G-Parking Functions and Monomial Ideals
Brice Huang
Mentored by Wuttisak Trongsiriwat
Fourth Annual MIT PRIMES Conference
May 17, 2014
Definitions
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Directed graph (digraph): collection of vertices and oriented
edges between pairs of vertices
1 q
0
q2
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@
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q
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Rq?
3
Definitions
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Directed graph (digraph): collection of vertices and oriented
edges between pairs of vertices
1 q
0
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q2
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Rq?
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Subtree of digraph: subgraph in which each vertex has a
unique path to a vertex known as the root
1q
0
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q
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Definitions
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Directed graph (digraph): collection of vertices and oriented
edges between pairs of vertices
1 q
0
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q2
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@
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q
@
Rq?
3
Subtree of digraph: subgraph in which each vertex has a
unique path to a vertex known as the root
1q
0
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q
q
3
Spanning tree of digraph: a subtree containing all vertices
1q
0
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Definitions
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Subforest of digraph: the union of one or more subtrees on
disjoint sets of vertices
1q
0
q?
q
3
Definitions
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Subforest of digraph: the union of one or more subtrees on
disjoint sets of vertices
1q
0
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q?
q
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Spanning forest of digraph: a subforest containing all vertices
1q
0
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Parking Functions
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A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
Parking Functions
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A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
I
Analogy: n drivers on a one-way road with parking spots
0, 1, . . . , n − 1
Parking Functions
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A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
I
Analogy: n drivers on a one-way road with parking spots
0, 1, . . . , n − 1
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Example: (1,0,3,0)
0
1
2
3
Parking Functions
I
A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
I
Analogy: n drivers on a one-way road with parking spots
0, 1, . . . , n − 1
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Example: (1,0,3,0)
1
0
1
2
3
Parking Functions
I
A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
I
Analogy: n drivers on a one-way road with parking spots
0, 1, . . . , n − 1
I
Example: (1,0,3,0)
0
1
0
1
2
3
Parking Functions
I
A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
I
Analogy: n drivers on a one-way road with parking spots
0, 1, . . . , n − 1
I
Example: (1,0,3,0)
0
1
0
1
3
2
3
Parking Functions
I
A classical parking function is an n-tuple of nonnegative
integers (b1 , . . . , bn ) that, when sorted in decreasing order, is
termwise less than (n, n − 1, . . . , 1)
I
Analogy: n drivers on a one-way road with parking spots
0, 1, . . . , n − 1
I
Example: (1,0,3,0)
0
1
0
3
0
1
2
3
Parking Functions
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There are (n + 1)n−1 classical parking functions of size n
Parking Functions
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There are (n + 1)n−1 classical parking functions of size n
Theorem (Cayley)
The complete graph Kn+1 has (n + 1)n−1 spanning trees.
G -Parking Functions
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G is a digraph on vertices {0, 1, . . . , n}
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For a nonempty subset I ⊆ {1, . . . , n}, and vertex i ∈ I , let
dI (i) denote the number of edges from i to vertices outside I
1 q
0
q2
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q
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3
G -Parking Functions
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G is a digraph on vertices {0, 1, . . . , n}
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For a nonempty subset I ⊆ {1, . . . , n}, and vertex i ∈ I , let
dI (i) denote the number of edges from i to vertices outside I
1 q
0
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q2
@
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@
Rq?
?@
q
@
3
A G -parking function is an n-tuple (b1 , . . . , bn ) such that for
any nonempty subset I ⊆ {1, 2, . . . , n}, there exists i ∈ I such
that bi < dI (i)
G -Parking Functions
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G is a digraph on vertices {0, 1, . . . , n}
I
For a nonempty subset I ⊆ {1, . . . , n}, and vertex i ∈ I , let
dI (i) denote the number of edges from i to vertices outside I
1 q
0
q2
@
I
@
Rq?
?@
q
@
3
I
A G -parking function is an n-tuple (b1 , . . . , bn ) such that for
any nonempty subset I ⊆ {1, 2, . . . , n}, there exists i ∈ I such
that bi < dI (i)
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Example: (0, 1, 1) is a G -parking function, where G is the
graph above
G -Parking Functions
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Theorem
The number of G -parking functions equals the number of spanning
trees of G rooted at 0.
G -Parking Functions
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Theorem
The number of G -parking functions equals the number of spanning
trees of G rooted at 0.
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Classical parking functions are the special case G = Kn+1
G -Parking Functions
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Theorem
The number of G -parking functions equals the number of spanning
trees of G rooted at 0.
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Classical parking functions are the special case G = Kn+1
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Chebikin and Pylyavskyy constructed an explicit bijection
Chebikin-Pylyavskyy Bijection
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For every subtree T of G rooted at 0, assign an order π(T ) to
T ’s vertices. Let i <π(T ) j denote i being smaller than j in
this order
Chebikin-Pylyavskyy Bijection
I
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For every subtree T of G rooted at 0, assign an order π(T ) to
T ’s vertices. Let i <π(T ) j denote i being smaller than j in
this order
An choice of orders Π(G ) is a proper set of tree orders if for
each subtree T rooted at 0:
I
I
if an edge (i, j) ∈ T , then i >π(T ) j
if t is a subtree of T , then the orders π(t) and π(T ) are
consistent
Chebikin-Pylyavskyy Bijection
I
I
For every subtree T of G rooted at 0, assign an order π(T ) to
T ’s vertices. Let i <π(T ) j denote i being smaller than j in
this order
An choice of orders Π(G ) is a proper set of tree orders if for
each subtree T rooted at 0:
I
I
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if an edge (i, j) ∈ T , then i >π(T ) j
if t is a subtree of T , then the orders π(t) and π(T ) are
consistent
Example: Breadth-first search order
1q
0
0 <π(T ) 1 <π(T ) 3 <π(T ) 2
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q
q2
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3
Chebikin-Pylyavskyy Bijection
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Fix a proper set of tree orders Π(G )
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For each spanning tree T , let e(T , i) be the edge out of i in T
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Given a subtree T and order π(T ), for each vertex i, order the
edges from i to T such that (i, j1 ) <π(T ) (i, j2 ) if j1 <π(T ) j2
Chebikin-Pylyavskyy Bijection
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Fix a proper set of tree orders Π(G )
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For each spanning tree T , let e(T , i) be the edge out of i in T
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Given a subtree T and order π(T ), for each vertex i, order the
edges from i to T such that (i, j1 ) <π(T ) (i, j2 ) if j1 <π(T ) j2
Theorem (Chebikin, Pylyavskyy)
Map each spanning tree T to (b1 , . . . , bn ), where bi is the number
of edges e from i such that e <π(T ) e(T , i). This mapping is a
bijection between G ’s spanning trees rooted at 0 and G -parking
functions.
Chebikin-Pylyavskyy Bijection - An Example
1
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G -parking functions: (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0),
(0,1,1)
Chebikin-Pylyavskyy Bijection - An Example
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G -parking functions: (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0),
(0,1,1)
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Spanning trees:
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(0,0,0)
3 0
r
r2 1 r
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@r?
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(1,0,0)
3 0
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r
(0,1,0)
r2 1 r
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@
r2 1 r
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r? r? @r
3 0
(0,0,1)
3 0
r2 1 r
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r R
@r
(1,1,0)
3 0
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r
r2
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@r?
(0,1,1)
3
Monomial Ideals
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K[x1 , . . . , xn ] is the polynomial ring in variables x1 , . . . , xn on
a fixed field K of characteristic 0
Monomial Ideals
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K[x1 , . . . , xn ] is the polynomial ring in variables x1 , . . . , xn on
a fixed field K of characteristic 0
I
For each nonempty subset I ⊆ {1, . . . , n}, define
Y d (i)
mI =
xi I
i∈I
and let the ideal IG = hmI i as I ranges over all nonempty
subsets of {1, . . . , n}
Monomial Ideals
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K[x1 , . . . , xn ] is the polynomial ring in variables x1 , . . . , xn on
a fixed field K of characteristic 0
I
For each nonempty subset I ⊆ {1, . . . , n}, define
Y d (i)
mI =
xi I
i∈I
and let the ideal IG = hmI i as I ranges over all nonempty
subsets of {1, . . . , n}
I
(b1 , . . . , bn ) is a G -parking function if and only if x1b1 · · · xnbn
does not vanish in K[x1 , . . . , xn ]/IG
Monomial Ideals - An Example
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IG = hx12 , x22 , x32 , x12 x2 , x1 x3 , x2 x32 , x1 x20 x3 i
Monomial Ideals - An Example
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0
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IG = hx12 , x22 , x32 , x12 x2 , x1 x3 , x2 x32 , x1 x20 x3 i
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Non-vanishing monomials: 1, x1 , x2 , x3 , x1 x2 , x2 x3
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G -parking functions: (0,0,0), (1,0,0), (0,1,0), (0,0,1), (1,1,0),
(0,1,1)
Almost-G -Parking Functions
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For each nonempty subset I = {i1 < · · · < ik } ⊆ {1, . . . , n},
define
Y d (i)
m̂I = xi1
xi I
i∈I
and let the ideal ÎG = hm̂I i as I ranges over all nonempty
subsets of {1, . . . , n}
I
(b1 , . . . , bn ) is an almost-G -parking function if x1b1 · · · xnbn does
not vanish in K[x1 , . . . , xn ]/ÎG
Almost-G -Parking Functions
I
For each nonempty subset I = {i1 < · · · < ik } ⊆ {1, . . . , n},
define
Y d (i)
m̂I = xi1
xi I
i∈I
and let the ideal ÎG = hm̂I i as I ranges over all nonempty
subsets of {1, . . . , n}
I
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(b1 , . . . , bn ) is an almost-G -parking function if x1b1 · · · xnbn does
not vanish in K[x1 , . . . , xn ]/ÎG
Theorem (Postnikov, Shapiro, Shapiro)
When G = Kn+1 , the number of almost-G -parking functions
equals the number of (undirected) spanning forests of G .
Almost-G -Parking Functions and Spanning Forests
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We explicity construct a bijection between almost-G -parking
functions and spanning forests of G whose connected
components are rooted at their smallest vertices
Almost-G -Parking Functions and Spanning Forests
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We explicity construct a bijection between almost-G -parking
functions and spanning forests of G whose connected
components are rooted at their smallest vertices
I
For every subtree T of G rooted at its numerically smallest
vertex, assign an order π̂(T ). Let i <π̂(T ) j denote i being
smaller than j in this order
Almost-G -Parking Functions and Spanning Forests
I
We explicity construct a bijection between almost-G -parking
functions and spanning forests of G whose connected
components are rooted at their smallest vertices
I
For every subtree T of G rooted at its numerically smallest
vertex, assign an order π̂(T ). Let i <π̂(T ) j denote i being
smaller than j in this order
I
A choice of orders Π̂(G ) is a super-proper set of tree orders if
for each subtree T rooted at its numerically smallest vertex:
I
I
if an edge (i, j) ∈ T , then i >π̂(T ) j
if t is a subtree of T with the same root, then the orders π̂(t)
and π̂(T ) are consistent
Almost-G -Parking Functions and Spanning Forests
I
We explicity construct a bijection between almost-G -parking
functions and spanning forests of G whose connected
components are rooted at their smallest vertices
I
For every subtree T of G rooted at its numerically smallest
vertex, assign an order π̂(T ). Let i <π̂(T ) j denote i being
smaller than j in this order
I
A choice of orders Π̂(G ) is a super-proper set of tree orders if
for each subtree T rooted at its numerically smallest vertex:
I
I
I
if an edge (i, j) ∈ T , then i >π̂(T ) j
if t is a subtree of T with the same root, then the orders π̂(t)
and π̂(T ) are consistent
Example: Breadth-first search order
Almost-G -Parking Functions and Spanning Forests
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Given a super-proper set of tree orders Π̂(G ), for every
spanning forest F of G whose connected components are
rooted at their numerically smallest vertices, assign an order
π(F ) such that i <π(F ) j if:
I
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the root of i’s connected component is smaller than the root
of j’s connected component, or
i and j are in the same connected component T , and i <π̂(T ) j
Almost-G -Parking Functions and Spanning Forests
I
I
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Fix a super-proper set of tree orders Π̂(G )
Let e(F , i) be the edge out of i in the spanning forest F , if it
exists
Given a subforest F and order π(F ), for each vertex i, order
the edges from i to F such that (i, j1 ) <π(F ) (i, j2 ) if
j1 <π(F ) j2
Almost-G -Parking Functions and Spanning Forests
I
I
I
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Fix a super-proper set of tree orders Π̂(G )
Let e(F , i) be the edge out of i in the spanning forest F , if it
exists
Given a subforest F and order π(F ), for each vertex i, order
the edges from i to F such that (i, j1 ) <π(F ) (i, j2 ) if
j1 <π(F ) j2
Map each spanning forest F whose connected components are
rooted at their numerically smallest vertices to (b1 , . . . , bn ),
where bi is:
I
I
the number of edges from i to vertices smaller than i in π(F ),
if i is the root of its connected component
the number of edges e from i such that e <π(F ) e(F , i),
otherwise
Almost-G -Parking Functions and Spanning Forests
I
I
I
I
Fix a super-proper set of tree orders Π̂(G )
Let e(F , i) be the edge out of i in the spanning forest F , if it
exists
Given a subforest F and order π(F ), for each vertex i, order
the edges from i to F such that (i, j1 ) <π(F ) (i, j2 ) if
j1 <π(F ) j2
Map each spanning forest F whose connected components are
rooted at their numerically smallest vertices to (b1 , . . . , bn ),
where bi is:
I
I
the number of edges from i to vertices smaller than i in π(F ),
if i is the root of its connected component
the number of edges e from i such that e <π(F ) e(F , i),
otherwise
Theorem
This mapping is a bijection between almost-G -parking functions
and G ’s spanning forests whose connected components are rooted
at their numerically smallest vertices.
An Example
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This corresponds to the almost-G -parking function (0,2,1)
Modified Monomial Ideals
For each nonempty I ⊆ {1, . . . , n}, choose any kI ∈ I and let
Y d (i)
m̂I0 = xkI
xi I
i∈I
Let ÎG0 = hm̂I0 i as I ranges over all nonempty subsets of {1, . . . , n},
and let Â0G = K[x1 , . . . , xn ]/ÎG0
Theorem
If G = Kn+1 , then dim Â0G is independent of the choices of kI .
Future Directions
Conjecture
dim Â0G is independent of the choices of kI for all choices of kI
that preserve the monotonicity of the ideal ÎG0 (i.e. if I ⊂ J, then
for any i ∈ I , degxi m̂I0 ≥ degxi m̂J0 ).
It would also be interesting to find a combinatorial interpretation
of ideals in which the mI are modified by multiplication by more
than one variable
Acknowledgements
Many thanks to:
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My family, for always supporting me
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Wuttisak Trongsiriwat and Professor Alexander Postnikov, for
their patience and guidance
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Dr. Slava Gerovitch, Dr. Tanya Khovanova, and the
MIT-PRIMES staff, for giving me this opportunity