Towards a bijective enumeration of spanning trees of the hypercube - PDF document

Towards a bijective enumeration of spanning trees of the hypercube Jeremy L. Martin and Victor Reiner University of Minnesota Full paper “Factorization of some weighted spanning tree enumerators” at http://math.umn.edu/ ∼ martin/math/pubs.html

✞ ☞✌ ✝ ✝✞ ✖✗ ✟ ✟✠ ✠ ✡ ✡☛ ☛ ☞ ✌ ☎✆ ✍ ✍✎ ✎ ✏ ✏✑ ✑ ✒ ✒✓ ✓ ✔ ✔✕ ✆ ☎ ✖ � ✛ ✚✛ ✚ ✙ ✘✙ ✘ ✗ � � � � ✄ � ✁ ✁ ✁ ✁ ✁ ✁ ✂ ✂ ✂ ✂✄ ✕ The hypercube Q n V ( Q n ) = { v = v 1 v 2 . . . v n : v i ∈ { 0 , 1 }} E ( Q n ) = { vw : v i = w i for all but one i } ❀✢❀✢❀ ❁✢❁✢❁ ❁✢❁✢❁ ❀✢❀✢❀ ❁✢❁✢❁ ❀✢❀✢❀ ❀✢❀✢❀ ❁✢❁✢❁ ✺✢✺ ✻✢✻ ✹✢✹ ✸✢✸ ✺✢✺ ✸✢✸ ✻✢✻ ✹✢✹ ✺✢✺ ✻✢✻ ✸✢✸ ✹✢✹ ✻✢✻ ✺✢✺ ✸✢✸ ✹✢✹ ✰✢✰ ✱✢✱ ✮✢✮✢✮ ✯✢✯✢✯ ✬✢✬ ✭✢✭ ✰✢✰ ✱✢✱ ✯✢✯✢✯ ✮✢✮✢✮ ✬✢✬ ✭✢✭ ✱✢✱ ✰✢✰ ✯✢✯✢✯ ✮✢✮✢✮ ✭✢✭ ✬✢✬ ✰✢✰ ✱✢✱ ✮✢✮✢✮ ✯✢✯✢✯ ✭✢✭ ✬✢✬ ✶✢✶ ✶✢✶ ✶✢✶ ✴✢✴ ✵✢✵ ✳✢✳ ✲✢✲ ✪✢✪ ✫✢✫ ✶✢✶ ✷✢✷ ✴✢✴ ✵✢✵ ✳✢✳ ✲✢✲ ✫✢✫ ✪✢✪ ✷✢✷ ✴✢✴ ✲✢✲ ✪✢✪ ✵✢✵ ✳✢✳ ✫✢✫ ✷✢✷ ✵✢✵ ✴✢✴ ✳✢✳ ✲✢✲ ✫✢✫ ✪✢✪ ✷✢✷ ✦✢✦✢✦ ★✢★ ✦✢✦✢✦ ★✢★ ✦✢✦✢✦ ★✢★ ✤✢✤ ✥✢✥ ✧✢✧✢✧ ✦✢✦✢✦ ✩✢✩ ★✢★ ✤✢✤ ✥✢✥ ✧✢✧✢✧ ✩✢✩ ✤✢✤ ✥✢✥ ✧✢✧✢✧ ✩✢✩ ✤✢✤ ✥✢✥ ✧✢✧✢✧ ✩✢✩ ✾✢✾ ✾✢✾ ✾✢✾ ✼✢✼ ✽✢✽ ✿✢✿ ✾✢✾ ✼✢✼ ✽✢✽ ✿✢✿ ✼✢✼ ✽✢✽ ✿✢✿ ✽✢✽ ✼✢✼ ✿✢✿ ✜✢✜✢✜ ✣✢✣✢✣ ✣✢✣✢✣ ✜✢✜✢✜ ✜✢✜✢✜ ✣✢✣✢✣ ✜✢✜✢✜ ✣✢✣✢✣

Spanning trees of Q n Tree( G ) = { spanning trees of a graph G } τ ( G ) = | Tree( G ) | [ n ] = { 1 , 2 , . . . , n } Theorem 0 (Stanley, Enumerative Combinatorics, vol. 2, p. 62) n � � k ( n 2 2 n − n − 1 k ) . τ ( Q n ) = 2 | S | = k =1 S ⊂ [ n ] | S |≥ 2 E.g., τ ( Q 3 ) = 2 |{ 1 , 2 }| · 2 |{ 1 , 3 }| · 2 |{ 2 , 3 }| · 2 |{ 1 , 2 , 3 }| = 4 · 4 · 4 · 6 = 384. Bijective proof??

The model: K n and the Pr¨ ufer code K n = complete graph on n vertices τ ( K n ) = n n − 2 Cayley’s Formula: bijection → [ n ] n − 2 Tree( K n ) − − − − − Pr¨ ufer code: 1 2 3 ⑦ ⑦ ⑦ 4 5 6 Pr¨ ufer code ← → ⑦ ⑦ ⑦ 4255458 7 8 9 ⑦ ⑦ ⑦ • deg T ( i ) = 1 + number of i ’s in Pr¨ ufer code of T Cayley-Pr¨ ufer Formula: � x deg T ( i ) · · · x deg T ( n ) x 1 · · · x n ( x 1 + · · · + x n ) n − 2 = 1 n T ∈ Tree( K n )

Weighted enumeration and bijections Suppose that you know Cayley’s formula τ ( K n ) = n n − 2 . . . • . . . and can prove it using the Matrix-Tree Theorem. . . . . . but are looking for a bijective proof. • Knowing the Cayley-Pr¨ ufer Formula � x deg T ( i ) · · · x deg T ( n ) x 1 · · · x n ( x 1 + · · · + x n ) n − 2 = 1 n T ∈ Tree( K n ) might be an important clue, enabling you to reproduce the Pr¨ ufer code (or a similar bijection). • Goal: Do the same thing for Q n by finding a weighted analogue of the formula � τ ( Q n ) = 2 | S | S ⊂ [ n ] | S |≥ 2

Weighted enumeration of spanning trees of Q n • Assign a monomial weight wt( e ) to each edge e ∈ Q n , � define wt( T ) = wt( e ) for T ∈ Tree( Q n ), e ∈ T and consider the generating function � wt( T ) . T ∈ Tree( Q n ) First attempt: Keep track of vertex degrees (` a la Pr¨ ufer). Weight each edge vw ∈ E ( Q n ) by wt( vw ) = y v y w so that � y deg T ( v ) wt( T ) = v v ∈ V ( Q n ) • Unfortunately, this does not factor nicely. E.g., for n = 3, it is x 000 · x 001 · · · x 111 · (some irreducible degree-6 nightmare) .

☛ ✂ ✟ ✞ ✝✞ ✝ ✆ ☎✆ ☎ ✄ ✂✄ ✁ ✠ �✁ � ✡ ✡☛ ☞ ☞✌ ✌ ✍ ✍✎ ✎ ✟✠ Directions of edges • Weight each edge vw by q i , where i = dir( vw ) is the unique index for which v i � = w i . So n � wt( T ) = q dir( T ) = q |{ edges of T in direction i }| i i =1 (0,1,1) (1,1,1) (0,0,1) (1,0,1) (0,1,0) (1,1,0) (0,0,0) (1,0,0) Theorem 1 � � � � � 2 2 n − n − 1 q 1 · · · q n q dir( T ) = q i T ∈ Tree( Q n ) S ⊂ [ n ] i ∈ S | S | ≥ 2

✝ ✟ ✆ ☎✆ ☎ ✄ ✂✄ ✂ ✁ �✁ � ✟✠ ✝✞ ✠ ✡ ✡☛ ☛ ☞ ☞✌ ✌ ✍ ✍✎ ✎ ✞ Decoupled vertex degrees • For each edge e = vw not in direction i , either v i = w i = 0 or v i = w i = 1 . Weight e by x i or x − 1 accordingly. E.g., for e = { 0 10 , 1 10 } , i wt( e ) = q dir( e ) x dd( e ) = q 1 x 2 x − 1 3 . • Equivalently, record which Q n − 1 ⊂ Q n the edge e belongs to. (0,1,1) (1,1,1) (0,0,1) (1,0,1) (0,1,0) (1,1,0) (0,0,0) (1,0,0)

The main result Theorem 2 �� � � � q dir( T ) x dd( T ) q i ( x − 1 = q 1 . . . q n + x i ) i i ∈ S T ∈ Tree( Q n ) S ⊂ [ n ] � �� � | S | ≥ 2 f S � � q dir( T ) = x dd( T ) = x dd( e ) . where q dir( e ) , e ∈ T e ∈ T Compare Theorem 1: � � � � � 2 2 n − n − 1 q 1 · · · q n q dir( T ) = q i i ∈ S T ∈ Tree( Q n ) S ⊂ [ n ] | S | ≥ 2 and Theorem 0: � τ ( Q n ) = 2 | S | S ⊂ [ n ] | S |≥ 2

Sketch of the proof Weighted Matrix-Tree Theorem Let L = ( L vw ) v,w ∈ V ( G ) be the weighted Laplacian:   0 v � = w and vw �∈ E ( G )      − wt( vw ) vw ∈ E ( G ) L vw =   �  wt( e ) v = w    e ∋ v Then � T ∈ Tree( G ) wt( T ) = det ˆ L , where ˆ L is obtained by deleting the v th row and v th column of L . Identification of Factors Lemma (Krattenthaler) f | det ˆ ˆ L ⇐ ⇒ L has a nullvector in Q [ q, x ] / ( f ) . • Use a computer algebra package (e.g., Macaulay) to compute “witness” nullvectors for factors f = f S • Experimentally, the witnesses have a nice form, reducing the proof to calculation • Same method can be used for threshold graphs (specializing a result of Remmel and Williamson) and products of K n ’s