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Signed posets and a B -symmetric generalization of Stanleys - - PowerPoint PPT Presentation

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Signed graphs Colorings Acyclic orientations References Signed posets and a B -symmetric generalization of Stanleys acyclicity theorem Jake Huryn, Kat Husar, and Hannah Johnson joint work with Eric Fawcett, Torey Hilbert and Mikey Reilly

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1/22 Signed graphs Colorings Acyclic orientations References

Signed posets and a B-symmetric generalization

  • f Stanley’s acyclicity theorem

Jake Huryn, Kat Husar, and Hannah Johnson

joint work with Eric Fawcett, Torey Hilbert and Mikey Reilly under Sergei Chmutov

The Ohio State University

August 16, 2020

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Our Goal

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Graphs and Posets

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Graphs and Posets

Poset: A partially ordered set b < a < c b < d < c b < d < e Arrow Convention:

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Graphs and Posets

Poset: A partially ordered set b < a < c b < d < c b < d < e Arrow Convention:

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Signed Graphs

Possible Edges:

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Signed Graph

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Signed Graph

How can we tell if it’s acyclic?

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Covering Graphs

Signed Graph:

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Covering Graphs

Signed Graph: Covering Graph:

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Covering Graphs

Signed Graph: Covering Graph:

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Covering Graphs

Signed Graph: Covering Graph:

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Covering Graph Examples

Signed Graph:

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Covering Graph Examples

Signed Graph: Covering Graph:

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Covering Graph Examples

Signed Graph: Covering Graph:

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Covering Graph Examples

Signed Graph: Covering Graph:

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Signed Posets

Signed Graph:

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Signed Posets

Signed Graph: Covering Graph:

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Signed Posets

Signed Graph: Covering Graph: Signed Posets:

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Signed Posets

Signed Graph: Covering Graph: Signed Posets:

+c < −d < +b < +a −d > −e −c > +d > −b > −a +d < +e

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Proper coloring

Proper coloring: A function κ: V (G) → Z such that for all adjacent vertices v, w, κ(v) ≠ σ(v, w)κ(w).

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Proper coloring

Proper coloring: A function κ: V (G) → Z such that for all adjacent vertices v, w, κ(v) ≠ σ(v, w)κ(w). Examples of improper coloring: +1 −1 − +1 +1 + −1 −1 + −

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B-symmetric chromatic function

YG(. . . , x−2, x−1, x0, x1, x2, . . . ) :=

  • κ:V (G)→Z

proper

  • v∈V (G)

xκ(v)

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B-symmetric chromatic function

YG(. . . , x−2, x−1, x0, x1, x2, . . . ) :=

  • κ:V (G)→Z

proper

  • v∈V (G)

xκ(v) Examples: Y

− = · · · + x−2 + x−1 + x1 + x2 + · · ·

Y −

+

=

  • i,j

xixj −

  • i

x2

i − 2

  • i

xix−i + 2x2

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Linear extension of a signed poset

Poset P: Lift ˜ P: v

w

v −v w −w

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Linear extension of a signed poset

Poset P: Lift ˜ P: v

w

v −v w −w B-symmetric linear extensions: −w −v v w ω −w −v v w α −w −v v w β

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Order-preserving coloring

If v < w in the lift ˜ P and κ: ˜ P → Z is order-preserving, then κ(v) < κ(w). Note: κ(v) = −κ(−v)

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Order-preserving coloring

If v < w in the lift ˜ P and κ: ˜ P → Z is order-preserving, then κ(v) < κ(w). Note: κ(v) = −κ(−v) Some examples: −b −a a b

−w −v v w

a −a −w

−v v w

a −a 0 −w

−v v w

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What did we want to do again?

Goal: Given a signed graph G and a nonnegative integer k,

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What did we want to do again?

Goal: Given a signed graph G and a nonnegative integer k, use YG to compute sinkG(k), the number of acyclic orientations

  • f G with k sinks.
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What did we want to do again?

Goal: Given a signed graph G and a nonnegative integer k, use YG to compute sinkG(k), the number of acyclic orientations

  • f G with k sinks.

Specifically, we will find a linear map ϕ: BSym → Z[t] such that ϕ(YG) =

∞

  • k=0

sinkG(k)tk.

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What did we want to do again?

Goal: Given a signed graph G and a nonnegative integer k, use YG to compute sinkG(k), the number of acyclic orientations

  • f G with k sinks.

Specifically, we will find a linear map ϕ: BSym → Z[t] such that ϕ(YG) =

∞

  • k=0

sinkG(k)tk. For example, if G = − , then ϕ(YG) = 1 + 3t:

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Step-by-step

Step 1: Decompose YG into a sum, over signed posets, of quasi-B-symmetric functions: YG =

  • P is an acyclic
  • rientation of G

YP

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Step-by-step

Step 1: Decompose YG into a sum, over signed posets, of quasi-B-symmetric functions: YG =

  • P is an acyclic
  • rientation of G

YP Step 2: Find a convenient expression for YP as a sum over the linear extensions of P: YP =

  • α is a linear

extension of P

QA(α,ω),ε(α)

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Step-by-step

Step 1: Decompose YG into a sum, over signed posets, of quasi-B-symmetric functions: YG =

  • P is an acyclic
  • rientation of G

YP Step 2: Find a convenient expression for YP as a sum over the linear extensions of P: YP =

  • α is a linear

extension of P

QA(α,ω),ε(α) Step 3: Use the convenient expression to find a linear map ϕ: QBSym → Z[t] such that for any signed poset P with k sinks, ϕ(YP) = tk.

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Step 1: YP

Given a signed poset P with vertices v1, . . . , vn, define YP =

  • κ is an order-preserving

coloring of P

xκ(v1) · · · xκ(vn). YP is quasi-B-symmetric.

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Step 1: YP

Given a signed poset P with vertices v1, . . . , vn, define YP =

  • κ is an order-preserving

coloring of P

xκ(v1) · · · xκ(vn). YP is quasi-B-symmetric.

Lemma

For any signed graph G, YG =

  • P is an acyclic
  • rientation of G

YP.

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Step 1: YP

Given a signed poset P with vertices v1, . . . , vn, define YP =

  • κ is an order-preserving

coloring of P

xκ(v1) · · · xκ(vn). YP is quasi-B-symmetric.

Lemma

For any signed graph G, YG =

  • P is an acyclic
  • rientation of G

YP.

Proof.

Given a coloring, induce the unique acyclic orientation of G which makes the coloring order-preserving. ◻

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Step 2: A convenient expression

Our expression for YP is a sum over all order-preserving colorings, but we want to write it as a (finite) sum over just the linear

  • extensions. But how?
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Step 2: A convenient expression

Our expression for YP is a sum over all order-preserving colorings, but we want to write it as a (finite) sum over just the linear

  • extensions. But how?

Compare linear extensions to encode how they can be “averaged” to give order-preserving colorings.

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Step 2: A convenient expression, continued

Fix a linear extension ω. Given a linear extension α, how do we determine what α should contribute to YP?

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Step 2: A convenient expression, continued

Fix a linear extension ω. Given a linear extension α, how do we determine what α should contribute to YP? Look at the disagreements between two linear extensions: −b −a a b

−w −v v w

ω

−w −v v w

α

  • −1
  • 1
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Step 2: A convenient expression, continued

Fix a linear extension ω. Given a linear extension α, how do we determine what α should contribute to YP? Look at the disagreements between two linear extensions: −b −a a b

−w −v v w

ω

−w −v v w

α

  • −1
  • 1

↝

−w −v v w

c −c

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Step 2: A convenient expression, continued

Fix a linear extension ω. Given a linear extension α, how do we determine what α should contribute to YP? Look at the disagreements between two linear extensions: −b −a a b

−w −v v w

ω

−w −v v w

α

  • −1
  • 1

↝

−w −v v w

c −c

  • 0<a<b

x−axb +

  • 0<c

x−cxc

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Step 2: A convenient expression, continued

Fix a linear extension ω. Given a linear extension α, how do we determine what α should contribute to YP? Look at the disagreements between two linear extensions: −b −a a b

−w −v v w

ω

−w −v v w

α

  • −1
  • 1

↝

−w −v v w

c −c

  • 0<a<b

x−axb +

  • 0<c

x−cxc =

  • 0<a≤b

x−axb

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Step 2: A convenient expression, continued

Fix a linear extension ω. Given a linear extension α, how do we determine what α should contribute to YP? Look at the disagreements between two linear extensions: −b −a a b

−w −v v w

ω

−w −v v w

α

  • −1
  • 1

↝

−w −v v w

c −c

  • 0<a<b

x−axb +

  • 0<c

x−cxc =

  • 0<a≤b

x−axb = Q{0},−+

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Step 2: A convenient expression, continued

For S ⊆ {0, . . . , n − 1} and ε ∈ {−1, 1}n, define QS,ε :=

  • 0≤i1≤···≤in

s∈S = ⇒ is<is+1 0∈S = ⇒ 0<i1

xε1i1 · · · xεnin.

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Step 2: A convenient expression, continued

For S ⊆ {0, . . . , n − 1} and ε ∈ {−1, 1}n, define QS,ε :=

  • 0≤i1≤···≤in

s∈S = ⇒ is<is+1 0∈S = ⇒ 0<i1

xε1i1 · · · xεnin.

Lemma

Let P be a signed poset. Then YP =

  • α is a linear

extension of P

QA(α,ω),ε(α).

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Step 3: An awful function

Let S ⊆ {0, . . . , n − 1} and ε ∈ {−1, 1}n. Then ϕ(QS,ε) :=

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Step 3: An awful function

Let S ⊆ {0, . . . , n − 1} and ε ∈ {−1, 1}n. Then ϕ(QS,ε) :=

                            

t(t − 1)k S = {0, . . . , n − k − 1} and εi > 0 for each i ∈ {n − k, . . . , n} (t − 1)k S = {0, . . . , n − k − 1}, εn−k < 0, and εi > 0 for each i ∈ {n − k + 1, . . . , n} (t − 1)n S = ∅ and εi > 0 for each i ∈ {1, . . . , n}

  • therwise.
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Step 3: An awful function

Let S ⊆ {0, . . . , n − 1} and ε ∈ {−1, 1}n. Then ϕ(QS,ε) :=

                            

t(t − 1)k S = {0, . . . , n − k − 1} and εi > 0 for each i ∈ {n − k, . . . , n} (t − 1)k S = {0, . . . , n − k − 1}, εn−k < 0, and εi > 0 for each i ∈ {n − k + 1, . . . , n} (t − 1)n S = ∅ and εi > 0 for each i ∈ {1, . . . , n}

  • therwise.

Obnoxious obstruction: The QS,ε’s aren’t linearly independent! Q{0},−+ − Q{0,1},−+ = Q{0},+− − Q{0,1},+−.

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The conclusion

Theorem

For any signed graph G, ϕ(YG) =

∞

  • k=0

sinkG(k)tk.

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The conclusion

Theorem

For any signed graph G, ϕ(YG) =

∞

  • k=0

sinkG(k)tk.

Proof.

We have ϕ(YG) =

  • Y is an acyclic
  • rientation of G

ϕ(YP) =

  • Y is an acyclic
  • rientation of G

tsink(P) ◻

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What now?

Can we find a natural basis for BSym on which ϕ acts nicely?

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What now?

Can we find a natural basis for BSym on which ϕ acts nicely? Can this result be refined/modified by choosing a better ϕ?

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What now?

Can we find a natural basis for BSym on which ϕ acts nicely? Can this result be refined/modified by choosing a better ϕ? What other information about G is “linear” in YG?

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What now?

Can we find a natural basis for BSym on which ϕ acts nicely? Can this result be refined/modified by choosing a better ϕ? What other information about G is “linear” in YG? What other nice properties does YG have?

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What now?

Can we find a natural basis for BSym on which ϕ acts nicely? Can this result be refined/modified by choosing a better ϕ? What other information about G is “linear” in YG? What other nice properties does YG have? What variations on YG might have nice properties?

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References

Richard P. Stanley. “Acyclic orientations of graphs”. In: Discrete mathematics 5 (1973), pp. 171–178. Richard P. Stanley. “A symmetric function generalization of the chromatic polynomial of a graph”. In: Advances in mathematics 111 (1995), pp. 166–194. Thomas Zaslavsky. “Orientations of signed graphs”. In: European Journal of Combinatorics 12 (1991), pp. 361–375.