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The cycle matroid complexes of bi-coned graphs Lee, Kang-Ju (SNU) Joint work with Kook, Woong (SNU) The 4th KIAS Combinatorics Workshop, KIAS May 30, 2014
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Matroid Complexes
Graph G =(V, E)
Simplicial Complex ∆ = (V, K) where K ⊂ 2V ..
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Matroid Complexes
Graph G =(V, E)
Simplicial Complex ∆ = (V, K) where K ⊂ 2V ..
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Matroid Complexes
Graph G =(V, E)
Simplicial complex ∆ = (V, K) where K ⊂ 2V
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Matroid Complexes
. Definition . A simplicial complex is a pair ∆ = (V, K) where K ⊂ 2V satisfies (1) ∅ ∈ K , (2) if F ∈ K and G ⊂ F , then G ∈ K . .
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Matroid Complexes . Definition . A matroid M is a pair (E, I) where I ⊂ 2E , called the independent sets, satisfies (1) ∅ ∈ I ,
(2) if I ∈ I and I ′ ⊂ I , then I ′ ∈ I , and
(3) if I and I ′ are in I and |I| > |I ′ |, then there is e ∈ I \ I ′ such that I′ ∪ e ∈ I. .
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Matroid Complexes . Definition . A matroid M is a pair (E, I) where I ⊂ 2E , called the independent sets, satisfies (1) ∅ ∈ I ,
(2) if I ∈ I and I ′ ⊂ I , then I ′ ∈ I , and
(3) if I and I ′ are in I and |I| > |I ′ |, then there is e ∈ I \ I ′ such that I′ ∪ e ∈ I. . A matroid complex is denoted by ∆(M) .
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Matroid Complexes . Definition . A matroid M is a pair (E, I) where I ⊂ 2E , called the independent sets, satisfies (1) ∅ ∈ I ,
(2) if I ∈ I and I ′ ⊂ I , then I ′ ∈ I , and
(3) if I and I ′ are in I and |I| > |I ′ |, then there is e ∈ I \ I ′ such that I′ ∪ e ∈ I. . A matroid complex is denoted by ∆(M) . (3) implies that every maximal face has the same dimension.
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Matroid Complexes . Definition . Given a graph G , the cycle matroid M(G) is the pair (E(G), I(G)) .where I ∈ I(G) if and only if I is acyclic.
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Matroid Complexes . Definition . Given a graph G , the cycle matroid M(G) is the pair (E(G), I(G)) .where I ∈ I(G) if and only if I is acyclic. . Example .
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Matroid Complexes . Definition . Given a graph G , the cycle matroid M(G) is the pair (E(G), I(G)) .where I ∈ I(G) if and only if I is acyclic. . Example .
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Matroid Complexes . Definition . Given a graph G , the cycle matroid M(G) is the pair (E(G), I(G)) .where I ∈ I(G) if and only if I is acyclic. . Example .
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h-vector of simplicial complexes
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h-vector of simplicial complexes . Definition . Let ∆ be a simplicial complex of dim n − 1 . Define
f∆ (x) =
n ∑
fi x n−i
i=0
where fi is the number of faces of dim i − 1 and f0 = 1. Define
h∆ (x) =
n ∑
hi x n−i = f∆ (x − 1)
i=0
.and define h-vector as (h0 , h1 , · · · , hn ).
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h-vector of simplicial complexes . Definition . Let ∆ be a simplicial complex of dim n − 1. Define
f∆ (x) =
n ∑
fi x n−i
i=0
where fi is the number of faces of dim i − 1 and f0 = 1. Define
h∆ (x) =
n ∑
hi x n−i = f∆ (x − 1)
i=0
and h-vector as (h0 , h1 , · · · , hn ). .
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h-vector of coned graphs
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h-vector of coned graphs . Theorem (W. Kook, 2011) . .h-vector of a coned graph ← partially edge-rooted forests.
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h-vector of coned graphs . Theorem (W. Kook, 2011) . .h-vector of a coned graph ← partially edge-rooted forests.
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h-vector of coned graphs . Theorem (W. Kook, 2011) . .h-vector of a coned graph ← partially edge-rooted forests.
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h-vector of coned graphs Let G be a graph with |V (G)| = n.
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h-vector of coned graphs Let G be a graph with |V (G)| = n.
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . .h-vector of a bi-coned graph ← partially B-edge-rooted forests.
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . .h-vector of a bi-coned graph ← partially B-edge-rooted forests.
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . .h-vector of a bi-coned graph ← partially B-edge-rooted forests.
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . .h-vector of a bi-coned graph ← partially B-edge-rooted forests.
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . .h-vector of a bi-coned graph ← partially B-edge-rooted forests.
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h-vector of bi-coned graphs Let G be a bipartite graph with a bipartition ([m], [n′ ]).
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h-vector of bi-coned graphs Let G be a bipartite graph with a bipartition ([m], [n′ ]).
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h-vector of bi-coned graphs Let G be a bipartite graph with a bipartition ([m], [n′ ]).
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . An exponential generating function
∑ m,n≥0
h∆(M(Km+1,n+1 ) (x)
ym zn m! n!
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h-vector of bi-coned graphs . Theorem (W. Kook, K. Lee, 2014+) . An exponential generating function
∑ m,n≥0
h∆(M(Km+1,n+1 ) (x)
ym zn m! n!
=x exp(xT B (y ,z)) exp(RB (y ,z))+exp(xT B (y ,z)) exp(RB (y ,z))B(y ,z) ∑ ∑ m n m where T B (y ,z)= mn−1 nm−1 ym! zn! , RB (y ,z)= (m+n−1)mn−1 nm−1 ym! ∑ m zn y n−1 m−1 . m! n! .B(y ,z)= (mn)m n
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zn n!
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Stanley’s M-vector Conjecture
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Stanley’s M-vector Conjecture . Definition . A multicomplex M is a set of monomials such that if u ∈ M and v |u , then v ∈ M. .
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Stanley’s M-vector Conjecture . Definition . A multicomplex M is a set of monomials such that if u ∈ M and v |u , then v ∈ M. Define h-vector (h0 , h1 , · · · , ) to be h . i = (the number of monomials of deg i ).
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Stanley’s M-vector Conjecture . Definition . A multicomplex M is a set of monomials such that if u ∈ M and v |u , then v ∈ M. Define h-vector (h0 , h1 , · · · , ) to be h . i = (the number of monomials of deg i ). . Conjecture (Stanley’s M-vector Conjecture, 1977) . The h-vector of a matroid complex is the h-vector of a pure multicomplex. .
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Stanley’s M-vector Conjecture . Definition . A multicomplex M is a set of monomials such that if u ∈ M and v |u , then v ∈ M. Define h-vector (h0 , h1 , · · · , ) to be h . i = (the number of monomials of deg i ). . Conjecture (Stanley’s M-vector Conjecture, 1977) . The h-vector of a matroid complex is the h-vector of a pure multicomplex. . For cycle matroid complexes, planar graphs (C. Merino, 2001) and coned graphs (W. Kook, 2012).
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Stanley’s M-vector Conjecture . Theorem (W. Kook, K. Lee, 2014+) . This conjecture is true for the case of the cycle matroid for bi-coned graphs. .
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Stanley’s M-vector Conjecture . Theorem (W. Kook, K. Lee, 2014+) . This conjecture is true for the case of the cycle matroid for bi-coned graphs. .
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α-invariant
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn .
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn . Then
α(G) = |χ(∆(M(G)))| ˜
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn . Then
α(G) = |χ(∆(M(G)))| ˜ ˜n−1 (∆(M(G))) = rk H
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn . Then
α(G) = |χ(∆(M(G)))| ˜ ˜n−1 (∆(M(G))) = rk H = (the top Betti number of JG∗ )
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn . Then
α(G) = |χ(∆(M(G)))| ˜ ˜n−1 (∆(M(G))) = rk H = (the top Betti number of JG∗ ) = (Möbius invariant of CG⊥ ) .
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn . Then
α(G) = |χ(∆(M(G)))| ˜ ˜n−1 (∆(M(G))) = rk H = (the top Betti number of JG∗ ) = (Möbius invariant of CG⊥ ) = µ⊥ (G) .
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α-invariant . Definition . Let G be a connected graph with n + 1 vertices. Define α(G) as hn . Then
α(G) = |χ(∆(M(G)))| ˜ ˜n−1 (∆(M(G))) = rk H = (the top Betti number of JG∗ ) = (Möbius invariant of CG⊥ ) .
= µ⊥ (G) = TG (0, 1).
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α-invariant . Problem (D. Bayer, S. Popescu, B. Sturmfels, 2001) . In the context of commutative algebra, the problem of finding formulas for . α(G) was posed.
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α-invariant . Problem (D. Bayer, S. Popescu, B. Sturmfels, 2001) . In the context of commutative algebra, the problem of finding formulas for . α(G) was posed. . Theorem (I. Novik, A. Postnikov, B. Sturmfels, 2002) .
α(Km+1 ) = (m − 1)Hm−2 (m) where Hn (x) is the Hermite polynomial.
α(Km+1,n+1 ) = mn · Hm−1,n−1 (n, m) where Hm,n (x, y ) is an bipartitie analogue of the Hermite polynomial. .
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α-invariant . Problem (D. Bayer, S. Popescu, B. Sturmfels, 2001) . In the context of commutative algebra, the problem of finding formulas for . α(G) was posed. . Theorem (I. Novik, A. Postnikov, B. Sturmfels, 2002) .
α(Km+1 ) = (m − 1)Hm−2 (m) where Hn (x) is the Hermite polynomial.
α(Km+1,n+1 ) = mn · Hm−1,n−1 (n, m) where Hm,n (x, y ) is an bipartitie analogue of the Hermite polynomial. . They used the recurrence relations. ..
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α-invariant of Km+1 . Theorem (W. Kook, K. Lee, 2014+) . m
α(Km+1 ) =
[2] ( ∑ m) k≥1
2k
(2k · mm−1−2k )(2k − 1)!!
where k -th term is
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α-invariant of Km+1,n+1 . Theorem (W. Kook, K. Lee, 2014+) . min (m−1,n−1) ( )( ) ∑ m n α(Km+1,n+1 ) = k!(m − k)(n − k)nm−1−k mn−1−k k k k=0
where k -th term is
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Edge-rooted forest, B-edge-rooted forest These objects are not simply combinatorial interpretations.
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Edge-rooted forest, B-edge-rooted forest These objects are not simply combinatorial interpretations.
. Theorem . These are “code” for constructing homology cycles. Cycle matroid of a coned graph (W. Kook, 2007) .
Cycle matroid of a bi-coned graph (W. Kook, K. Lee, 2014+) ..
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The cycle matroid of bi-coned graphs
Thank you!
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