Combinatorics of Complete Non-Ambiguous Trees Thomas Selig joint - - PowerPoint PPT Presentation

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Combinatorics of Complete Non-Ambiguous Trees Thomas Selig joint - - PowerPoint PPT Presentation

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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

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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

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Permutations

Permutation = Bijection {1, . . . , n} → {1, . . . , n}.

Thomas Selig Combinatorics of CNATs

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Permutations

Permutation = Bijection {1, . . . , n} → {1, . . . , n}. One-line notation π = 561243 ∈ S6.

Thomas Selig Combinatorics of CNATs

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Permutations

Permutation = Bijection {1, . . . , n} → {1, . . . , n}. One-line notation π = 561243 ∈ S6. Diagram representation: 1 2 3 4 5 6 1 2 3 4 5 6

Thomas Selig Combinatorics of CNATs

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Inversions

Inversions are pairs (πi, πj) s.t. i < j and πi > πj.

Thomas Selig Combinatorics of CNATs

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Inversions

Inversions are pairs (πi, πj) s.t. i < j and πi > πj. 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

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Inversions

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

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Inversions

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

Thomas Selig Combinatorics of CNATs

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Permutation graphs

Permutation graph Gπ for π ∈ Sn: Vertex set [n] := {1, . . . , n}; Edge set Inv(π).

Thomas Selig Combinatorics of CNATs

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Permutation graphs

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

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Permutation graphs

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

Thomas Selig Combinatorics of CNATs

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Example

π = 561243 ∈ S6. 3 4 1 2 5 6

Thomas Selig Combinatorics of CNATs

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Connectivity

π ∈ Sn is decomposable if ∃k < n, {π1, . . . , πk} = {1, . . . , k}, indecomposable otherwise.

Thomas Selig Combinatorics of CNATs

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Connectivity

π ∈ Sn is decomposable if ∃k < n, {π1, . . . , πk} = {1, . . . , k}, indecomposable otherwise. Theorem A permutation graph Gπ is connected if, and only if, π is indecomposable.

Thomas Selig Combinatorics of CNATs

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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

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Underlying binary tree

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Underlying binary tree

Thomas Selig Combinatorics of CNATs

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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

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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

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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

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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

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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). This talk: refine this enumeration.

Thomas Selig Combinatorics of CNATs

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Properties of CNATs

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Properties of CNATs

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Properties of CNATs

Permutation?

Thomas Selig Combinatorics of CNATs

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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

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Counting CNATs

Definition The Tutte polynomial of a connected graph G = (V , E) is defined by TG(x, y) :=

  • S⊆E

(x − 1)cc(S)−1(y − 1)cc(S)+|S|−|V |, where cc(S) = number of connected components of (V , S).

Thomas Selig Combinatorics of CNATs

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Counting CNATs

Definition The Tutte polynomial of a connected graph G = (V , E) is defined by TG(x, y) :=

  • S⊆E

(x − 1)cc(S)−1(y − 1)cc(S)+|S|−|V |, 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}
  • = TGπ(1, 0).

Thomas Selig Combinatorics of CNATs

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External activity

Definition G = (V , E) a graph, T ⊆ E a spanning tree of G, <E a total

  • rder 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.

Thomas Selig Combinatorics of CNATs

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External activity

Definition G = (V , E) a graph, T ⊆ E a spanning tree of G, <E a total

  • rder 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 TG(1, 0) =

  • {T spanning tree of G; ext(T) = 0}
  • .

Thomas Selig Combinatorics of CNATs

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Edges of permutation graph

1 2 3 4 5 6 1 2 3 4 5 6 c c′ Define f (i, j) = (i, πj).

Thomas Selig Combinatorics of CNATs

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Edges of permutation graph

1 2 3 4 5 6 1 2 3 4 5 6 c c′ Define f (i, j) = (i, πj). f (c) = (4, 6), f (c′) = (5, 2).

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Edges of permutation graph

1 2 3 4 5 6 1 2 3 4 5 6 c c′ Define f (i, j) = (i, πj). 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 (Gπ).

Thomas Selig Combinatorics of CNATs

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Proof of theorem

1 2 3 4 5 6 1 2 3 4 5 6 3 4 1 2 5 6

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Proof of theorem

1 2 3 4 5 6 1 2 3 4 5 6 3 4 1 2 5 6 3 4 1 2 5 6 ˆ f

Thomas Selig Combinatorics of CNATs

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Proof of theorem

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

Thomas Selig Combinatorics of CNATs

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Open questions

Other combinatorial interpretations of TG(1, 0) (acyclic

  • rientations, Abelian sandpile model). Other interpretations of

CNATs?

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Open questions

Other combinatorial interpretations of TG(1, 0) (acyclic

  • rientations, Abelian sandpile model). Other interpretations of

CNATs? All spanning trees: a “multi-rooted” generalisation of CNATs?

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The decreasing case

A CNAT N is decreasing if Perm(N) = (n + 1)n · · · 1.

Thomas Selig Combinatorics of CNATs

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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

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The decreasing case

A CNAT N is decreasing if Perm(N) = (n + 1)n · · · 1. 1 2 3 4 5 6 G = Kn+1 and TG(1, 0) = n!.

Thomas Selig Combinatorics of CNATs

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The decreasing case

A CNAT N is decreasing if Perm(N) = (n + 1)n · · · 1. 1 2 3 4 5 6 G = Kn+1 and TG(1, 0) = n!. Bijective proof?

Thomas Selig Combinatorics of CNATs

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Bottom row decomposition

1 2 3 4 5 6 7 = 1 2 5 7 + 3 4 6 7

Thomas Selig Combinatorics of CNATs

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The bijection

Defined recursively:

Thomas Selig Combinatorics of CNATs

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The bijection

Defined recursively: Ψ( 1 2 3 4 5 6 7 ) = Ψ( 3 4 6 7 ) · Ψ( 1 2 5 7 )

Thomas Selig Combinatorics of CNATs

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The bijection

Defined recursively: Ψ( 1 2 3 4 5 6 7 ) = Ψ( 3 4 6 7 ) · Ψ( 1 2 5 7 ) Ψ( 3 4 6 7 ) = 3 · Ψ( 4 6 7 )

Thomas Selig Combinatorics of CNATs

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The bijection

Defined recursively: Ψ( 1 2 3 4 5 6 7 ) = Ψ( 3 4 6 7 ) · Ψ( 1 2 5 7 ) Ψ( 3 4 6 7 ) = 3 · Ψ( 4 6 7 ) Ψ( ) = ∅

Thomas Selig Combinatorics of CNATs

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The result

Theorem (S., Steingr´ ımsson ++) Ψ : {N ∈ CNAT; Perm(N) = (n + 1)n · · · 1} → Sn is a bijection. Moreover: dots in the bottom row of N → left-to-right minima in Ψ(N);

  • {empty rows of N}
  • = 1 +
  • {descents of Ψ(N)}
  • .

Thomas Selig Combinatorics of CNATs

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Open questions

Other properties of Ψ.

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Open questions

Other properties of Ψ. Non-recursive definition of Ψ.

Thomas Selig Combinatorics of CNATs

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Open questions

Other properties of Ψ. Non-recursive definition of Ψ. Extend to the multi-rooted case.

Thomas Selig Combinatorics of CNATs

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Open questions

Other properties of Ψ. Non-recursive definition of Ψ. Extend to the multi-rooted case. Other special cases, e.g. π = (m + 1)(m + 2) · · · (m + n)12 · · · m (complete bipartite graph).

Thomas Selig Combinatorics of CNATs

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Thank you!

Thomas Selig Combinatorics of CNATs