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Permutation patterns The “Galois” connections Pat () - Comp (n) and gPat () - gComp (n) The Galois closures gComp (n) G and G Permutation groups, permutation patterns, and Galois connections Erkko Lehtonen and Reinhard P¨ oschel Technische Universit¨ at Dresden Institute of Algebra AAA92 Arbeitstagung Allgemeine Algebra Workshop on General Algebra Praha 27.5.2016 AAA92, Praha, May 27, 2016 R. P¨oschel, Permutation groups, permutation patterns, and Galois connections (1/21)
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Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Jan 18, 2021

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Page 1: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation groups, permutation patterns,and Galois connections

Erkko Lehtonen and Reinhard Poschel

Technische Universitat DresdenInstitute of Algebra

AAA92Arbeitstagung Allgemeine Algebra

Workshop on General AlgebraPraha 27.5.2016

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (1/21)

Page 2: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Outline

Permutation patterns

The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)

The Galois closures gComp(n) G and Galois kernels gPat(`)H

Remarks and references

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (2/21)

Page 3: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Outline

Permutation patterns

The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)

The Galois closures gComp(n) G and Galois kernels gPat(`)H

Remarks and references

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (3/21)

Page 4: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 5: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

5 43 21

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 6: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

3

5 421

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 7: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

31

5 42

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 8: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

1

3

5 42

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 9: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

1

53

42

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 10: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

1

53

2

4

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 11: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

21

53

4

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 12: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

53

21 4

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 13: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

53 2 41

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 14: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

53 2 π41

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 15: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

53 2 π41

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoided

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 16: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Some motivationWhich number sequences (permutations) can be sorted by a stack?

53 2 π41

Proposition

A permutation π can be sorted by a stack if and only if it does notcontain a subsequence . . . a . . . b . . . c . . . with c < a < b.

i.e., such “patterns” abc (like 352) must be avoidedAAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (4/21)

Page 17: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:

π1 . . . πn := (π(1), . . . , π(n)).

e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4

graphical representation:

π = 31524 ∈ S5

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)

Page 18: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:

π1 . . . πn := (π(1), . . . , π(n)).

e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4

graphical representation:

π = 31524 ∈ S5

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)

Page 19: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:

π1 . . . πn := (π(1), . . . , π(n)).

e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4

graphical representation:

π = 31524 ∈ S5

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)

Page 20: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

PermutationsA permutation π ∈ Sn (bijection π ∈ [n][n], [n] := {1, . . . , n}) willbe considered as a word (n-tuple π ∈ [n]n) of length n:

π1 . . . πn := (π(1), . . . , π(n)).

e.g. π = 31524 ∈ [5]5 is the permutation1 7→ 3, 2 7→ 1, 3 7→ 5, 4 7→ 2, 5 7→ 4

graphical representation:π = 31524 ∈ S5

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (5/21)

Page 21: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

π = 31524 ∈ S5

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 22: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

π = 31524 ∈ S5

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 23: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 24: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

1 2 3

3

2

1

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 25: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

σ = 231 ∈ S3

1 2 3

3

2

1

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 26: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

σ ≤ πσ is pattern of π

π involves σ

π = 31524 ∈ S5

σ = 231 ∈ S3

1 2 3

3

2

1

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 27: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns (Example)

352

231

231 = red(352)

reduced form

substring

π = 31524 ∈ S5

1 2 3

3

2

1

1 2 3 4 5

1

5

4

3

2

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (6/21)

Page 28: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns

` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `

such that σ = red(u)(σ is `-pattern of π, or π involves σ)

π avoids σ :⇐⇒ σ � π.

Pat(`) π := {σ ∈ S` | σ ≤ π}

The pattern involvement relation ≤ is a partial order on the setP :=

⋃n≥1 Sn of all finite permutations.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)

Page 29: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns

` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `

such that σ = red(u)(σ is `-pattern of π, or π involves σ)

π avoids σ :⇐⇒ σ � π.

Pat(`) π := {σ ∈ S` | σ ≤ π}

The pattern involvement relation ≤ is a partial order on the setP :=

⋃n≥1 Sn of all finite permutations.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)

Page 30: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns

` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `

such that σ = red(u)(σ is `-pattern of π, or π involves σ)

π avoids σ :⇐⇒ σ � π.

Pat(`) π := {σ ∈ S` | σ ≤ π}

The pattern involvement relation ≤ is a partial order on the setP :=

⋃n≥1 Sn of all finite permutations.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)

Page 31: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Permutation patterns

` ≤ n, σ ∈ S`, π ∈ Snσ ≤ π :⇐⇒ there exists a substring u of π of length `

such that σ = red(u)(σ is `-pattern of π, or π involves σ)

π avoids σ :⇐⇒ σ � π.

Pat(`) π := {σ ∈ S` | σ ≤ π}

The pattern involvement relation ≤ is a partial order on the setP :=

⋃n≥1 Sn of all finite permutations.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (7/21)

Page 32: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Outline

Permutation patterns

The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)

The Galois closures gComp(n) G and Galois kernels gPat(`)H

Remarks and references

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (8/21)

Page 33: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection Pat(`)−Comp(n)

The relation σ � π (π avoids σ) induces a Galois connectionbetween subsets of S` and Sn. The corresponding “monotoneGalois connection” (residuation) is given by the following operators(monotone w.r.t. ⊆):

For S ⊆ S`, T ⊆ Sn (` ≤ n) let

Comp(n) S := {τ ∈ Sn | Pat(`) τ ⊆ S} = {τ ∈ Sn | ∀σ′ ∈ S` \ S : σ′ � τ},

Pat(`) T :=⋃τ∈T

Pat(`) τ = S` \ {σ′ ∈ S` | ∀τ ∈ T : σ′ � τ}.

In particular we have the defining property of a monotone Galoisconnection:

Pat(`) T ⊆ S ⇐⇒ T ⊆ Comp(n) S

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (9/21)

Page 34: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection Pat(`)−Comp(n)

The relation σ � π (π avoids σ) induces a Galois connectionbetween subsets of S` and Sn. The corresponding “monotoneGalois connection” (residuation) is given by the following operators(monotone w.r.t. ⊆):

For S ⊆ S`, T ⊆ Sn (` ≤ n) let

Comp(n) S := {τ ∈ Sn | Pat(`) τ ⊆ S} = {τ ∈ Sn | ∀σ′ ∈ S` \ S : σ′ � τ},

Pat(`) T :=⋃τ∈T

Pat(`) τ = S` \ {σ′ ∈ S` | ∀τ ∈ T : σ′ � τ}.

In particular we have the defining property of a monotone Galoisconnection:

Pat(`) T ⊆ S ⇐⇒ T ⊆ Comp(n) S

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (9/21)

Page 35: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection Pat(`)−Comp(n)

The relation σ � π (π avoids σ) induces a Galois connectionbetween subsets of S` and Sn. The corresponding “monotoneGalois connection” (residuation) is given by the following operators(monotone w.r.t. ⊆):

For S ⊆ S`, T ⊆ Sn (` ≤ n) let

Comp(n) S := {τ ∈ Sn | Pat(`) τ ⊆ S} = {τ ∈ Sn | ∀σ′ ∈ S` \ S : σ′ � τ},

Pat(`) T :=⋃τ∈T

Pat(`) τ = S` \ {σ′ ∈ S` | ∀τ ∈ T : σ′ � τ}.

In particular we have the defining property of a monotone Galoisconnection:

Pat(`) T ⊆ S ⇐⇒ T ⊆ Comp(n) S

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (9/21)

Page 36: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Connection to permutation groups

Proposition

If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.

(The converse does not hold)

Sketch of the proof.

Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:

Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.

Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)

Page 37: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Connection to permutation groups

Proposition

If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.

(The converse does not hold)

Sketch of the proof.

Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:

Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.

Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)

Page 38: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Connection to permutation groups

Proposition

If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.

(The converse does not hold)

Sketch of the proof.

Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:

Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.

Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)

Page 39: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Connection to permutation groups

Proposition

If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.

(The converse does not hold)

Sketch of the proof.

Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:

Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.

Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)

Page 40: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Connection to permutation groups

Proposition

If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.

(The converse does not hold)

Sketch of the proof.

Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:

Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.

Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)

Page 41: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Connection to permutation groups

Proposition

If S is a subgroup of S`, then Comp(n) S is a subgroup of Sn.

(The converse does not hold)

Sketch of the proof.

Assume that S ≤ S`. Let π, τ ∈ Comp(n) S .Thus Pat(`) π,Pat(`) τ ⊆ S .Crucial observation:

Pat(`) π−1 = (Pat(`) π)−1 := {σ−1 | σ ∈ Pat(`) π},Pat(`) πτ ⊆ (Pat(`) π)(Pat(`) τ) := {σσ′ | σ ∈ Pat(`) π, σ′ ∈ Pat(`) τ}.

Consequently, π−1 and πτ also belong to Comp(n) S (sinceS ≤ S`). Thus Comp(n) S is a group.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (10/21)

Page 42: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection gPat(`)− gComp(n)

Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:

gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},

gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H

Pat(`) τ〉,

This gives also a monotone Galois connection since we have:

gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G

Thus

gPat(`) gComp(n) G ⊆ G (kernel operator),

H ⊆ gComp(n) gPat(`)H (closure operator),

gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),

gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)

Page 43: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection gPat(`)− gComp(n)

Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:

gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},

gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H

Pat(`) τ〉,

This gives also a monotone Galois connection since we have:

gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G

Thus

gPat(`) gComp(n) G ⊆ G (kernel operator),

H ⊆ gComp(n) gPat(`)H (closure operator),

gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),

gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)

Page 44: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection gPat(`)− gComp(n)

Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:

gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},

gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H

Pat(`) τ〉,

This gives also a monotone Galois connection since we have:

gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G

Thus

gPat(`) gComp(n) G ⊆ G (kernel operator),

H ⊆ gComp(n) gPat(`)H (closure operator),

gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),

gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)

Page 45: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection gPat(`)− gComp(n)

Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:

gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},

gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H

Pat(`) τ〉,

This gives also a monotone Galois connection since we have:

gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G

Thus

gPat(`) gComp(n) G ⊆ G (kernel operator),

H ⊆ gComp(n) gPat(`)H (closure operator),

gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),

gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)

Page 46: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection gPat(`)− gComp(n)

Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:

gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},

gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H

Pat(`) τ〉,

This gives also a monotone Galois connection since we have:

gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G

Thus

gPat(`) gComp(n) G ⊆ G (kernel operator),

H ⊆ gComp(n) gPat(`)H (closure operator),

gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),

gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)

Page 47: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

The (monotone) Galois connection gPat(`)− gComp(n)

Modification of Comp−Patfor permutation groups G ≤ S` and H ≤ Sn:

gComp(n) G := 〈Comp(n) G 〉 = Comp(n) G = {τ ∈ Sn | Pat(`) τ ⊆ G},

gPat(`)H := 〈Pat(`)H〉 = 〈⋃τ∈H

Pat(`) τ〉,

This gives also a monotone Galois connection since we have:

gPat(`)H ⊆ G ⇐⇒ H ⊆ gComp(n) G

Thus

gPat(`) gComp(n) G ⊆ G (kernel operator),

H ⊆ gComp(n) gPat(`)H (closure operator),

gComp(n) G = gComp(n) gPat(`) gComp(n) G (closures),

gPat(`)H = gPat(`) gComp(n) gPat(`)H (kernels).

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (11/21)

Page 48: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Outline

Permutation patterns

The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)

The Galois closures gComp(n) G and Galois kernels gPat(`)H

Remarks and references

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (12/21)

Page 49: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Question

How to characterize the Galois closures and the Galois kernels ofthe monotone Galois connection gComp(n)− gPat(`) (` < n) ?

Answer: as automorphism groups of special relations(pc-relations and pc-extended invariants, resp. )

Recall the (usual) Galois connection Aut− Inv (betweenpermutations π ∈ Sn and relations % ∈ Reln on {1, . . . , n}):

AutR := {π ∈ Sn | ∀% ∈ R : π . %} for R ⊆ Reln,

InvT := {% ∈ Reln | ∀π ∈ T : π . %} for T ⊆ Sn,

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (13/21)

Page 50: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Question

How to characterize the Galois closures and the Galois kernels ofthe monotone Galois connection gComp(n)− gPat(`) (` < n) ?

Answer: as automorphism groups of special relations(pc-relations and pc-extended invariants, resp. )

Recall the (usual) Galois connection Aut− Inv (betweenpermutations π ∈ Sn and relations % ∈ Reln on {1, . . . , n}):

AutR := {π ∈ Sn | ∀% ∈ R : π . %} for R ⊆ Reln,

InvT := {% ∈ Reln | ∀π ∈ T : π . %} for T ⊆ Sn,

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (13/21)

Page 51: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Question

How to characterize the Galois closures and the Galois kernels ofthe monotone Galois connection gComp(n)− gPat(`) (` < n) ?

Answer: as automorphism groups of special relations(pc-relations and pc-extended invariants, resp. )

Recall the (usual) Galois connection Aut− Inv (betweenpermutations π ∈ Sn and relations % ∈ Reln on {1, . . . , n}):

AutR := {π ∈ Sn | ∀% ∈ R : π . %} for R ⊆ Reln,

InvT := {% ∈ Reln | ∀π ∈ T : π . %} for T ⊆ Sn,

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (13/21)

Page 52: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-relations

For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).

Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):

hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.

Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.

For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as

%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.

% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.

For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)

Page 53: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-relations

For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).

Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):

hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.

Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.

For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as

%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.

% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.

For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)

Page 54: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-relations

For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).

Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):

hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.

Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.

For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as

%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.

% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.

For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)

Page 55: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-relations

For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).

Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):

hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.

Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.

For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as

%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.

% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.

For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)

Page 56: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-relations

For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).

Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):

hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.

Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.

For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as

%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.

% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.

For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)

Page 57: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-relations

For I ∈ P`(n) (`-element subsets of [n]), let hI : [`]→ I be theorder-isomorphism ([`],≤)→ (I ,≤).

Example ` = 3, n = 5, I := {3, 5, 2} = {2, 3, 5} ∈ P3(5):

hI : 1 7→ 2, 2 7→ 3, 3 7→ 5.

Thus, for s := (3, 1), r := (5, 2), we get hI (s) = r and h−1I (r) = s.

For k ≤ ` ≤ n, % ⊆ [n]k , σ ⊆ [`]k define %∨ ⊆ [`]k and σ∧ ⊆ [n]k as

%∨ := {h−1I (r) | r ∈ %, Im r ⊆ I ∈ P`(n)},σ∧ := {hJ(s) | s ∈ σ, J ∈ P`(n)}.

% ⊆ [n]k is called pattern closed relation (pc-relation) if %∨∧ = %.

For k = ` this means r ∈ % ∧ red(r) = red(s) =⇒ s ∈ %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (14/21)

Page 58: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Characterization of the Galois closures

The Galois closures of the closure operator gComp(n) gPat(`) canbe characterized by a single irreflexive k-ary pc-relation.

We have:

Theorem

(A) gComp(n) gPat(`)H = Aut pcInvH for H ≤ Sn.

(B) Let H be a subgroup of Sn. Then the following are equivalent:

(a) H is Galois closed, i.e., H = gComp(n) gPat(`)H,(a’) ∃G ≤ S` : H = gComp(n) G ,(b) H = Aut pcInvH,(c) ∃ k ≤ `∃ % ⊆ [n]k6= : % = %∨∧︸ ︷︷ ︸

% is pc-relation

∧ H = Aut %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (15/21)

Page 59: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Characterization of the Galois closures

The Galois closures of the closure operator gComp(n) gPat(`) canbe characterized by a single irreflexive k-ary pc-relation.

We have:

Theorem

(A) gComp(n) gPat(`)H = Aut pcInvH for H ≤ Sn.

(B) Let H be a subgroup of Sn. Then the following are equivalent:

(a) H is Galois closed, i.e., H = gComp(n) gPat(`)H,(a’) ∃G ≤ S` : H = gComp(n) G ,(b) H = Aut pcInvH,(c) ∃ k ≤ `∃ % ⊆ [n]k6= : % = %∨∧︸ ︷︷ ︸

% is pc-relation

∧ H = Aut %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (15/21)

Page 60: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Characterization of the Galois closures

The Galois closures of the closure operator gComp(n) gPat(`) canbe characterized by a single irreflexive k-ary pc-relation.

We have:

Theorem

(A) gComp(n) gPat(`)H = Aut pcInvH for H ≤ Sn.

(B) Let H be a subgroup of Sn. Then the following are equivalent:

(a) H is Galois closed, i.e., H = gComp(n) gPat(`)H,(a’) ∃G ≤ S` : H = gComp(n) G ,(b) H = Aut pcInvH,(c) ∃ k ≤ `∃ % ⊆ [n]k6= : % = %∨∧︸ ︷︷ ︸

% is pc-relation

∧ H = Aut %.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (15/21)

Page 61: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-extended invariants

Let G ≤ S`. A relation σ ⊆ [`]k (k ≤ `) is called apattern closed extended invariant (pc-extended invariant) of G if

σ∧∨ = σ and σ∧ ∈ Inv Aut γG∧.

where γG := {s | s ∈ G} - the elements of G ≤ S` = [`]`6= ⊆ [`]`

are viewed as `-tuples.

pcExtG : set of all pc-extended invariants of G .

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (16/21)

Page 62: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-extended invariants

Let G ≤ S`. A relation σ ⊆ [`]k (k ≤ `) is called apattern closed extended invariant (pc-extended invariant) of G if

σ∧∨ = σ and σ∧ ∈ Inv Aut γG∧.

where γG := {s | s ∈ G} - the elements of G ≤ S` = [`]`6= ⊆ [`]`

are viewed as `-tuples.

pcExtG : set of all pc-extended invariants of G .

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (16/21)

Page 63: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

pc-extended invariants

Let G ≤ S`. A relation σ ⊆ [`]k (k ≤ `) is called apattern closed extended invariant (pc-extended invariant) of G if

σ∧∨ = σ and σ∧ ∈ Inv Aut γG∧.

where γG := {s | s ∈ G} - the elements of G ≤ S` = [`]`6= ⊆ [`]`

are viewed as `-tuples.

pcExtG : set of all pc-extended invariants of G .

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (16/21)

Page 64: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Characterization of the Galois kernels

Now the Galois kernels of the kernel operator gPat(`) gComp(n) canbe characterized by pc-extended invariant relations:

Theorem

gPat(`) gComp(n) G = Aut pcExtG for G ≤ S`.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (17/21)

Page 65: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Characterization of the Galois kernels

Now the Galois kernels of the kernel operator gPat(`) gComp(n) canbe characterized by pc-extended invariant relations:

Theorem

gPat(`) gComp(n) G = Aut pcExtG for G ≤ S`.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (17/21)

Page 66: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Outline

Permutation patterns

The “Galois” connections Pat(`)−Comp(n) andgPat(`)− gComp(n)

The Galois closures gComp(n) G and Galois kernels gPat(`)H

Remarks and references

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (18/21)

Page 67: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Remarks

For G ⊆ S`, M.D. Atkinson and R. Beals [1999] consideredthe sequence

G ,Comp(`+1) G , . . . ,Comp(n) G ,Comp(n+1) G , . . .

in the group case, i.e., G ≤ Sn (then Comp(n) G = gComp(n) G ),and asked for the “asymptotic” behaviour of the above sequence.

Recent and much more detailed results about this sequence:Erkko Lehtonen [2015/16]

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (19/21)

Page 68: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Remarks

For G ⊆ S`, M.D. Atkinson and R. Beals [1999] consideredthe sequence

G ,Comp(`+1) G , . . . ,Comp(n) G ,Comp(n+1) G , . . .

in the group case, i.e., G ≤ Sn (then Comp(n) G = gComp(n) G ),and asked for the “asymptotic” behaviour of the above sequence.

Recent and much more detailed results about this sequence:Erkko Lehtonen [2015/16]

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (19/21)

Page 69: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

Remarks

For G ⊆ S`, M.D. Atkinson and R. Beals [1999] consideredthe sequence

gPat(1) G , . . . , gPat(`−1) G ,G ,Comp(`+1) G , . . . ,Comp(n) G ,Comp(n+1) G , . . .

in the group case, i.e., G ≤ Sn (then Comp(n) G = gComp(n) G ),and asked for the “asymptotic” behaviour of the above sequence.

Recent and much more detailed results about this sequence:Erkko Lehtonen [2015/16]

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (19/21)

Page 70: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

References

M.D. Atkinson and R. Beals, Permuting mechanismsand closed classes of permutations. In: Combinatorics,computation & logic ’99 (Auckland), vol. 21 of Aust. Comput.Sci. Commun., Springer, Singapore, 1999, pp. 117–127.

E. Lehtonen, Permutation groups arising from patterninvolvement. arXiv:1605.05571.

E. Lehtonen and R. Poschel, Permutation groups,pattern involvement, and Galois connections. arXiv:1605.04516.

N. Ruskuc, Classes of permutations avoiding 231 or 321.Lecture given at TU Dresden, Nov. 25, 2015.

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (20/21)

Page 71: Permutation groups, permutation patterns, and Galois ...aaa.karlin.mff.cuni.cz/presentations/poeschel.pdf · Permutation groups, permutation patterns, and Galois connections Erkko

Permutation patterns The “Galois” connections Pat(`) − Comp(n) and gPat(`) − gComp(n) The Galois closures gComp(n) G and Galois kernels gPat(`) H Remarks and references

AAA92, Praha, May 27, 2016 R. Poschel, Permutation groups, permutation patterns, and Galois connections (21/21)