S´eminaireFlajolet21/09/2017 VivianePons LatticeandHopfalgebraofintegerrelations

(1)

Motivation Poset interpretation Lattice and Hopf algebra

Lattice and Hopf algebra of integer relations

Viviane Pons

with Gr´ egory Chˆ atel and Vincent Pilaud Universit´ e Paris-Sud

S´ eminaire Flajolet 21/09/2017

Viviane Pons Lattice and Hopf algebra of integer relations

(2)

permutations binary trees binary sequences

Combinatorics

4321

4231 4312

3421 3412

3241 2431 4213 4132

1234

1324 1243

2134 2143

2314 3124 1342 1423

3142 2413 4123 1432

3214 2341

− − −

− + − + − −

− − +

− + + + − + + + −

+ + +

Algebra

Malvenuto-Reutenauer algebra Loday-Ronco algebra Descent Hopf algebra

→ All these objects (and more) can be interpreted in terms of integer posets.

(3)

permutations binary trees binary sequences

Combinatorics

4321

4231 4312

3421 3412

3241 2431 4213 4132

1234

1324 1243

2134 2143

2314 3124 1342 1423

3142 2413 4123 1432

3214 2341

− − −

− + − + − −

− − +

− + + + − + + + −

+ + +

Algebra

Malvenuto-Reutenauer algebra Loday-Ronco algebra Descent Hopf algebra

→ All these objects (and more) can be interpreted in terms of integer posets.

(4)

Weak order on permutations Integer posets and combinatorial objects Interval-posets

Weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(5)

Weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(6)

Weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(7)

Weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413

2413 ∨ 4213 = 4231

(8)

Weak order

123 213 132

231 312

321 1234 2134 1324 1243 2314 3124 2143 1342 1423 3214 2341 3142 2413 4123 1432

3241 2431 3412 4213 4132 3421 4231 4312

4321

2413 ∧ 4213 = 2413 2413 ∨ 4213 = 4231

(9)

Permutation poset 4312

1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(10)

Permutation poset 4312

1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(11)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(12)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(13)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(14)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(15)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(16)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(17)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(18)

Permutation poset

4312 1 2 3 4

3412

1 2 3 4

3142 1 2 3 4

(19)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(20)

Permutation poset

4312 1 2 3 4

3412 1 2 3 4

3142 1 2 3 4

(21)

Relation associated to a permutation

. . . x . . . y . . . ⇒ x R y 4312 1 2 3 4

Weak order

R ^Inc = {i R j , i < j } R ^Dec = {j R i , i < j }

R 4 S ⇔ R ^Inc ⊇ S ^Inc and R ^Dec ⊆ S ^Dec

(22)

Relation associated to a permutation

. . . x . . . y . . . ⇒ x R y 4312 1 2 3 4

Weak order

R ^Inc = {i R j , i < j } R ^Dec = {j R i , i < j }

R 4 S ⇔ R ^Inc ⊇ S ^Inc and R ^Dec ⊆ S ^Dec

(23)

Binary tree poset

1 2 3 4 5 6 7 8

→ Tamari lattice

(24)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(25)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(26)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(27)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(28)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(29)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(30)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(31)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(32)

Binary tree poset

4

2 5

1 3 7

6 8

1 2 3 4 5 6 7 8

→ Tamari lattice

(33)

Binary sequence poset

− − + − +

₁ ₂ ₃ ₄ ₅ ₆

→ Boolean lattice

(34)

Binary sequence poset

− − + − +

₁ ₂ ₃ ₄ ₅ ₆

→ Boolean lattice

(35)

Binary sequence poset

− − + − +

₁ ₂ ₃ ₄ ₅ ₆

→ Boolean lattice

(36)

Binary sequence poset

− − + − +

₁ ₂ ₃ ₄ ₅ ₆

→ Boolean lattice

(37)

Permutation intervals

σ ⁰ = 4231

2431 4213

2413 σ = 2143

1 2 3 4

(38)

Interval-poset characterization Posets (anti-symmetric, transitive) and

intervals of intervals of intervals of permutations binary trees binary sequence (weak order) (Tamari lattice) (boolean lattice)

a b c

⇒

a b c

or

a b c a b c

a b c

⇒

a b c

or

a b c a b c

a b c

(39)

Interval-poset characterization Posets (anti-symmetric, transitive) and

intervals of intervals of intervals of permutations binary trees binary sequence (weak order) (Tamari lattice) (boolean lattice)

a b c

⇒

a b c

or

a b c a b c

a b c

⇒

a b c

or

a b c a b c

a b c

(40)

Interval-poset characterization Posets (anti-symmetric, transitive) and

intervals of intervals of intervals of permutations binary trees binary sequence (weak order) (Tamari lattice) (boolean lattice)

a b c

⇒

a b c

or

a b c

⇒

a b c

or

a b c

(41)

Interval-poset characterization Posets (anti-symmetric, transitive) and

intervals of intervals of intervals of permutations binary trees binary sequence (weak order) (Tamari lattice) (boolean lattice)

a b c

⇒

a b c

or

a b c a b c

a b c

⇒

a b c

or

a b c a b c

a b c

(42)

Lattice of integer binary relation Hopf algebra

Integer Poset Other Hopf algebras

Binary relations on integer R is a binary relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i i R k

Family of size 2 ⁿ⁽ⁿ⁻¹⁾ .

(43)

Binary relations on integer R is a binary relation of size n.

1 2 ... i ... j ... k ... n

i R j

k R i i R k

Family of size 2 ⁿ⁽ⁿ⁻¹⁾ .

(44)

Binary relations on integer R is a binary relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i

i R k Family of size 2 ⁿ⁽ⁿ⁻¹⁾ .

(45)

Binary relations on integer R is a binary relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i i R k

Family of size 2 ⁿ⁽ⁿ⁻¹⁾ .

(46)

Binary relations on integer R is a binary relation of size n.

1 2 ... i ... j ... k ... n

i R j k R i i R k

Family of size 2 ⁿ⁽ⁿ⁻¹⁾ .

(47)

Weak order

Let R an integer binary relation

R ^Inc = {i R j , i < j } R ^Dec = {j R i , i < j }

Let R and S be two integer binary relations

R 4 S ⇔ R ^Inc ⊇ S ^Inc et R ^Dec ⊆ S ^Dec

(48)

Weak order

Let R an integer binary relation

R ^Inc = {i R j , i < j } R ^Dec = {j R i , i < j } Let R and S be two integer binary relations

R 4 S ⇔ R ^Inc ⊇ S ^Inc et R ^Dec ⊆ S ^Dec

(49)

Binary relations of size 2

1 2

1 2 1 2

1 2

(50)

Binary relations of size 2

1 2

1 2 1 2

1 2

(51)

Binary relations of size 2

1 2

(52)

Binary relations of size 2

1 2

1 2 1 2

1 2

(53)

Binary relations of size 2

1 2

1 2 1 2

1 2

(54)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 31 2 3

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

1 2 3

(55)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(56)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(57)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(58)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(59)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(60)

Meet and join

1 2 3 4 1 2 3 4

1 2 3 4

(61)

Hopf algebra

R × S = X

R S

(62)

1 2

×

¹

= X

1 2 3

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(63)

1 2 3

(64)

Coproduct

4R = X P ⊗ Q where R = P ∪ Q ∪ (P → Q)

(65)

4

¹ ² ³

=

1 2 3

⊗ ∅ +

¹

⊗

¹ ²

+

¹ ²

⊗

¹

+ ∅ ⊗

¹ ² ³

(66)

4

¹ ² ³

=

¹ ² ³

⊗ ∅

+

¹

⊗

¹ ²

+

¹ ²

⊗

¹

+ ∅ ⊗

¹ ² ³

(67)

4

¹ ² ³

=

¹ ² ³

⊗ ∅ +

¹

⊗

¹ ²

+

¹ ²

⊗

¹

+ ∅ ⊗

¹ ² ³

(68)

4

¹ ² ³

=

¹ ² ³

⊗ ∅ +

¹

⊗

¹ ²

+

¹ ²

⊗

¹

+ ∅ ⊗

¹ ² ³

(69)

Integer Posets

We restrict ourselves to transitive antisymmetric relations.

(70)

1 2 3

1 2 3 1 2 3

1 2 3

1 2 3 1 2 3

(71)

Hopf algebra of integer poset

Quotient of the integer relations Hopf algebra R ≡ 0 if R is not a poset.

(72)

1 2

×

¹

=

1 2 3

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(73)

1 2

×

¹

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(74)

1 2

×

¹

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(75)

1 2

×

¹

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(76)

1 2

×

¹

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(77)

1 2

×

¹

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(78)

1 2

×

¹

=

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+

¹ ² ³

+ . . .

(79)

1 2 3

(80)

To check: is it well defined?

P × 0 = 0 40 = 0

(81)

P × 0 = 0

1 2 3

×

¹ ²

=

1 2 3 4 5

+

¹ ² ³ ⁴ ⁵

+ . . .

(82)

P × 0 = 0

1 2 3

×

¹ ²

=

¹ ² ³ ⁴ ⁵

+

¹ ² ³ ⁴ ⁵

+ . . .

(83)

40 = 0

4

¹ ²

=

∅ ⊗

¹ ²

+

¹ ²

⊗ ∅

4

¹ ² ³

=

∅ ⊗

¹ ² ³

+

¹ ² ³

⊗ ∅

(84)

40 = 0

4

¹ ²

= ∅ ⊗

¹ ²

+

¹ ²

⊗ ∅ 4

¹ ² ³

=

∅ ⊗

¹ ² ³

+

¹ ² ³

⊗ ∅

(85)

40 = 0

4

¹ ²

= ∅ ⊗

¹ ²

+

¹ ²

⊗ ∅ 4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ² ³

⊗ ∅

(86)

Question: can we restrict to intervals of permutations?

Yes!

(87)

Question: can we restrict to intervals of permutations?

Yes!

(88)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

1 2 3 4

+

¹ ² ³ ⁴

+ . . .

Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(89)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

1 2 3 4

+

¹ ² ³ ⁴

+ . . .

Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(90)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

¹ ² ³ ⁴

+

¹ ² ³ ⁴

+ . . .

Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(91)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

¹ ² ³ ⁴

+

¹ ² ³ ⁴

+ . . . Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(92)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

¹ ² ³ ⁴

+

¹ ² ³ ⁴

+ . . . Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(93)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

¹ ² ³ ⁴

+

¹ ² ³ ⁴

+ . . .

Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(94)

Interval-poset of permutations

a b c

⇒

_a _b _c

or

_a _b _c

Product

1 2 3

×

¹

=

¹ ² ³ ⁴

+

¹ ² ³ ⁴

+ . . .

Coproduct

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ²

⊗

¹

+

¹ ² ³

⊗ ∅

(95)

Product with 0

1 2

×

¹ ² ³

=

1 2 3 4 5

+

¹ ² ³ ⁴ ⁵

+ . . .

Coproduct of 0

4

¹ ² ³

=

∅ ⊗

¹ ² ³

+

¹ ² ³

⊗ ∅

(96)

Product with 0

1 2

×

¹ ² ³

=

¹ ² ³ ⁴ ⁵

+

¹ ² ³ ⁴ ⁵

+ . . .

Coproduct of 0

4

¹ ² ³

=

∅ ⊗

¹ ² ³

+

¹ ² ³

⊗ ∅

(97)

Product with 0

1 2

×

¹ ² ³

=

¹ ² ³ ⁴ ⁵

+

¹ ² ³ ⁴ ⁵

+ . . .

Coproduct of 0

4

¹ ² ³

= ∅ ⊗

¹ ² ³

+

¹ ² ³

⊗ ∅

(98)

Question: can we restrict to intervals of binary trees (and binary sequences)?

Not exactly... But

Lattice: ok

Hopf algebra: as a sub algebra of the Hopf algebra on permutations intervals

(99)

Question: can we restrict to intervals of binary trees (and binary sequences)?

Not exactly... But

Lattice: ok

Hopf algebra: as a sub algebra of the Hopf algebra on permutations intervals

(100)

Question: can we restrict to intervals of binary trees (and binary sequences)?

Not exactly... But

Lattice: ok

Hopf algebra: as a sub algebra of the Hopf algebra on permutations intervals

(101)

References

I Chˆ atel, Pilaud, P. The weak order on integer posets arXiv:1701.07995

I Pilaud, P. Permutrees arXiv:1606.09643

I Pilaud, P. The Hopf algebra of integer posets (work in progress)