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Enumeration of Permutation Classes by Inflation of Independent Sets - - PowerPoint PPT Presentation

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Enumeration of Permutation Classes by Inflation of Independent Sets of Graphs mile Nadeau (based on joint work with Christian Bean and Henning Ulfarsson) Reykjavik University Permutations Patterns 2019 Staircase encoding For any permutation

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Enumeration of Permutation Classes by Inflation

  • f Independent Sets of Graphs

Émile Nadeau

(based on joint work with Christian Bean and Henning Ulfarsson)

Reykjavik University

Permutations Patterns 2019

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SLIDE 2

Staircase encoding

For any permutation π we can extract the left-to-right minima and place them on the diagonal of a square grid.

Example

π = ¯ 6¯ 598¯ 17432

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Staircase encoding

For any permutation π we can extract the left-to-right minima and place them on the diagonal of a square grid.

Example

π = ¯ 6¯ 598¯ 17432

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Staircase encoding

For any permutation π we can extract the left-to-right minima and place them on the diagonal of a square grid.

Example

π = ¯ 6¯ 598¯ 17432

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Staircase encoding

We can then record the permutations contained in each cell. We call this the staircase encoding of the permutation

Example

π = ¯ 6¯ 598¯ 17432

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Staircase encoding

We can then record the permutations contained in each cell. We call this the staircase encoding of the permutation

Example

π = ¯ 6¯ 598¯ 17432 21 1 321

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Staircase encoding

Many different permutations can have the same staircase encoding

Example

π′ = ¯ 6¯ 598¯ 14372 and π′′ = ¯ 6¯ 597¯ 18432 have the same staircase encoding has the permutation π. π π′ π′′

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Our goal

We will use the staircase encoding to describe the structure of permutation classes and give their generating functions.

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Our goal

We will use the staircase encoding to describe the structure of permutation classes and give their generating functions. Given a permutation class we need to be able to ◮ Describe the image of the class under the staircase encoding

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Our goal

We will use the staircase encoding to describe the structure of permutation classes and give their generating functions. Given a permutation class we need to be able to ◮ Describe the image of the class under the staircase encoding ◮ Find the number of permutations in the class that correspond to each staircase encoding in the image, i.e., the number of ways of interleaving rows and columns

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Permutations avoiding 123

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123 avoiders

We start with the example of Av(123) from Bean, Tannock and Ulfarsson in Pattern avoiding permutations and independent sets in graphs.

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123 avoiders

We start with the example of Av(123) from Bean, Tannock and Ulfarsson in Pattern avoiding permutations and independent sets in graphs.

Definition

We say that a cell of the staircase encoding is active if it contains a non-empty permutation.

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123 avoiders

We start with the example of Av(123) from Bean, Tannock and Ulfarsson in Pattern avoiding permutations and independent sets in graphs.

Definition

We say that a cell of the staircase encoding is active if it contains a non-empty permutation. To describe all the staircase encodings that can be obtained from 123 avoiders we follow a two step process

  • 1. Find all the possible sets of active cells for a staircase encoding
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123 avoiders

We start with the example of Av(123) from Bean, Tannock and Ulfarsson in Pattern avoiding permutations and independent sets in graphs.

Definition

We say that a cell of the staircase encoding is active if it contains a non-empty permutation. To describe all the staircase encodings that can be obtained from 123 avoiders we follow a two step process

  • 1. Find all the possible sets of active cells for a staircase encoding
  • 2. Find the permutations that can occupy any of those cells
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Sets of active cells

Avoiding 123 puts constraints on which pairs of cells can contain permutations. We encode those restriction by edges of a graph

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Sets of active cells

Avoiding 123 puts constraints on which pairs of cells can contain permutations. We encode those restriction by edges of a graph

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Sets of active cells

Avoiding 123 puts constraints on which pairs of cells can contain permutations. We encode those restriction by edges of a graph

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Sets of active cells

An independent set of the graph defines a subset of cells that contain permutations in the staircase encoding of a 123 avoider.

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Sets of active cells

An independent set of the graph defines a subset of cells that contain permutations in the staircase encoding of a 123 avoider. Let F(x, y) be the generating function such that the coefficient of xnyk is the number of independent sets of size k in a grid with n left-to-right minima. F(x, y) satisfies F(x, y) = 1 + xF(x, y) + xyF(x, y)2 1 − y(F(x, y) − 1).

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Sets of active cells

An independent set of the graph defines a subset of cells that contain permutations in the staircase encoding of a 123 avoider. Let F(x, y) be the generating function such that the coefficient of xnyk is the number of independent sets of size k in a grid with n left-to-right minima. F(x, y) satisfies F(x, y) = 1 + xF(x, y) + xyF(x, y)2 1 − y(F(x, y) − 1). Finally, the permutations in all cells of the staircase encoding must avoid 12.

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From encoding to permutations

Points in two cells in the same row of the grid cannot create 12. Hence, all points of the left cell are above the points of the right

  • cell. We say that the rows are decreasing.
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From encoding to permutations

Points in two cells in the same row of the grid cannot create 12. Hence, all points of the left cell are above the points of the right

  • cell. We say that the rows are decreasing.
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From encoding to permutations

Points in two cells in the same row of the grid cannot create 12. Hence, all points of the left cell are above the points of the right

  • cell. We say that the rows are decreasing.
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From encoding to permutations

Points in two cells in the same row of the grid cannot create 12. Hence, all points of the left cell are above the points of the right

  • cell. We say that the rows are decreasing.
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From encoding to permutations

Points in two cells in the same row of the grid cannot create 12. Hence, all points of the left cell are above the points of the right

  • cell. We say that the rows are decreasing.

Similarly columns are said to be decreasing. For each staircase encoding, only one permutation in Av(123) is mapped to it by the staircase encoding because only one interleaving is possible.

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The number of staircase encodings for 123 avoiders of length n is given by the generating function F

  • x,

x 1−x

  • .

The staircase encoding is a bijection for Av(123).

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The number of staircase encodings for 123 avoiders of length n is given by the generating function F

  • x,

x 1−x

  • .

The staircase encoding is a bijection for Av(123).

Theorem

The generating function of Av(123) is F

  • x,

x 1−x

  • .

Remark

A symmetric results can be stated for 132 avoiders.

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Avoiding 2314 and 3124

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Row-up and column-up patterns

We want to replace 123 with two new patterns: ru = 2314 and cu = 3124. ru cu We use this notation since ru forbids increasing sequences along rows while cu forbids increasing sequences along columns. Hence, all permutations in Av(2314, 3124) have a different staircase encoding.

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Sets of active cells

If we look at the left-to-right minima of the patterns as left-to-right minima on the grid we see the same constraints for sets of active cells as in the 123 case.

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Sets of active cells

If we look at the left-to-right minima of the patterns as left-to-right minima on the grid we see the same constraints for sets of active cells as in the 123 case.

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Sets of active cells

If we look at the left-to-right minima of the patterns as left-to-right minima on the grid we see the same constraints for sets of active cells as in the 123 case. F(x, y) also describes the independent sets.

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Set of active cells

Cells avoid 2314 and 3124.

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Set of active cells

Cells avoid 2314 and 3124. Each staircase encoding gives a permutation in the class ru cu

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Enumeration

Theorem

Let P be a set of skew-indecomposable permutations. The generating function of Av(2314, 3124, 1 ⊕ P) is F(x, B(x) − 1) where B(x) is the generating function of Av(2314, 3124, P).

Example

The generating function of Av(2314, 3124, 1234) is F

  • x, 1 − √1 − 4x

2x − 1

  • since Av(2314, 3124, 123) = Av(123).
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Enumeration

Example

A(x), the generation function of Av(2314, 3124) satisfies A(x) = F(x, A(x) − 1). Solving the equation gives A(x) = 3 − x − √ 1 − 6x + x2 2 .

Remark

A symmetric results can be derived for rd = 2143 and cd = 3142 and sum-indecomposable patterns. rd cd

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New cores

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We look at avoiders of ru = 2314, cu = 3124 and cd = 3142.

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We look at avoiders of ru = 2314, cu = 3124 and cd = 3142.

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We look at avoiders of ru = 2314, cu = 3124 and cd = 3142.

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We look at avoiders of ru = 2314, cu = 3124 and cd = 3142.

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Theorem

Let P be a set of skew-indecomposable permutations. Then the generating function for Av(ru, cu, cd, 1 ⊕ P) = Av(2314, 3124, 3142, 1 ⊕ P) is G(x, B(x) − 1) where B(x) is the generating function for Av(2314, 3124, 3124, P).

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Theorem

Let P be a set of skew-indecomposable permutations. Then the generating function for Av(ru, cu, cd, 1 ⊕ P) = Av(2314, 3124, 3142, 1 ⊕ P) is G(x, B(x) − 1) where B(x) is the generating function for Av(2314, 3124, 3124, P).

Remark

A symmetric version can be done for ru, cu, cd.

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Regarding the bases Av(rd, cd, cu) and Av(rd, cd, ru) some extra care is needed since cu and ru are sum-indecomposable.

. . .

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Regarding the bases Av(rd, cd, cu) and Av(rd, cd, ru) some extra care is needed since cu and ru are sum-indecomposable.

. . .

However this can be handled by ◮ tracking the number of rows/columns of the independent set ◮ using a different permutation class for the leftmost/topmost cell in each row

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Avoiding 2134 and 2413

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2134 rd = 2413

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2134 rd = 2413

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2134 rd = 2413

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2134 rd = 2413

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2134 rd = 2413

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2134 rd = 2413

Remark

Note that all the diagonal cells are disconnected from the graph.

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H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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H H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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H s y′ s y′ s y′ s y′ s z

◮ x for substitution with Av(2413, 2134) ◮ y for substitution with Av+(12), y′ = y + 1 ◮ z for substitution with Av+(2413, 2134) (with maximum remove) ◮ s for substitution with Av(213)

H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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H s y′ s y′ s y′ s y′ s z

◮ x for substitution with Av(2413, 2134) ◮ y for substitution with Av+(12), y′ = y + 1 ◮ z for substitution with Av+(2413, 2134) (with maximum remove) ◮ s for substitution with Av(213)

yzs H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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H s y′ s y′ s y′ s y′ s z

◮ x for substitution with Av(2413, 2134) ◮ y for substitution with Av+(12), y′ = y + 1 ◮ z for substitution with Av+(2413, 2134) (with maximum remove) ◮ s for substitution with Av(213)

yzs

1 1−s(y+1)

H(x, y, z, s) = 1 + xH(x, y, z, s) + yzsH(x, y, z, s) 1 − s(y + 1)

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We show that for a set of patterns P satisfying: for all π ∈ P ◮ π is skew-indecomposable, ◮ π avoids and ◮ π contains

  • r π = α ⊕ 1 with α skew-indecomposable.

Theorem

The generating function of Av(2134, 2413, 1 ⊕ P) is H

  • xB,

x 1 − x , B − 1, xC

  • where

◮ B(x) is the generating function of Av(2134, 2413, ×P), ◮ C(x) is the generating function of Av(213, ×P×),

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Example

A(x), the generating function of Av(2134, 2413) satisfies A(x) = H

  • xA(x),

x 1 − x , A(x) − 1, 1 − √1 − 4x 2 − 1

  • The equation can be solved explicitly.
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Unbalanced Wilf-equivalence

A(x) is the generating function of Av(2314, 3124, 13524, 12435). A(x) = F(x, B(x) − 1) where B(x) is the generating function of Av(2314, 3124, 2413, 1324).

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Unbalanced Wilf-equivalence

A(x) is the generating function of Av(2314, 3124, 13524, 12435). A(x) = F(x, B(x) − 1) where B(x) is the generating function of Av(2314, 3124, 2413, 1324). B(x) = G(x, C(x) − 1) where C(x) is the generating function of Av(2314, 3124, 2413, 213) = Av(213)

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Unbalanced Wilf-equivalence

A(x) is the generating function of Av(2314, 3124, 13524, 12435). A(x) = F(x, B(x) − 1) where B(x) is the generating function of Av(2314, 3124, 2413, 1324). B(x) = G(x, C(x) − 1) where C(x) is the generating function of Av(2314, 3124, 2413, 213) = Av(213) Computing A(x) gives the same generating function as for the class Av(2413, 2134).

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Conclusion

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Basis that can be handled

Basis Subclasses References 2314, 3124 8 Schröder number 2413, 3142 8 Schröder number 2314, 3124, 2413, 3142 64 Atkinson & Stitt (2002) 2314, 3124, 2413 8 Mansour & Shattuck (2017) 2314, 3124, 3142* 8 Mansour & Shattuck (2017) 2413, 3142, 2314 8 Callan, Mansour & Shattuck (2017) 2413, 3142, 3124* 8 Callan, Mansour & Shattuck (2017) 2413, 3124 4 Albert, Atkinson & Vatter (2014) 2314, 3142 4 Albert, Atkinson & Vatter (2014) 2134, 2413 2 Albert, Atkinson & Vatter (2014)

*Symmetry of an other class.

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Future work

◮ Using length five patterns with 3 left-to-right minima

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Future work

◮ Using length five patterns with 3 left-to-right minima ◮ Consider also the right-to-left maxima

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Future work

◮ Using length five patterns with 3 left-to-right minima ◮ Consider also the right-to-left maxima ◮ Other Wilf-equivalences and bijective proof