Extended boxed product and application to synchronized trees Olivier - PowerPoint PPT Presentation

Extended boxed product and application to synchronized trees Olivier Bodini 1 , Antoine Genitrini 2 et Nicolas Rolin 1 1 Universit´ e Paris 13 – LIPN 2 UPMC Paris – LIP6 GASCom 2016 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 1 / 32

1 Introduction 2 Adaptating boxed product 3 Specification of synchronized trees 4 Uniform sampling N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 2 / 32

Introduction Increasing tree Plane tree labeled with n nodes: Each label between 1 and n is taken Each path from the root to a leaf is a strictly increasing path ⇒ two nodes are sharing a label. • 1 • • 4 2 • • • 6 5 3 Shape of the tree Increasing tree N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 3 / 32

Introduction Synchronized tree Plane tree labeled with n + 1 nodes Each label between 1 and n is taken Each path from the root to a leaf is a strictly increasing path ⇒ two nodes are sharing a label. • 1 • • 4 2 • • • • 5 6 5 3 Shape of the tree Synchronized tree N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 4 / 32

Introduction Definitions A combinatorial class C is a set of objects, with a size function, denoted by | · | : C → N and such that for every integer n , the subset C n of objects of size n , is finite with cardinality C n . We define the exponential generating function of a combinatorial class C to be: z n � C ( z ) = C n n ! n ≥ 0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 5 / 32

Introduction Definitions A combinatorial class C is a set of objects, with a size function, denoted by | · | : C → N and such that for every integer n , the subset C n of objects of size n , is finite with cardinality C n . We define the exponential generating function of a combinatorial class C to be: z n � C ( z ) = C n n ! n ≥ 0 Small dictionary C = A + B → C ( z ) = A ( z ) + B ( z ) C = A ⋆ B → C ( z ) = A ( z ) · B ( z ) N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 5 / 32

Introduction Boxed product The boxed product on the labeled classes: C = ( A � ⋆ B ) , Means that C is the product of A and B , and that the element with the smallest label comes from the A component. Its translation to equation is: v = z dA � C = A � ⋆ B → C ( z ) = dv ( v ) B ( v ) dv . v =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 6 / 32

Introduction Specification of increasing trees The increasing trees class T verifies the following specification: T = Z � ⋆ Seq ( T ) Z T T . . . T T N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 7 / 32

Introduction Specification of increasing trees The increasing trees class T verifies the following specification: T = Z � ⋆ Seq ( T ) Z T T . . . T T Hence its generating function verifies: v = z 1 dv � 1 − T ( v ) or T ′ ( z ) = T ( z ) = 1 − T ( z ) and T (0) = 0 . v =0 So: √ T ( z ) = 1 − 1 − 2 z N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 7 / 32

Adaptating boxed product Increasing trees with a synchronization We can regroup the two identical nodes: • • • • • • • • • • • • • • • N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 8 / 32

Adaptating boxed product Increasing trees with a synchronization We can regroup the two identical nodes: • • • N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 8 / 32

Adaptating boxed product Extended boxed operator Let A , B and C combinatorial classes with no element of size 0. C E = A B can be expressed with boxed operator: E = C � ⋆ ( A ⋆ B ) t = z E ( z ) = � C ′ ( t ) A ( t ) B ( t ) dt . t =0 blanc blanc blanc blanc N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 9 / 32

Adaptating boxed product Extended boxed operator Let A , B and C combinatorial classes with no element of size 0. C A B E = D = A B C can be expressed with boxed operator: can be expressed with boxed operator: E = C � ⋆ ( A ⋆ B ) D = A � ⋆ ( B � ⋆ C ) + B � ⋆ ( A � ⋆ C ) t = z D ( z ) = E ( z ) = � C ′ ( t ) A ( t ) B ( t ) dt . z x � � t =0 A ′ ( x ) B ′ ( y ) C ( y ) dxdy + blanc x =0 y =0 blanc z y � � A ′ ( y ) B ′ ( x ) C ( x ) dxdy . blanc y =0 x =0 blanc N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 9 / 32

Adaptating boxed product A more complicated example P k , p = A We now look at the class P k , p . X 1 Y 1 The previous method gives: P k , p = A � ⋆ X � 1 ⋆ ( Y � 1 ⋆ ( . . . ) + X � 2 ⋆ ( . . . )) . . . . . . + A � ⋆ Y � 1 ⋆ ( X � 1 ⋆ ( . . . ) + Y � 2 ⋆ ( . . . )) The two branches interlace. X p − 1 Y k − 1 � k + p � ⇒ The number of terms in the sum is k X p Y k R N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 10 / 32

Adaptating boxed product Stanley method Stanley showed that computing the number of linear extensions of a partial order reduces to compute the volume of convex polytopes. z # { linear extension of ≻} = 1 z z z a b 4! � � � � dzdtdxdy = 2 z =0 t =0 x = t y = t t Graph portraying the partial order ≻ : z ≻ a , z ≻ b , a ≻ t , b ≻ t N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 11 / 32

Adaptating boxed product Simple example : factorization Factorize the generating function: y z x z � � � � A ′ ( x ) B ′ ( y ) C ( y ) dxdy + A ′ ( x ) B ′ ( y ) C ( x ) dxdy D ( z ) = x =0 y =0 y =0 x =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 12 / 32

Adaptating boxed product Simple example : factorization Factorize the generating function: y z x z � � � � A ′ ( x ) B ′ ( y ) C ( y ) dxdy + A ′ ( x ) B ′ ( y ) C ( x ) dxdy D ( z ) = x =0 y =0 y =0 x =0 we swap the integration order of y and x on the right term z x z z � � � � A ′ ( x ) B ′ ( y ) C ( y ) dxdy + A ′ ( x ) B ′ ( y ) C ( x ) dxdy = y = x x =0 y =0 x =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 12 / 32

Adaptating boxed product Simple example : factorization Factorize the generating function: y z x z � � � � A ′ ( x ) B ′ ( y ) C ( y ) dxdy + A ′ ( x ) B ′ ( y ) C ( x ) dxdy D ( z ) = x =0 y =0 y =0 x =0 we swap the integration order of y and x on the right term z x z z � � � � A ′ ( x ) B ′ ( y ) C ( y ) dxdy + A ′ ( x ) B ′ ( y ) C ( x ) dxdy = y = x x =0 y =0 x =0 we replace y and x by min( x , y ) in C z z � � = A ′ ( x ) B ′ ( y ) C (min( x , y )) dxdy x =0 y =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 12 / 32

Adaptating boxed product A more complicated example: factorization Y 1 · · · Y k − 1 Y k A R · · · X p − 1 X p X 1 x p − 1 y k − 1 min( x p , y k ) z u x 1 u y 1 � � � � � � � � P k , p ( z ) = . . . . . . u =0 x 1 =0 x 2 =0 x p =0 y 1 =0 y 2 =0 y k =0 t =0 A ′ ( u ) X ′ 1 ( x 1 ) X ′ 2 ( x 2 ) . . . X ′ p ( x p ) Y ′ 1 ( y 1 ) Y ′ 2 ( y 2 ) . . . Y ′ k ( y k ) R ′ ( t ) dudtdx 1 . . . N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 13 / 32

Adaptating boxed product Simple example: reordering We change the integration order: min( x , y ) z z � � � A ′ ( x ) B ′ ( y ) C ′ ( t ) dxdydt D ( z ) = x =0 y =0 t =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 14 / 32

Adaptating boxed product Simple example: reordering We change the integration order: min( x , y ) z z � � � A ′ ( x ) B ′ ( y ) C ′ ( t ) dxdydt D ( z ) = x =0 y =0 t =0 min( x , z ) z z � � � A ′ ( x ) B ′ ( y ) C ′ ( t ) dxdydt D ( z ) = y = t x =0 t =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 14 / 32

Adaptating boxed product Simple example: reordering We change the integration order: min( x , y ) z z � � � A ′ ( x ) B ′ ( y ) C ′ ( t ) dxdydt D ( z ) = x =0 y =0 t =0 min( x , z ) z z � � � A ′ ( x ) B ′ ( y ) C ′ ( t ) dxdydt D ( z ) = y = t x =0 t =0 z z z � � � A ′ ( x ) B ′ ( y ) C ′ ( t ) dxdydt D ( z ) = x = t y = t t =0 N. ROLIN (Universit´ e Paris 13 – LIPN) Extended boxed product GASCom 2016 14 / 32