B), c in Stab(G). Set ˜ c := c + 1A, where A is the set of neighbours of the sink, and 1A

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

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Thomas Selig Permutations and the ASM on Ferrers graphs

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r 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

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K1,m1,...,mk with dominating sink (Cori, Poulalhon 02).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K1,m1,...,mk with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs Km,n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes.

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 24

Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K1,m1,...,mk with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs Km,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

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SLIDE 25

Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K1,m1,...,mk with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs Km,n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes. Graph decompositions (Dukes, S. 16). Count them: LevelG(x) = TG(1, x) where TG =Tutte polynomial (Merino 97, Cori, Le Borgne 03, Bernardi 08).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Recurrent configurations

Question: how to find all recurrent configurations for a given graph

  • r family of graphs?

Trees and cycles are easy. Complete graphs (Cori, Rossin 00), parking functions. Wheel graphs (Cori, Dartois 04). Complete multipartite graphs K1,m1,...,mk with dominating sink (Cori, Poulalhon 02). Complete bipartite graphs Km,n (Dukes, Le Borgne 13, Dukes et al. 14), parallelogram polyominoes. Graph decompositions (Dukes, S. 16). Count them: LevelG(x) = TG(1, x) where TG =Tutte polynomial (Merino 97, Cori, Le Borgne 03, Bernardi 08). This talk: Ferrers graphs.

Thomas Selig Permutations and the ASM on Ferrers graphs

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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

(a) an EWT

1 1 1 1 1 1 1 1 1 1 1

(b) not an EWT

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and acyclic orientations

F G(F)

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EWTs and acyclic orientations

F G(F) 1 1 1 1 1 1 1 1 1 1 1

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EWTs and acyclic orientations

F G(F) 1 1 1 1 1 1 1 1 1 1 1

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and acyclic orientations

F G(F) 1 1 1 1 1 1 1 1 1 1 1 (0 =↓= |, 1 =↑= |)

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and acyclic orientations

F G(F) 1 1 1 1 1 1 1 1 1 1 1 (0 =↓= |, 1 =↑= |) EWT(F) ↔ {Ac. Or. of G(F) where top-left vertex = unique sink}.

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and minimal recurrent configurations

For T ∈ EWT(F), define c(T ) = (c1, . . . , cn) by: ci =

  • |{1′s in row i} if i ∈ RL(F)

|{0′s in column i} if i ∈ CL(F) .

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and minimal recurrent configurations

For T ∈ EWT(F), define c(T ) = (c1, . . . , cn) by: ci =

  • |{1′s in row i} if i ∈ RL(F)

|{0′s in column i} if i ∈ CL(F) . 1 1 1 1 1 1 1 1 1 1 1 1

9 6 3 1 8 7 5 4 2 10

c(T ) = (3, 1, 1, 0, 2, 2, 3, 1, 0, 2).

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and minimal recurrent configurations

For T ∈ EWT(F), define c(T ) = (c1, . . . , cn) by: ci =

  • |{1′s in row i} if i ∈ RL(F)

|{0′s in column i} if i ∈ CL(F) .

Thomas Selig Permutations and the ASM on Ferrers graphs

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EWTs and minimal recurrent configurations

For T ∈ EWT(F), define c(T ) = (c1, . . . , cn) by: ci =

  • |{1′s in row i} if i ∈ RL(F)

|{0′s in column i} if i ∈ CL(F) . Theorem (SSS 17) For any Ferrers diagram F, the map Φ : EWT(F) → Recmin(G(F)) T → c(T ) is a bijection.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Canonical toppling

Let G = G(n, B) and c ∈ Rec. Dhar’s criterion: starting from c, after toppling the sink, every vertex topples exactly once, stabilising to c.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Canonical toppling

Let G = G(n, B) and c ∈ Rec. Dhar’s criterion: starting from c, after toppling the sink, every vertex topples exactly once, stabilising to c. Definition Let c ∈ Rec(G). The canonical toppling of c is the sequence CanonTop(c) = (B1 = {0}), T1, B2, T2, . . . s.t. Ti (resp. Bi) is exactly the set of unstable vertices after toppling those of B1, T1, . . . , Bi (resp. B1, T1, . . . , Ti−1). N.B.: We have [n] =

i≥1

(Bi ⊔ Ti).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Example

1 3 6 9 10 8 7 5 4 2

3 1 2 2 1 3 2 1 c = (3, 1, 1, 0, 2, 2, 3, 1, 0, 2).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Example

1 3 6 9 10 8 7 5 4 2

3 1 2 2 1 3 2 1 c = (3, 1, 1, 0, 2, 2, 3, 1, 0, 2). CanonTop(c) = 0 − 2 5 7 − 6 1 − 8 10 − 9 3 − 4.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Permutations

Use one-line notation π = π1 · · · πn. Let ˜ π := 0 · π = ˜ π0˜ π1 · · · ˜ πn.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Permutations

Use one-line notation π = π1 · · · πn. Let ˜ π := 0 · π = ˜ π0˜ π1 · · · ˜ πn. For π ∈ Sn, let AscTop(π) := {πi; ˜ πi > ˜ πi−1} (ascent tops). DesBot(π) = [n] \ AscTop(π) (descent bottoms, includes 0). For B ⊆ [n − 1] with 0 ∈ B, Sn(B) := {π ∈ Sn; DesBot(π) = B}.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Permutations

Use one-line notation π = π1 · · · πn. Let ˜ π := 0 · π = ˜ π0˜ π1 · · · ˜ πn. For π ∈ Sn, let AscTop(π) := {πi; ˜ πi > ˜ πi−1} (ascent tops). DesBot(π) = [n] \ AscTop(π) (descent bottoms, includes 0). For B ⊆ [n − 1] with 0 ∈ B, Sn(B) := {π ∈ Sn; DesBot(π) = B}. Definition The run decomposition of π ∈ Sn is the sequence of blocks, where each block is a maximal set of consecutive ascent tops or descent bottoms of ˜ π, denoted RunDec(π).

Thomas Selig Permutations and the ASM on Ferrers graphs

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Permutations

Use one-line notation π = π1 · · · πn. Let ˜ π := 0 · π = ˜ π0˜ π1 · · · ˜ πn. For π ∈ Sn, let AscTop(π) := {πi; ˜ πi > ˜ πi−1} (ascent tops). DesBot(π) = [n] \ AscTop(π) (descent bottoms, includes 0). For B ⊆ [n − 1] with 0 ∈ B, Sn(B) := {π ∈ Sn; DesBot(π) = B}. Definition The run decomposition of π ∈ Sn is the sequence of blocks, where each block is a maximal set of consecutive ascent tops or descent bottoms of ˜ π, denoted RunDec(π). If π = 2 5 7 6 1 8 10 9 3 4, then RunDec(π) = 0 − 2 5 7 − 6 1 − 8 10 − 9 3 − 4.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Permutations

Use one-line notation π = π1 · · · πn. Let ˜ π := 0 · π = ˜ π0˜ π1 · · · ˜ πn. For π ∈ Sn, let AscTop(π) := {πi; ˜ πi > ˜ πi−1} (ascent tops). DesBot(π) = [n] \ AscTop(π) (descent bottoms, includes 0). For B ⊆ [n − 1] with 0 ∈ B, Sn(B) := {π ∈ Sn; DesBot(π) = B}. Definition The run decomposition of π ∈ Sn is the sequence of blocks, where each block is a maximal set of consecutive ascent tops or descent bottoms of ˜ π, denoted RunDec(π). If π = 2 5 7 6 1 8 10 9 3 4, then RunDec(π) = 0 − 2 5 7 − 6 1 − 8 10 − 9 3 − 4. N.B.: RunDec(π) uniquely determines π.

Thomas Selig Permutations and the ASM on Ferrers graphs

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Minimal recurrent configurations

For π ∈ S, define c = c(π) = (c1, . . . , cn) by ci =

  • |{j ∈ AscTop(π); π = · · · i · · · j · · · and i < j}| if i ∈ DesBot(π)

|{j ∈ DesBot(π); π = · · · i · · · j · · · and i > j}| if i ∈ AscTop(π)

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 47

Minimal recurrent configurations

For π ∈ S, define c = c(π) = (c1, . . . , cn) by ci =

  • |{j ∈ AscTop(π); π = · · · i · · · j · · · and i < j}| if i ∈ DesBot(π)

|{j ∈ DesBot(π); π = · · · i · · · j · · · and i > j}| if i ∈ AscTop(π) If π = 2 5 7 6 1 8 10 9 3 4, then c(π) = (3, 1, 1, 0, 2, 2, 3, 1, 0, 2).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 48

Minimal recurrent configurations

For π ∈ S, define c = c(π) = (c1, . . . , cn) by ci =

  • |{j ∈ AscTop(π); π = · · · i · · · j · · · and i < j}| if i ∈ DesBot(π)

|{j ∈ DesBot(π); π = · · · i · · · j · · · and i > j}| if i ∈ AscTop(π) If π = 2 5 7 6 1 8 10 9 3 4, then c(π) = (3, 1, 1, 0, 2, 2, 3, 1, 0, 2). Theorem (DSSS ++) For any B ⊆ [n − 1] with 0 ∈ B, the map Ψ : Sn(B) → Recmin(G(n, B)) π → c(π) is a bijection. Moreover, RunDec(π) = CanonTop(c(π)).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 49

Minimal recurrent configurations

For π ∈ S, define c = c(π) = (c1, . . . , cn) by ci =

  • |{j ∈ AscTop(π); π = · · · i · · · j · · · and i < j}| if i ∈ DesBot(π)

|{j ∈ DesBot(π); π = · · · i · · · j · · · and i > j}| if i ∈ AscTop(π) If π = 2 5 7 6 1 8 10 9 3 4, then c(π) = (3, 1, 1, 0, 2, 2, 3, 1, 0, 2). Theorem (DSSS ++) For any B ⊆ [n − 1] with 0 ∈ B, the map Ψ : Sn(B) → Recmin(G(n, B)) π → c(π) is a bijection. Moreover, RunDec(π) = CanonTop(c(π)). N.B.: ci is the number of neighbours of i in G(n, B) which appear to the right of i in π.

Thomas Selig Permutations and the ASM on Ferrers graphs

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“Proof”

π = π1 · · · πn, c = c(π), G = G(n, DesBot(π)).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 51

“Proof”

π = π1 · · · πn, c = c(π), G = G(n, DesBot(π)). Starting from c and toppling the sink, the vertices π1, . . . , πn can be toppled in that order. So c ∈ Rec(G) and CanonTop(c) = RunDec(π).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 52

“Proof”

π = π1 · · · πn, c = c(π), G = G(n, DesBot(π)). Starting from c and toppling the sink, the vertices π1, . . . , πn can be toppled in that order. So c ∈ Rec(G) and CanonTop(c) = RunDec(π). level(c) = 0, so c is minimal: Ψ is well defined.

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 53

“Proof”

π = π1 · · · πn, c = c(π), G = G(n, DesBot(π)). Starting from c and toppling the sink, the vertices π1, . . . , πn can be toppled in that order. So c ∈ Rec(G) and CanonTop(c) = RunDec(π). level(c) = 0, so c is minimal: Ψ is well defined. CanonTop(c) uniquely determines c ∈ Recmin(G), so Ψ is a bijection.

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 54

Find all recurrents?

If c ∈ Rec(G), ∃ˆ c ∈ Recmin(G), ˆ c c.

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 55

Find all recurrents?

If c ∈ Rec(G), ∃ˆ c ∈ Recmin(G), ˆ c c. Problem: may be many such ˆ c’s.

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 56

Find all recurrents?

If c ∈ Rec(G), ∃ˆ c ∈ Recmin(G), ˆ c c. Problem: may be many such ˆ c’s. Idea: use canonical toppling, which uniquely determines minimal recurrent configurations! I.e. ∀c ∈ Rec(G), ∃!ˆ c ∈ Recmin(G), CanonTop(c) = CanonTop(ˆ c) (also satisfies ˆ c c).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 57

Find all recurrents?

If c ∈ Rec(G), ∃ˆ c ∈ Recmin(G), ˆ c c. Problem: may be many such ˆ c’s. Idea: use canonical toppling, which uniquely determines minimal recurrent configurations! I.e. ∀c ∈ Rec(G), ∃!ˆ c ∈ Recmin(G), CanonTop(c) = CanonTop(ˆ c) (also satisfies ˆ c c). Thus, if R(ˆ c) := {c ∈ Rec(G); CanonTop(c) = CanonTop(ˆ c)}, we have Rec(G) =

  • ˆ

c∈Recmin(G)

R(ˆ c).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 58

Find all recurrents?

If c ∈ Rec(G), ∃ˆ c ∈ Recmin(G), ˆ c c. Problem: may be many such ˆ c’s. Idea: use canonical toppling, which uniquely determines minimal recurrent configurations! I.e. ∀c ∈ Rec(G), ∃!ˆ c ∈ Recmin(G), CanonTop(c) = CanonTop(ˆ c) (also satisfies ˆ c c). Thus, if R(ˆ c) := {c ∈ Rec(G); CanonTop(c) = CanonTop(ˆ c)}, we have Rec(G) =

  • ˆ

c∈Recmin(G)

R(ˆ c). It remains to describe R(ˆ c).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 59

The excess vector

Definition For π ∈ Sn, write RunDec(π) = B1, A1, B2, A2, . . .. Define the excess vector (of π) λ(π) = (λ1, . . . , λn) by: λi =

  • |{j ∈ Bk; j < i}|

if i ∈ Ak |{j ∈ Ak−1; j > i}| if i ∈ Bk .

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 60

The excess vector

Definition For π ∈ Sn, write RunDec(π) = B1, A1, B2, A2, . . .. Define the excess vector (of π) λ(π) = (λ1, . . . , λn) by: λi =

  • |{j ∈ Bk; j < i}|

if i ∈ Ak |{j ∈ Ak−1; j > i}| if i ∈ Bk . N.B.: λi ≥ 1 for all 1 ≤ i ≤ n.

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 61

The excess vector

Definition For π ∈ Sn, write RunDec(π) = B1, A1, B2, A2, . . .. Define the excess vector (of π) λ(π) = (λ1, . . . , λn) by: λi =

  • |{j ∈ Bk; j < i}|

if i ∈ Ak |{j ∈ Ak−1; j > i}| if i ∈ Bk . N.B.: λi ≥ 1 for all 1 ≤ i ≤ n. If RunDec(π) = 0 − 2 5 7 − 6 1 − 8 10 − 9 3 − 4, then λ(π) = (3, 1, 2, 1, 1, 1, 1, 2, 1, 2).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 62

The excess vector

Definition For π ∈ Sn, write RunDec(π) = B1, A1, B2, A2, . . .. Define the excess vector (of π) λ(π) = (λ1, . . . , λn) by: λi =

  • |{j ∈ Bk; j < i}|

if i ∈ Ak |{j ∈ Ak−1; j > i}| if i ∈ Bk . N.B.: λi ≥ 1 for all 1 ≤ i ≤ n. If RunDec(π) = 0 − 2 5 7 − 6 1 − 8 10 − 9 3 − 4, then λ(π) = (3, 1, 2, 1, 1, 1, 1, 2, 1, 2). Theorem(DSSS ++) Let B ⊆ [n − 1] with 0 ∈ B, G = G(n, B). For any ˆ c ∈ Recmin(G), R(ˆ c) = {c ∈ Zn; 0 ≤ ci − ˆ ci ≤ λi − 1 for all 1 ≤ i ≤ n}, where λ = (λ1, . . . , λn) is the excess vector of π = Ψ−1(ˆ c).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 63

Consequences

|R(ˆ c)| =

n

  • i=1

λi (with λ = λ

  • Ψ(−1)(ˆ

c)

  • .

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-64
SLIDE 64

Consequences

|R(ˆ c)| =

n

  • i=1

λi (with λ = λ

  • Ψ(−1)(ˆ

c)

  • .

Bijection {(π, ℓ); π ∈ Sn(B), 0 ≤ ℓ ≤ λ(π) − 1} ↔ {c ∈ Rec(G(n, B))} s.t. RunDec(π) = CanonTop(c).

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-65
SLIDE 65

Consequences

|R(ˆ c)| =

n

  • i=1

λi (with λ = λ

  • Ψ(−1)(ˆ

c)

  • .

Bijection {(π, ℓ); π ∈ Sn(B), 0 ≤ ℓ ≤ λ(π) − 1} ↔ {c ∈ Rec(G(n, B))} s.t. RunDec(π) = CanonTop(c). LevelG(n,B)(x) =

  • π∈Sn(B)
  • 1 + x + · · · + xλ1(π)−1

· · ·

  • 1 + · · · + xλn(π)−1

.

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-66
SLIDE 66

Consequences

|R(ˆ c)| =

n

  • i=1

λi (with λ = λ

  • Ψ(−1)(ˆ

c)

  • .

Bijection {(π, ℓ); π ∈ Sn(B), 0 ≤ ℓ ≤ λ(π) − 1} ↔ {c ∈ Rec(G(n, B))} s.t. RunDec(π) = CanonTop(c). LevelG(n,B)(x) =

  • π∈Sn(B)
  • 1 + x + · · · + xλ1(π)−1

· · ·

  • 1 + · · · + xλn(π)−1

.

  • 0∈B⊆[n−1]

|Rec(G(n, B))| is the number of alternating trees (or locally binary search trees) on n vertices (bijective proof).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 67

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case).

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-68
SLIDE 68

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case). Avalanche polynomial (done for Kn,2 and its subgraphs).

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-69
SLIDE 69

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case). Avalanche polynomial (done for Kn,2 and its subgraphs). Restriction to permutation subclasses.

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-70
SLIDE 70

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case). Avalanche polynomial (done for Kn,2 and its subgraphs). Restriction to permutation subclasses. Generalisations:

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-71
SLIDE 71

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case). Avalanche polynomial (done for Kn,2 and its subgraphs). Restriction to permutation subclasses. Generalisations:

Ferrers graphs with multiple edges (done).

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-72
SLIDE 72

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case). Avalanche polynomial (done for Kn,2 and its subgraphs). Restriction to permutation subclasses. Generalisations:

Ferrers graphs with multiple edges (done). multi-dimensional Ferrers graphs.

Thomas Selig Permutations and the ASM on Ferrers graphs

slide-73
SLIDE 73

Open questions

Structure of the sandpile group. c ⊕ c′ := σ(c + c′) (known in complete bipartite case). Avalanche polynomial (done for Kn,2 and its subgraphs). Restriction to permutation subclasses. Generalisations:

Ferrers graphs with multiple edges (done). multi-dimensional Ferrers graphs. More general bipartite graphs (doing).

Thomas Selig Permutations and the ASM on Ferrers graphs

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SLIDE 74

Thank you!

Thomas Selig Permutations and the ASM on Ferrers graphs