Symmetries in trees Olivier Bernardi (Brandeis University) First - PowerPoint PPT Presentation

Symmetries in trees Olivier Bernardi (Brandeis University) First part is joint work with Alejandro Morales (UQAM) 3 5 1 2 1 3 2 1 4 6 2 1 4 3 2 McGill University, September 2013

Part I Multitype Cayley trees

Cayley trees A tree is a connected acyclic graph. A Cayley tree of size n is a tree with vertex set [ n ] := { 1 , 2 , . . . , n } . 4 8 3 2 5 6 1 7

Cayley trees A tree is a connected acyclic graph. A Cayley tree of size n is a tree with vertex set [ n ] := { 1 , 2 , . . . , n } . Theorem [Sylvester 1857] . The number of Cayley trees of size n is n n − 2 .

Cayley trees A tree is a connected acyclic graph. A Cayley tree of size n is a tree with vertex set [ n ] := { 1 , 2 , . . . , n } . Theorem [Sylvester 1857] . The number of Cayley trees of size n is n n − 2 . Example . There are 16 Cayley trees of size n = 4 . 2 1 2 2 1 3 2 3 3 1 4 3 4 4 4 1 1 3 3 2 3 3 4 3 3 1 3 2 4 3 3 3 2 3 1 3 3 3 3 4 2 3 1 3 3 4 3 3 3 3 1 3 3 2 3 4 3 3 3 1 4 3 3 2 1 3 3 3 3 2 4 3 3 1 3 3 4 3 2 3 1 3 3 4 3 2 3 3 3 1 4 3 3 3 2 3 3 4 3 1 3 2 3 3 4 3 1 3 3 3 3 2

Multitype Cayley trees A multitype Cayley tree of size ( n 1 , . . . , n d ) is a tree with vertex set V = { ( t, i ) , t ∈ [ d ] , i ∈ [ n t ] } .

Multitype Cayley trees A multitype Cayley tree of size ( n 1 , . . . , n d ) is a tree with vertex set V = { ( t, i ) , t ∈ [ d ] , i ∈ [ n t ] } . 3 5 1 2 size = (4 , 3 , 6 , 2) type 1 1 3 1 2 type 2 type 3 2 1 4 6 type 4 vertex (3 , 6) 4 3 2 root vertex

Multitype Cayley trees A multitype Cayley tree of size ( n 1 , . . . , n d ) is a tree with vertex set V = { ( t, i ) , t ∈ [ d ] , i ∈ [ n t ] } . 2 3 3 5 1 2 type 1 2 1 1 3 type 2 type 3 4 4 1 6 type 4 4 3 2 root vertex Theorem [Knuth 68, Bousquet-M´ elou, Chapuy 12] . The number of multitype Cayley trees of size ( n 1 , . . . , n d ) with root-vertex of type ρ such that the type of adjacent vertices differ by ± 1 is d n ρ � n i ( n i − 1 + n i +1 ) n i − 1 , n 1 n d i =1 with n 0 = n d +1 = 0 .

Warm up: counting Cayley trees by degrees Theorem [Cayley 1889] : The generating function of Cayley trees of size n counted according to their degrees is x deg(1) x deg(2) � · · · x deg( n ) = x 1 x 2 · · · x n ( x 1 + x 2 + . . . + x n ) n − 2 . 1 2 n T

Warm up: counting Cayley trees by degrees Theorem [Cayley 1889] : The generating function of Cayley trees of size n counted according to their degrees is x deg(1) x deg(2) � · · · x deg( n ) = x 1 x 2 · · · x n ( x 1 + x 2 + . . . + x n ) n − 2 . 1 2 n T Known proofs : Recurrences, matrix-tree theorem, Prufer’s code, Joyal’s endofunction approach, Pitman’s double counting argument.

Warm up: counting Cayley trees by degrees For α = ( α 1 , . . . , α n ) we denote by T α the set of Cayley trees of size n where vertex i has degree α i . � We must take α 1 , . . . , α n > 0 such that α i = 2 n − 2 . i

Warm up: counting Cayley trees by degrees For α = ( α 1 , . . . , α n ) we denote by T α the set of Cayley trees of size n where vertex i has degree α i . Claim. If β i = α i − 1 , β j = α j + 1 and β k = α k for k � = i, j , then ( α i − 1) |T α | = ( β j − 1) |T β | .

Warm up: counting Cayley trees by degrees For α = ( α 1 , . . . , α n ) we denote by T α the set of Cayley trees of size n where vertex i has degree α i . Claim. If β i = α i − 1 , β j = α j + 1 and β k = α k for k � = i, j , then ( α i − 1) |T α | = ( β j − 1) |T β | . Proof. bijection i i j j Tree in T α with a marked edge Tree in T β with a marked edge incident to i not on the path i � j . incident to j not on the path i � j .

Warm up: counting Cayley trees by degrees For α = ( α 1 , . . . , α n ) we denote by T α the set of Cayley trees of size n where vertex i has degree α i . Claim. If β i = α i − 1 , β j = α j + 1 and β k = α k for k � = i, j , then ( α i − 1) |T α | = ( β j − 1) |T β | . ( n − 2)! n − 2 � � Corollary. |T α | = ( α 1 − 1)! ··· ( α n − 1)! |T ( n − 1 , 1 , 1 ,..., 1) | = . α 1 − 1 ,...,α n − 1 1

Warm up: counting Cayley trees by degrees For α = ( α 1 , . . . , α n ) we denote by T α the set of Cayley trees of size n where vertex i has degree α i . Claim. If β i = α i − 1 , β j = α j + 1 and β k = α k for k � = i, j , then ( α i − 1) |T α | = ( β j − 1) |T β | . ( n − 2)! n − 2 � � Corollary. |T α | = ( α 1 − 1)! ··· ( α n − 1)! |T ( n − 1 , 1 , 1 ,..., 1) | = . α 1 − 1 ,...,α n − 1 Hence, x deg(1) x deg(2) � · · · x deg( n ) = x 1 x 2 · · · x n ( x 1 + x 2 + . . . + x n ) n − 2 . 1 2 n T

Warm up: counting Cayley trees by degrees For α = ( α 1 , . . . , α n ) we denote by T α the set of Cayley trees of size n where vertex i has degree α i . Claim. If β i = α i − 1 , β j = α j + 1 and β k = α k for k � = i, j , then ( α i − 1) |T α | = ( β j − 1) |T β | . ( n − 2)! n − 2 � � Corollary. |T α | = ( α 1 − 1)! ··· ( α n − 1)! |T ( n − 1 , 1 , 1 ,..., 1) | = . α 1 − 1 ,...,α n − 1 Hence, x deg(1) x deg(2) � · · · x deg( n ) = x 1 x 2 · · · x n ( x 1 + x 2 + . . . + x n ) n − 2 . 1 2 n T Rooted version: x ch(1) x ch(2) � . . . x ch( n ) = ( x 1 + x 2 + · · · + x n ) n − 1 , 1 2 n rooted Cayley tree where ch( i ) is the number of children of vertex i .

Main result Let n = ( n 1 , . . . , n d ) be a tuple of positive integers. Let T ρ ( n ) be the set of multitype Cayley trees of size n , and root-type ρ .

Main result Let n = ( n 1 , . . . , n d ) be a tuple of positive integers. Let T ρ ( n ) be the set of multitype Cayley trees of size n , and root-type ρ . x ch s ( t,i ) � The weight of a tree T is w ( T ) = , s,t,i s,t ∈ [ d ] ,i ∈ [ n t ] where ch s ( t, i ) = number of children of type s of the vertex ( t, i ) . 3 5 1 2 1 3 1 2 4 2 1 6 4 3 Vertex ( t, i ) = (2 , 3) gives weight x 1 , 2 , 3 x 2 3 , 2 , 3 . 2 root vertex

Main result Let n = ( n 1 , . . . , n d ) be a tuple of positive integers. Let T ρ ( n ) be the set of multitype Cayley trees of size n , and root-type ρ . x ch s ( t,i ) � The weight of a tree T is w ( T ) = . s,t,i s,t ∈ [ d ] ,i ∈ [ n t ] Theorem [B., Morales] . � d � n t n s − 1   d n t � � � � � � � � � w ( T ) = x s,t,i × x s,t,i .   s =1 t =1 i =1 A ∈ Cayley i =1 T ∈T ρ ( n ) ( s,t ) ∈ A d,ρ Cayley trees with vertex-set [ d ] and root-vertex ρ .

Main result Let n = ( n 1 , . . . , n d ) be a tuple of positive integers. Let T ρ ( n ) be the set of multitype Cayley trees of size n , and root-type ρ . x ch s ( t,i ) � The weight of a tree T is w ( T ) = . s,t,i s,t ∈ [ d ] ,i ∈ [ n t ] Theorem [B., Morales] . � d � n t n s − 1   d n t � � � � � � � � � w ( T ) = x s,t,i × x s,t,i .   s =1 t =1 i =1 A ∈ Cayley i =1 T ∈T ρ ( n ) ( s,t ) ∈ A d,ρ det( L ρ ) − � � L s,t = i ∈ [ n t ] x s,t,i if s � = t � L ρ = ( L s,t ) s,t ∈ [ d ] \{ ρ } , with � � L s,s = t � = s,i ∈ [ n t ] x s,t,i . �

Main result x ch s ( t,i ) � The weight of a tree T is w ( T ) = . s,t,i s,t ∈ [ d ] ,i ∈ [ n t ] Theorem [B., Morales] . � d � n t n s − 1   d n t � � � � � � � � � w ( T ) = x s,t,i × x s,t,i .   s =1 t =1 i =1 i =1 A ∈ Cayley T ∈T ρ ( n ) ( s,t ) ∈ A d,ρ Example: Spanning trees of K m,n rooted on a black vertex. Set d = 2 , ρ = 1 , n 1 = m, n 2 = n, x 1 , 1 ,i = x 2 , 2 ,j = 0 . Denote x 2 , 1 ,i = x i , x 1 , 2 ,j = y j . ch( w j ) x ch( b i ) � � � = ( x 1 + · · · + x m ) n ( y 1 + · · · + y n ) m − 1 . y i j T ⊂ K m,n i ∈ [ m ] j ∈ [ n ]

Sketch of proof. For α = ( α s,t,i ) s,t ∈ [ d ] ,i ∈ [ n d ] we denote by T α the set of Cayley trees such that vertex ( t, i ) has α s,t,i children of type s . Claim. If β s,t,i = α s,t,i − 1 and β s,t,j = α s,t,j + 1 but everything else is the same, then α s,t,i |T α | = β s,t,j |T β | .

Sketch of proof. For α = ( α s,t,i ) s,t ∈ [ d ] ,i ∈ [ n d ] we denote by T α the set of Cayley trees such that vertex ( t, i ) has α s,t,i children of type s . Claim. If β s,t,i = α s,t,i − 1 and β s,t,j = α s,t,j + 1 but everything else is the same, then α s,t,i |T α | = β s,t,j |T β | . Proof. vertex ( t, j ) vertex ( t, i ) vertex ( t, j ) vertex ( t, j ) � � Φ Φ vertex ( t, i ) vertex ( t, i ) vertex ( t, i ) vertex ( t, j ) root-vertex root-vertex root-vertex root-vertex (a) (b)

Sketch of proof. For α = ( α s,t,i ) s,t ∈ [ d ] ,i ∈ [ n d ] we denote by T α the set of Cayley trees such that vertex ( t, i ) has α s,t,i children of type s . Claim. If β s,t,i = α s,t,i − 1 and β s,t,j = α s,t,j + 1 but everything else is the same, then α s,t,i |T α | = β s,t,j |T β | . Corollary: It suffices to count trees in which vertices not labelled 1 are leaves (the “star-trees”).