Signed graphs Colorings Acyclic orientations References Signed posets and a B -symmetric generalization of 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 1/22
Signed graphs Colorings Acyclic orientations References Our Goal 2/22
Signed graphs Colorings Acyclic orientations References Graphs and Posets 3/22
Signed graphs Colorings Acyclic orientations References Graphs and Posets Arrow Convention: Poset: A partially ordered set b < a < c b < d < c b < d < e 3/22
Signed graphs Colorings Acyclic orientations References Graphs and Posets Arrow Convention: Poset: A partially ordered set b < a < c b < d < c b < d < e 3/22
Signed graphs Colorings Acyclic orientations References Signed Graphs Possible Edges: 4/22
Signed graphs Colorings Acyclic orientations References Signed Graph 5/22
Signed graphs Colorings Acyclic orientations References Signed Graph How can we tell if it’s acyclic? 5/22
Signed graphs Colorings Acyclic orientations References Covering Graphs Signed Graph: 6/22
Signed graphs Colorings Acyclic orientations References Covering Graphs Covering Graph: Signed Graph: 6/22
Signed graphs Colorings Acyclic orientations References Covering Graphs Covering Graph: Signed Graph: 6/22
Signed graphs Colorings Acyclic orientations References Covering Graphs Covering Graph: Signed Graph: 6/22
Signed graphs Colorings Acyclic orientations References Covering Graph Examples Signed Graph: 7/22
Signed graphs Colorings Acyclic orientations References Covering Graph Examples Covering Graph: Signed Graph: 7/22
Signed graphs Colorings Acyclic orientations References Covering Graph Examples Covering Graph: Signed Graph: 7/22
Signed graphs Colorings Acyclic orientations References Covering Graph Examples Covering Graph: Signed Graph: 7/22
Signed graphs Colorings Acyclic orientations References Signed Posets Signed Graph: 8/22
Signed graphs Colorings Acyclic orientations References Signed Posets Covering Graph: Signed Graph: 8/22
Signed graphs Colorings Acyclic orientations References Signed Posets Covering Graph: Signed Graph: Signed Posets: 8/22
Signed graphs Colorings Acyclic orientations References Signed Posets Covering Graph: Signed Graph: Signed Posets: + c < − d < + b < + a − d > − e − c > + d > − b > − a + d < + e 8/22
Signed graphs Colorings Acyclic orientations References 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 ). 9/22
Signed graphs Colorings Acyclic orientations References 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 0 9/22
Signed graphs Colorings Acyclic orientations References B-symmetric chromatic function � � Y G ( . . . , x − 2 , x − 1 , x 0 , x 1 , x 2 , . . . ) : = x κ ( v ) κ : V ( G ) → Z v ∈ V ( G ) proper 10/22
Signed graphs Colorings Acyclic orientations References B-symmetric chromatic function � � Y G ( . . . , x − 2 , x − 1 , x 0 , x 1 , x 2 , . . . ) : = x κ ( v ) κ : V ( G ) → Z v ∈ V ( G ) proper Examples: − = · · · + x − 2 + x − 1 + x 1 + x 2 + · · · Y � � x 2 � x i x − i + 2 x 2 i − 2 Y − x i x j − = 0 i , j i i + 10/22
Signed graphs Colorings Acyclic orientations References Linear extension of a signed poset Lift ˜ Poset P : P : v − v w v w − w 11/22
Signed graphs Colorings Acyclic orientations References Linear extension of a signed poset Lift ˜ Poset P : P : v − v w v w − w B-symmetric linear extensions: β ω α − v w − v w − v − w w v − w v − w v 11/22
Signed graphs Colorings Acyclic orientations References 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 ) 12/22
Signed graphs Colorings Acyclic orientations References 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: − v − v w − v a a b w a w 0 − w − a − w − a − w − a − b v v v 12/22
Signed graphs Colorings Acyclic orientations References What did we want to do again? Goal: Given a signed graph G and a nonnegative integer k , 13/22
Signed graphs Colorings Acyclic orientations References What did we want to do again? Goal: Given a signed graph G and a nonnegative integer k , use Y G to compute sink G ( k ) , the number of acyclic orientations of G with k sinks. 13/22
Signed graphs Colorings Acyclic orientations References What did we want to do again? Goal: Given a signed graph G and a nonnegative integer k , use Y G to compute sink G ( k ) , the number of acyclic orientations of G with k sinks. Specifically, we will find a linear map ϕ : BSym → Z [ t ] such that ∞ � sink G ( k ) t k . ϕ ( Y G ) = k = 0 13/22
Signed graphs Colorings Acyclic orientations References What did we want to do again? Goal: Given a signed graph G and a nonnegative integer k , use Y G to compute sink G ( k ) , the number of acyclic orientations of G with k sinks. Specifically, we will find a linear map ϕ : BSym → Z [ t ] such that ∞ � sink G ( k ) t k . ϕ ( Y G ) = k = 0 − For example, if G = , then ϕ ( Y G ) = 1 + 3 t : 13/22
Signed graphs Colorings Acyclic orientations References Step-by-step Step 1: Decompose Y G into a sum, over signed posets, of quasi- B -symmetric functions: � Y G = Y P P is an acyclic orientation of G 14/22
Signed graphs Colorings Acyclic orientations References Step-by-step Step 1: Decompose Y G into a sum, over signed posets, of quasi- B -symmetric functions: � Y G = Y P P is an acyclic orientation of G Step 2: Find a convenient expression for Y P as a sum over the linear extensions of P : � Y P = Q A ( α,ω ) ,ε ( α ) α is a linear extension of P 14/22
Signed graphs Colorings Acyclic orientations References Step-by-step Step 1: Decompose Y G into a sum, over signed posets, of quasi- B -symmetric functions: � Y G = Y P P is an acyclic orientation of G Step 2: Find a convenient expression for Y P as a sum over the linear extensions of P : � Y P = Q A ( α,ω ) ,ε ( α ) α is a linear extension of P 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 ( Y P ) = t k . 14/22
Signed graphs Colorings Acyclic orientations References Step 1: Y P Given a signed poset P with vertices v 1 , . . . , v n , define � x κ ( v 1 ) · · · x κ ( v n ) . Y P = κ is an order-preserving coloring of P Y P is quasi- B -symmetric. 15/22
Signed graphs Colorings Acyclic orientations References Step 1: Y P Given a signed poset P with vertices v 1 , . . . , v n , define � x κ ( v 1 ) · · · x κ ( v n ) . Y P = κ is an order-preserving coloring of P Y P is quasi- B -symmetric. Lemma For any signed graph G, � Y G = Y P . P is an acyclic orientation of G 15/22
Signed graphs Colorings Acyclic orientations References Step 1: Y P Given a signed poset P with vertices v 1 , . . . , v n , define � x κ ( v 1 ) · · · x κ ( v n ) . Y P = κ is an order-preserving coloring of P Y P is quasi- B -symmetric. Lemma For any signed graph G, � Y G = Y P . P is an acyclic orientation of G Proof. Given a coloring, induce the unique acyclic orientation of G which makes the coloring order-preserving. ◻ 15/22
Signed graphs Colorings Acyclic orientations References Step 2: A convenient expression Our expression for Y P 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? 16/22
Signed graphs Colorings Acyclic orientations References Step 2: A convenient expression Our expression for Y P 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. 16/22
Signed graphs Colorings Acyclic orientations References 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 Y P ? 17/22
Signed graphs Colorings Acyclic orientations References 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 Y P ? Look at the disagreements between two linear extensions: ω α − v w b � w − v 1 a 0 − a � − w v − 1 − b v − w 17/22
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