Analytic combinatorics of chord and hyperchord diagrams with k crossings Vincent Pilaud Juanjo Rué CNRS & LIX, FU Berlin École Polytechnique AofA’14, Paris
Planar chord configurations Structural properties The simplicial complex of crossing-free chord diagrams is the boundary complex of the associahedron Enumerative properties Theorem [Flajolet & Noy ’99] √ π n − 3 / 2 ρ − n . Λ # chord configurations in the following families ∼ n →∞ dissections partitions graphs conn. graphs forests trees √ √ √ ρ − 1 27 3 + 2 2 4 6 + 4 2 6 3 8 . 2246 √ √ 4 √ √ √ √ √ − 140 + 99 − 140 + 99 2 2 6 2 3 Λ 1 9 − 0 . 07465 4 4 6 27
Nearly-planar chord configurations Crossing-free chord configurations have relevant enumerative and structural properties Enumerative/structural properties of nearly planar chord configurations? chord hyperchord matchings partitions diagrams diagrams
Nearly-planar chord configurations Crossing-free chord configurations have relevant enumerative and structural properties Enumerative/structural properties of nearly planar chord configurations? chord hyperchord matchings partitions diagrams diagrams Possible constraints... ◮ at most k crossings ◮ no ( k + 1 ) -crossings ◮ each chord crosses at most k others ◮ become crossing-free when removing at most k chords
Nearly-planar chord configurations Crossing-free chord configurations have relevant enumerative and structural properties Enumerative/structural properties of nearly planar chord configurations? chord hyperchord matchings partitions diagrams diagrams Possible constraints... ... on the crossing graph ◮ at most k crossings edges ◮ no ( k + 1 ) -crossings cliques ◮ each chord crosses at most k others degrees ◮ become crossing-free when removing at most k chords covers
A zoom on ( k + 1 ) -crossing-free chord diagrams chord diagrams with no k + 1 mutually crossing chords have a rich combinatorial structure Theorem [Jonsson ’03] The simplicial complex of ( k + 1 ) -crossing-free chord diagrams is a sphere. Maximal ( k + 1 ) -crossing-free chord diagrams are k -triangulations They can be decomposed into a complex of k -stars [P. & Santos ’09] star decomposition flip k -triangulations are counted by a Hankel determinant of Catalan numbers [Jonsson ’05]
Our results on configurations with k crossings C family of configurations among chord hyperchord matchings partitions diagrams diagrams C ( n , m , k ) = # confs with n vertices, m (hyper)chords, and k crossings n , m ∈ N |C ( n , m , k ) | x n y m generating function C k ( x , y ) = �
Our results on configurations with k crossings C family of configurations among chord hyperchord matchings partitions diagrams diagrams Theorem (Rationality) The generating function C k ( x , y ) of configurations in C with exactly k crossings is a rational function of the generating function C 0 ( x , y ) of planar configurations in C and of the variables x and y. partial results in [Bona, Partitions with k crossings, ’00]
Our results on configurations with k crossings C family of configurations among chord hyperchord matchings partitions diagrams diagrams Theorem (Rationality) The generating function C k ( x , y ) of configurations in C with exactly k crossings is a rational function of the generating function C 0 ( x , y ) of planar configurations in C and of the variables x and y. Theorem (Asymptotics) For k ≥ 1 , the number of conf. in C with k crossings and n vertices is n →∞ Λ n α ρ − n ( 1 + o ( 1 )) , [ x n ] C k ( x , 1 ) = for certain constants Λ , α, ρ ∈ R depending on C and k.
Constants Theorem (Asymptotics) For k ≥ 1 , the number of conf. in C with k crossings and n vertices is n →∞ Λ n α ρ − n ( 1 + o ( 1 )) , [ x n ] C k ( x , 1 ) = for certain constants Λ , α, ρ ∈ R depending on C and k. exp. α sing. ρ − 1 family constant Λ √ 2 ( 2 k − 3 )!! k − 3 matchings 2 4 k − 1 k ! Γ � k − 1 � 2 2 ( 2 k − 3 )!! k − 3 partitions 4 2 3 k − 1 k ! Γ � k − 1 � 2 2 √ √ � 3 k � � √ − 2 + 3 2 − 140 + 99 2 ( 2 k − 3 )!! k − 3 chord √ 2 6 + 4 2 2 3 k + 1 ( 3 − 4 2 ) k − 1 k ! Γ( k − 1 diagrams 2 ) ≃ 1 . 034 3 k 0 . 003655 ( 2 k − 3 )!! k − 3 hyperchord 2 ≃ 64 . 97 0 . 03078 k − 1 k ! Γ( k − 1 diagrams 2 )
Matchings with k crossings M = { perfect matchings with endpoints on the unit circle } All matchings are “rooted” and “up to deformation” M ( n , k ) = number of matchings with n vertices and k crossings generating function M k ( x ) = � n ∈ N |M ( n , k ) | x n
Core matchings Core of a matching M = submatching M ⋆ formed by all chords involved in at least one crossing There are only finitely many core matchings with k crossings
Core matching polynomial 1 � � x n i ( K ) KM k ( x 1 , . . . , x k ) = i n ( K ) K k -core i ∈ [ k ] matching n i ( K ) = # regions of the complement of K with i boundary arcs n ( K ) = � i n i ( K ) = # of vertices of K KM 3 ( x 1 , x 2 , x 3 ) = 1 6 + 3 8 + 3 8 x 2 + 1 8 x 2 2 + 3 x 1 9 x 3 6 x 1 2 x 1 2 x 1 3 x 1
Computing core matching polynomials Core matchings can be decomposed into connected matchings
Computing core matching polynomials Core matchings can be decomposed into connected matchings a b i g 5 h e 4 4 c d 3 1 2 2 f 2 3 level of an arc α of M = graph distance between α and the leftmost arc in the crossing graph of M
Computing core matching polynomials Core matchings can be decomposed into connected matchings a b i g 5 h e 4 4 c d 3 1 2 2 f 2 3 level of an arc α of M = graph distance between α and the leftmost arc in the crossing graph of M To generate all possible connected matchings, start from a single arc and add arcs one by one. If the last constructed arc ( i , j ) was at level ℓ , then (i) either add a new arc ( u , v ) in the current level ℓ , with u > i and crossing at least one arc at level ℓ − 1, and no arc at level < ℓ − 1 (ii) or add an new arc ( u , v ) at a new level ℓ + 1 with u > 1 and crossing at least one arc at level ℓ and no at level < ℓ
Generating function of matchings with k crossings Proposition For k ≥ 1 , the generating function M k ( x ) of the perfect matchings with k crossings is given by x i d i − 1 M k ( x ) = x d � �� x i − 1 M 0 ( x ) � dx KM k x i ← ( i − 1 )! dx i − 1 In particular, M k ( x ) is a rational function of M 0 ( x ) and x
Generating function of matchings with k crossings Proposition For k ≥ 1 , the generating function M k ( x ) of the perfect matchings with k crossings is given by x i d i − 1 M k ( x ) = x d � �� x i − 1 M 0 ( x ) � dx KM k x i ← ( i − 1 )! dx i − 1 In particular, M k ( x ) is a rational function of M 0 ( x ) and x Choose a core matching with k crossings Replace each region with i boundaries by a crossing-free matching with a root and i − 1 additional marks Reroot to obtain a rooted matching
Asymptotic analysis �� n i ( K ) x i d i − 1 M k ( x ) = x d 1 � � � x i − 1 M 0 ( x ) � dx i − 1 dx n ( K ) ( i − 1 )! K k -core i ≥ 1 matching
Asymptotic analysis �� n i ( K ) x i d i − 1 M k ( x ) = x d 1 � � � x i − 1 M 0 ( x ) � dx i − 1 dx n ( K ) ( i − 1 )! K k -core i ≥ 1 matching M 0 ( x ) has two singularities around x = 1 2 and x = − 1 2 . Denote X + = √ 1 − 2 x around x = 1 2 , then √ � 2 � M 0 ( x ) = 2 − 2 2 X + + O X + x ∼ 1 2 √ d i 1 − 2 i + O � 2 − 2 i � dx i M 0 ( x ) = 2 2 ( 2 i − 3 )!! X + X + , x ∼ 1 2 where ( 2 i − 3 )!! : = ( 2 i − 3 ) · ( 2 i − 5 ) · · · 3 · 1.
Asymptotic analysis �� n i ( K ) x i d i − 1 M k ( x ) = x d 1 � � � x i − 1 M 0 ( x ) � dx i − 1 dx n ( K ) ( i − 1 )! K k -core i ≥ 1 matching � √ � n i ( K ) φ ( K ) 2 ( 2 i − 5 )!! X − φ ( K ) − 2 � � = ( 1 + O ( X + )) , 4 i − 1 ( i − 1 )! + 2 n ( K ) x ∼ 1 2 K k -core i > 1 matching where φ ( K ) = � i > 1 ( 2 i − 3 ) n i ( K )
Asymptotic analysis �� n i ( K ) x i d i − 1 M k ( x ) = x d 1 � � � x i − 1 M 0 ( x ) � dx i − 1 dx n ( K ) ( i − 1 )! K k -core i ≥ 1 matching � √ � n i ( K ) φ ( K ) 2 ( 2 i − 5 )!! X − φ ( K ) − 2 � � = ( 1 + O ( X + )) , 4 i − 1 ( i − 1 )! + 2 n ( K ) x ∼ 1 2 K k -core i > 1 matching where φ ( K ) = � i > 1 ( 2 i − 3 ) n i ( K ) is maximized by the core matchings with n 1 ( K ) = 3 k and n k ( K ) = 1:
Asymptotic analysis Proposition For k ≥ 1 , the number of perfect matchings with k crossings and n = 2 m vertices is ( 2 k − 3 )!! 2 4 m ( 1 + o ( 1 )) , � m k − 3 [ x 2 m ] M k ( x ) = 2 k − 1 k ! Γ � k − 1 m →∞ 2 where ( 2 k − 3 )!! : = ( 2 k − 3 ) · ( 2 k − 5 ) · · · 3 · 1 . Dominant core matchings maximize φ ( K ) = � i > 1 ( 2 i − 3 ) n i ( K )
Probabilities core matchings 1 0 . 8 probabilities 0 . 6 0 . 4 0 . 2 0 20 40 60 80 100 120 140 number of vertices
Extension to partitions S = subset of N ∗ distinct from { 1 } P S = { partitions with parts of size in S} crossing = two crossing chords that belong to distinct parts P S ( n , m , k ) = # partitions with n vert., m parts, and k crossings n , m ∈ N |P S ( n , m , k ) | x n y m generating function P S k ( x , y ) = �
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