Analytic combinatorics of chord and hyperchord diagrams with k - - PowerPoint PPT Presentation

Analytic combinatorics

• f chord and hyperchord diagrams

with k crossings

Vincent Pilaud

CNRS & LIX, École Polytechnique

Juanjo Rué

FU Berlin

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]

# chord configurations in the following families ∼

n→∞ Λ √π n−3/2 ρ−n.

dissections partitions graphs

• conn. graphs

forests trees ρ−1 3 + 2 √ 2 4 6 + 4 √ 2 6 √ 3 8.2246

27 4

Λ √

−140+99 √ 2 4

1 √

−140+99 √ 2 4 √ 6 9 − √ 2 6

0.07465

√ 3 27

Nearly-planar chord configurations

Crossing-free chord configurations have relevant enumerative and structural properties Enumerative/structural properties of nearly planar chord configurations? matchings partitions chord diagrams hyperchord diagrams

Nearly-planar chord configurations

Crossing-free chord configurations have relevant enumerative and structural properties Enumerative/structural properties of nearly planar chord configurations? matchings partitions chord diagrams hyperchord 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? matchings partitions chord diagrams hyperchord 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 matchings partitions chord diagrams hyperchord diagrams C(n, m, k) = # confs with n vertices, m (hyper)chords, and k crossings generating function Ck(x, y) =

n,m∈N |C(n, m, k)| xn y m

Our results on configurations with k crossings

C family of configurations among matchings partitions chord diagrams hyperchord diagrams

Theorem (Rationality)

The generating function Ck(x, y) of configurations in C with exactly k crossings is a rational function of the generating function C0(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 matchings partitions chord diagrams hyperchord diagrams

Theorem (Rationality)

The generating function Ck(x, y) of configurations in C with exactly k crossings is a rational function of the generating function C0(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 [xn] Ck(x, 1) =

n→∞ Λ nα ρ−n (1 + o(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 [xn] Ck(x, 1) =

n→∞ Λ nα ρ−n (1 + o(1)),

for certain constants Λ, α, ρ ∈ R depending on C and k. family constant Λ

• exp. α sing. ρ−1

matchings √ 2 (2k − 3)!! 4k−1 k! Γ

• k − 1

2

• k − 3

2 2 partitions (2k − 3)!! 23k−1 k! Γ

• k − 1

2

• k − 3

2 4 chord diagrams

• −2 + 3

√ 2 3k −140 + 99 √ 2 (2k − 3)!! 23k+1 (3 − 4 √ 2)k−1 k! Γ(k − 1

2)

k − 3 2 6 + 4 √ 2 hyperchord diagrams ≃ 1.0343k 0.003655 (2k − 3)!! 0.03078k−1 k! Γ(k − 1

2)

k − 3 2 ≃ 64.97

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 Mk(x) =

n∈N |M(n, k)| xn

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

KMk(x1, . . . , xk) =

• K k-core

matching

1 n(K)

• i∈[k]

xni (K)

i

ni(K) = # regions of the complement of K with i boundary arcs n(K) =

i ni(K) = # of vertices of K

KM3(x1, x2, x3) = 1 6 x1

6 + 3

2 x1

8 + 3

2 x1

8 x2 2 + 3 x1 8 x2 + 1

3 x1

9 x3

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 4 4 3 1 2 2 2 3 h e c d f

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 4 4 3 1 2 2 2 3 h e c d f

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 Mk(x) of the perfect matchings with k crossings is given by Mk(x) = x d dx KMk

• xi ←

xi (i − 1)! di−1 dxi−1

• xi−1M0(x)
• In particular, Mk(x) is a rational function of M0(x) and x

Generating function of matchings with k crossings

Proposition

For k ≥ 1, the generating function Mk(x) of the perfect matchings with k crossings is given by Mk(x) = x d dx KMk

• xi ←

xi (i − 1)! di−1 dxi−1

• xi−1M0(x)
• In particular, Mk(x) is a rational function of M0(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

Mk(x) = x d dx

• K k-core

matching

1 n(K)

• i≥1
• xi

(i − 1)! di−1 dxi−1

• xi−1M0(x)

ni (K)

Asymptotic analysis

Mk(x) = x d dx

• K k-core

matching

1 n(K)

• i≥1
• xi

(i − 1)! di−1 dxi−1

• xi−1M0(x)

ni (K) M0(x) has two singularities around x = 1

2 and x = − 1 2.

Denote X+ = √1 − 2x around x = 1

2, then

M0(x) =

x∼ 1

2

2 − 2 √ 2 X+ + O

• X+

2

di dxi M0(x) =

x∼ 1

2

2 √ 2 (2i − 3)!! X+

1−2i + O

• X+

2−2i

, where (2i − 3)!! := (2i − 3) · (2i − 5) · · · 3 · 1.

Asymptotic analysis

Mk(x) = x d dx

• K k-core

matching

1 n(K)

• i≥1
• xi

(i − 1)! di−1 dxi−1

• xi−1M0(x)

ni (K) =

x∼ 1

2

• K k-core

matching

φ(K) 2n(K)

• i>1

√ 2 (2i − 5)!! 4i−1 (i − 1)! ni (K) X −φ(K)−2

+

(1 + O(X+)), where φ(K) =

i>1(2i − 3)ni(K)

Asymptotic analysis

Mk(x) = x d dx

• K k-core

matching

1 n(K)

• i≥1
• xi

(i − 1)! di−1 dxi−1

• xi−1M0(x)

ni (K) =

x∼ 1

2

• K k-core

matching

φ(K) 2n(K)

• i>1

√ 2 (2i − 5)!! 4i−1 (i − 1)! ni (K) X −φ(K)−2

+

(1 + O(X+)), where φ(K) =

i>1(2i − 3)ni(K) is maximized by the core matchings

with n1(K) = 3k and nk(K) = 1:

Asymptotic analysis

Proposition

For k ≥ 1, the number of perfect matchings with k crossings and n = 2m vertices is [x2m] Mk(x) =

m→∞

(2k − 3)!! 2k−1 k! Γ

• k − 1

2

mk− 3

2 4m (1 + o(1)),

where (2k − 3)!! := (2k − 3) · (2k − 5) · · · 3 · 1. Dominant core matchings maximize φ(K) =

i>1(2i − 3)ni(K)

Probabilities core matchings

20 40 60 80 100 120 140 0.2 0.4 0.6 0.8 1 number of vertices probabilities

Extension to partitions

S = subset of N∗ distinct from {1} PS = {partitions with parts of size in S} crossing = two crossing chords that belong to distinct parts PS(n, m, k) = # partitions with n vert., m parts, and k crossings generating function PS

k (x, y) = n,m∈N |PS(n, m, k)| xn y m

Core partitions

Core of a partition P = subpartition P⋆ formed by all parts involved in at least one crossing There are only finitely many core partitions with k crossings Encoded in the core partition polynomial KPS

k (x1, . . . , xk)

Generating function

Proposition

For k ≥ 1, the generating function PS

k (x, y) of partitions with k crossings

and where the size of each block belongs to S is PS

k (x, y) = x d

dx KPS

k

• xi ←

xi (i − 1)! di−1 dxi−1

• xi−1PS

0 (x, y)

• , y
• .

If S is finite or ultimately periodic, then PS

k (x, y) is a rational function

• f PS

0 (x, y) and x.

Generating function

Proposition

For k ≥ 1, the generating function PS

k (x, y) of partitions with k crossings

and where the size of each block belongs to S is PS

k (x, y) = x d

dx KPS

k

• xi ←

xi (i − 1)! di−1 dxi−1

• xi−1PS

0 (x, y)

• , y
• .

If S is finite or ultimately periodic, then PS

k (x, y) is a rational function

• f PS

0 (x, y) and x.

Two difficulties for the asymptotic:

◮ minimal singularity and singular behavior of PS 0 (x, 1) ◮ characterize dominant k-core partitions

Difficulty 1: Singular behavior of PS

0 (x, 1)

Proposition

For S = {1}, the generating function PS

0 (x, 1) satisfies

PS

0 (x, 1)

=

x∼ρS αS − βS

• 1 − x

ρS + O

• 1 − x

ρS

• ,

where ρS, αS and βS are defined by

• s∈S

(s − 1)τS

s = 1,

ρS := τS

• s∈S sτSs ,

αS := 1 +

• s∈S

τS

s,

and βS :=

• 2
• s∈S sτSs3
• s∈S s(s − 1)τSs .

Singular behavior of generating functions defined by a smooth implicit-function schema (Meir & Moon)

Asymptotic analysis

Proposition

For k ≥ 1, and S = {1}, the number of partitions with k crossings, n vertices, and where the size of each block belongs to S is [xn] PS

k (x, 1)

=

n→∞ gcd(S)|n

ΛS n

ψ(k,S) 2

ρS

−n (1 + o(1)),

where ψ(k, S) = maximum of φ(K) :=

i>1(2i − 3) ni(K) and

ΛS := gcd(S) ψ(k, S) 2 Γ ψ(k,S)

2

+ 1

• K∈PS

φ(K)=ψ(k,S)

τSn1(K) n(K)

• i>1

ρSi βS (2i − 5)!! 2i−1 (i − 1)! ni (K) .

Difficulty 2: Dominant k-core partitions

Only determined for specific instances:

◮ all partitions: S = N∗

Proposition

For k ≥ 1, the number of partitions with k crossings and n vertices is [xn] PN∗

k (x, 1)

=

n→∞

(2k − 3)!! 23k−1 k! Γ

• k − 1

2

nk− 3

2 4n (1 + o(1)).

◮ q-uniform partitions: S = {q} and k = k′(q − 1)2

[xqm] P{q}

k′(q−1)2(x, 1)

=

m→∞ Λ{q} k′

mk′− 3

2

• qq

(q − 1)q−1 m (1 + o(1)).

◮ q-multiple partitions: S = qN and k = k′(q − 1)2

[xqm] PqN∗

k′(q−1)2(x, 1)

=

m→∞ ΛqN k′ mk′− 3

2

(q + 1)q+1 qq m (1 + o(1)).

Our results on configurations with k crossings

C family of configurations among matchings partitions chord diagrams hyperchord diagrams

Theorem (Rationality)

The generating function Ck(x, y) of configurations in C with exactly k crossings is a rational function of the generating function C0(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 [xn] Ck(x, 1) =

n→∞ Λ nα ρ−n (1 + o(1)),

for certain constants Λ, α, ρ ∈ R depending on C and k.