Pattern Avoiding and Quasisymetric function

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Pattern Avoiding and Quasisymetric function

Food for thought talk 2016

Kuang Sittipong Thamrongpairoj

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Outline

What is pattern avoiding permutation?

What are quasisymmetric functions?

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Outline

What is pattern avoiding permutation? What are quasisymmetric functions?

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Outline

What is pattern avoiding permutation? What are quasisymmetric functions?

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Acknowledgement

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Pattern avoiding permutation

What is a pattern in permutation? Consider σ = 21534 ∈ S5

Then, we get rid of some numbers in σ σ0 = 254

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Pattern avoiding permutation

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Pattern avoiding permutation

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Pattern avoiding permutation

If we can get τ from σ via the process, we say σ contains pattern τ .

We say that σ avoids τ , if σ does not contain τ . Let Sn(τ ) be the set of σ ∈ Sn avoiding τ

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Pattern avoiding permutation

If we can get τ from σ via the process, we say σ contains pattern τ . We say that σ avoids τ , if σ does not contain τ .

Let Sn(τ ) be the set of σ ∈ Sn avoiding τ

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Pattern avoiding permutation

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Pattern avoiding permutation

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Pattern avoiding permuation

.

Pop quiz

:

what is |Sn(12)|?

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Pattern avoiding permuation

.

Pop quiz

: what is |Sn(12)|?

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Pattern avoiding permuation

.

Pop quiz

: what is |Sn(12)|?

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Pattern avoiding permutation

Given two patterns τ1 and τ2, we say that τ1 and τ2 are Wilf-equivalent, if

|S_n(τ1)| = |Sn(τ2)| for all n.

Let’s look at equivalence classes of pattern of length 3. First, it is clear that 123 and 321 are equivalent

Proof: If σ avoids 123, the “reverse” of σ will avoid 321. Second, 132, 231, 312, 213 are equivalent.

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Pattern avoiding permutation

Let’s look at equivalence classes of pattern of length 3.

First, it is clear that 123 and 321 are equivalent

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Pattern avoiding permutation

Proof: If σ avoids 123, the “reverse” of σ will avoid 321.

Second, 132, 231, 312, 213 are equivalent.

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Pattern avoiding permutation

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Pattern avoiding permutation

Theorem (Knuth, 1973)

123 and 132 are equivalent. Therefore, there is only one equivlance class for patterns of length 3.

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Pattern avoiding permutation

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Pattern avoiding permutation

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Pattern avoiding permutation

Before we move on...

For any σ ∈ Sn, let Des(σ) = {i | σi > σi +1}

For example, if σ = 51423, then Des(σ) = {1, 3}

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Pattern avoiding permutation

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Pattern avoiding permutation

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Pattern avoiding permutation

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