Algebraic Constructions on Set Partitions 22/03/2007 1 De 16

Algebraic constructions on set partitions 22/03/2007 1 de 16

Algebraic constructions on set partitions

Maxime Rey

Laboratoire d’Informatique de l’Institut Gaspard-Monge Universit´eParis-Est

2007 July 6 Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

Several works mixed combinatorial algorithms, Hopf algebras, partial orders.

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

Several works mixed combinatorial algorithms, Hopf algebras, partial orders. permutations FQSym permutohedron binary trees PBT associahedron Std. Young tableaux FSym weak order on SYT compositions NCSF hypercube ordered set partitions WQSym pseudo-permutohedron plane trees T D quotient of pp segmented compositions TC quotient of pp

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

combinatorial algorithms : Hopf algebra : partial order :

permutations FQSym permutohedron binary trees PBT associahedron Std. Young tableaux FSym weak order on SYT compositions NCSF hypercube ordered set partitions WQSym pseudo-permutohedron plane trees T D quotient of pp segmented compositions TC quotient of pp

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

combinatorial algorithms : Robinson-Schensted correspondence and jeu de taquin, Hopf algebra : partial order : permutations FQSym permutohedron binary trees PBT associahedron Std. Young tableaux FSym weak order on SYT compositions NCSF hypercube ordered set partitions WQSym pseudo-permutohedron plane trees T D quotient of pp segmented compositions TC quotient of pp

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

combinatorial algorithms : Robinson-Schensted correspondence and jeu de taquin, Hopf algebra : FSym (Poirier-Reutenauer Hopf algebra), partial order : permutations FQSym permutohedron binary trees PBT associahedron Std. Young tableaux FSym weak order on SYT compositions NCSF hypercube ordered set partitions WQSym pseudo-permutohedron plane trees T D quotient of pp segmented compositions TC quotient of pp

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

combinatorial algorithms : Robinson-Schensted correspondence and jeu de taquin, Hopf algebra : FSym (Poirier-Reutenauer Hopf algebra), partial order : weak order on SYT. permutations FQSym permutohedron binary trees PBT associahedron Std. Young tableaux FSym weak order on SYT compositions NCSF hypercube ordered set partitions WQSym pseudo-permutohedron plane trees T D quotient of pp segmented compositions TC quotient of pp

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 2 de 16

Preamble

Goal : Similar construction over set partitions. Algebraic constructions on set partitions 22/03/2007 3 de 16

Related works

M. Rosas, B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, to appear. N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree preprint math.CO/0509265. NCSym is a non-commutative and co-commutative Hopf algebra. Algebraic constructions on set partitions 22/03/2007 4 de 16

Results

P, a self-dual Hopf algebra on set partitions Bidendriform ⇒ free, co-free and self-dual Multiplicative bases PBT ֒→ P ֒→ FQSym Bell order, a partial order on set partitions Intervals of the Bell order describe the product of P Scrolling, a jeu de taquin-like on set partitions Robinson-Schensted analogue on set partitions Plactic-like monoid structure Algebraic constructions on set partitions 22/03/2007 4 de 16

Results

P, a self-dual Hopf algebra on set partitions Bidendriform ⇒ free, co-free and self-dual Multiplicative bases PBT ֒→ P ֒→ FQSym Bell order, a partial order on set partitions Intervals of the Bell order describe the product of P Scrolling, a jeu de taquin-like on set partitions Robinson-Schensted analogue on set partitions Plactic-like monoid structure A. Burstein and I. Lankham ”Combinatorics of Patience Sorting Piles”, SLC54A. Algebraic constructions on set partitions 22/03/2007 4 de 16

Results

Combinatorial algorithm Robinson-Schensted analogue σ = 586714923

∅

A. Burstein and I. Lankham ”Combinatorics of Patience Sorting Piles”, SLC54A. Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

∅ 5 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

5 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

5 8

5 < 8 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

5 8

5 < 8 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

5 8 6

5 < 6 < 8 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 5 ↑

5 < 6 < 8 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 5 6

5 < 6 < 8 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 5 6 7 7

6 < 7 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

↑ 8 5 6 7 1

1 < 5 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

5 8 1 6 7

1 < 5 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 5 6 1 4 7 4

1 < 4 < 6 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 5 6 1 4 7 9 9

7 < 9 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 6 5 4 1 2 7 9 2 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 6 5 4 7 1 2 3 9 3 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 6 5 4 7 1 2 3 9

< Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 6 V 5 4 7 1 2 3 9

< Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 P (σ) 6 V = 5 4 7 ←→ {{5, 1}, {8, 6, 4, 2}, {7, 3}, {9}} 1 2 3 9

< Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

σ = 586714923

8 Q(σ)= P (σ−1) 6 V = 5 3 9 ←→ {{5, 1}, {8, 6, 3, 2}, {9, 4}, {7}} 1 2 4 7

< Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue σ = 586714923

8 Q(σ)= P (σ−1) 6 V = 5 3 9 ←→ {{5, 1}, {8, 6, 3, 2}, {9, 4}, {7}} 1 2 4 7

< σ 7−→ (P (σ),Q(σ)) is not a surjection. Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

There is no permutation σ such that:

3 3 P (σ)= 1 2 Q(σ)= 1 2 Algebraic constructions on set partitions 22/03/2007 5 de 16

Combinatorial algorithm Robinson-Schensted analogue

Deﬁnition For σ, π ∈ S, we write σ ≡ π , if P (σ)= P (π). Then, σ and π belong to the same Bell class. Algebraic constructions on set partitions 22/03/2007 6 de 16

Hopf algebra on set partitions

Deﬁnition We consider the following elements of FQSym :

P F δ := X σ , P (σ)=δ

for every set partition δ. Algebraic constructions on set partitions 22/03/2007 7 de 16

Hopf algebra on set partitions Associative algebra

Theorem

The elements (Pδ)δ∈SP form a subalgebra of FQSym . Algebraic constructions on set partitions 22/03/2007 7 de 16

Hopf algebra on set partitions Associative algebra

Theorem

The elements (Pδ)δ∈SP form a subalgebra of FQSym .

12 ⋒ 21 = 1243 + 1423 + 4123 + 1432 + 4132 + 4312 Algebraic constructions on set partitions 22/03/2007 7 de 16

Hopf algebra on set partitions Associative algebra

Theorem

The elements (Pδ)δ∈SP form a subalgebra of FQSym .

12 ⋒ 21 = 1243 + 1423 + 4123 + 1432 + 4132 + 4312 Algebraic constructions on set partitions 22/03/2007 7 de 16

Hopf algebra on set partitions Associative algebra

Theorem

The elements (Pδ)δ∈SP form a subalgebra of FQSym .

12 ⋒ 21 = 1243 + 1423 + 4123 + 1432 + 4132 + 4312 Algebraic constructions on set partitions 22/03/2007 7 de 16

Hopf algebra on set partitions Associative algebra

Theorem

The elements (Pδ)δ∈SP form a subalgebra of FQSym .

1⋒(213+231) = (1324+1342)+(3124)+(3214+3241+3421)+(3142+3412)

↓ {1} × {21|3} = {1|32|4} + {31|2|4} + {321|4} + {31|42} Algebraic constructions on set partitions 22/03/2007 7 de 16

Hopf algebra on set partitions Associative algebra

Theorem

The elements (Pδ)δ∈SP form a subalgebra of FQSym .

1⋒(213+231) = (1324+1342)+(3124)+(3214+3241+3421)+(3142+3412) ↓ {1} × {21|3} = {1|32|4} + {31|2|4} + {321|4} + {31|42}

Proposition (Restriction to intervals)

Let σ, π ∈ S. If σ ≡ π, then Std(σ|I ) ≡ Std(π|I ). Algebraic constructions on set partitions 22/03/2007 8 de 16

Hopf algebra on set partitions Coalgebra

Theorem

The elements (Pδ)δ∈SP form a subcoalgebra of FQSym .

Proposition Let u, v, u′, v′ be words. If u ≡ v and u′ ≡ v′, then u · v ≡ u′ · v′. Algebraic constructions on set partitions 22/03/2007 8 de 16

Hopf algebra on set partitions Coalgebra

Theorem

The elements (Pδ)δ∈SP form a subcoalgebra of FQSym .

∆(¯ (4213+4231))=1⊗(213+231) +21⊗12+21⊗21+321⊗1+312⊗1

↓ ∆(¯ {421|3})= {1}⊗{21|3}+{21}⊗({1|2}+{21})+({321}+{31|2})⊗{1}

Proposition Let u,v,u′, v′ be words. If u ≡ v and u′ ≡ v′, then u · v ≡ u′ · v′. Algebraic constructions on set partitions 22/03/2007 8 de 16

Hopf algebra on set partitions Coalgebra

Theorem

The elements (Pδ)δ∈SP form a subcoalgebra of FQSym .

∆(¯ (4213+4231))=1⊗(213+231) +21⊗12+21⊗21+321⊗1+312⊗1

↓ ∆(¯ {421|3})= {1}⊗{21|3}+{21}⊗({1|2}+{21})+({321}+{31|2})⊗{1}

Proposition Let u,v,u′, v′ be words. If u ≡ v and u′ ≡ v′, then u · v ≡ u′ · v′. Algebraic constructions on set partitions 22/03/2007 9 de 16

Hopf algebra on set partitions

Theorem

The elements (Pδ)δ∈SP form a Hopf subalgebra of FQSym .

Proposition B is a bidendriform bialgebra. Hence, B is free, cofree and self-dual. Algebraic constructions on set partitions 22/03/2007 9 de 16

Hopf algebra on set partitions

Theorem

The elements (Pδ)δ∈SP form a Hopf subalgebra of FQSym .

Proposition B is a bidendriform bialgebra. Hence, B is free, cofree and self-dual. Algebraic constructions on set partitions 22/03/2007 10 de 16

Bell classes Canonical permutation

6 3 4 1 2 5

Column reading. Algebraic constructions on set partitions 22/03/2007 10 de 16

Bell classes Canonical permutation

6 3 4 1 2 5 31

Column reading. Algebraic constructions on set partitions 22/03/2007 10 de 16

Bell classes Canonical permutation

6 3 4 1 2 5 31 642

Column reading. Algebraic constructions on set partitions 22/03/2007 10 de 16

Bell classes Canonical permutation

6 3 4 1 2 5 31 642 5

Column reading. Algebraic constructions on set partitions 22/03/2007 10 de 16

Bell classes Canonical permutation 316425

361425 316452 6 3 4 1 2 5 364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 10 de 16

Bell classes Canonical permutation

Proposition In every Bell class, 1 the canonical element is the unique permutation of minimal length,

2 there is a unique element of maximal length. Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals

Deﬁnition Rewriting rule:

..b ... ac.. −→ ..b ... ca.. |{z}>b |{z}>b

Theorem ∗ ∀σ ∈ S, P (σ) −→ σ. Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals

Deﬁnition Rewriting rule:

..b ... ac.. −→ ..b ... ca.. |{z}>b |{z}>b

Theorem ∗ ∀σ ∈ S, P (σ) −→ σ. Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

361452 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

361452 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals 316425

361425 316452

364125 361452

364152

364512 Algebraic constructions on set partitions 22/03/2007 11 de 16

Bell classes Intervals

Theorem Bell classes are intervals of the permutohedron. Algebraic constructions on set partitions 22/03/2007 12 de 16

1234 Intervals on S4 1234

3 4 1 2 4 1 2 3 2 1 3 4 2134 1324 1243

2 4 3 1 3 4 1 2 4 1 2 3 2314 3124 2143 1342 1423

3 4 4 1 2 4 3 1 2 3 1 2 2341 3214 2413 3142 4123 1432

3241 2431 3412 4213 4132 3 2 4 3 1 4 1 2

3421 4231 4312 4 4 2 3 1 3 1 2

4321 4 3 2 1 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 5 4 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 5 4 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 5 4 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 1 < 2 < 4 < 5 5 4 1 2 3 7

5 4 6 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 5 4 1 2 3 7

5 6 4 6 5 4 1 2 3 7 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

∞ 6 5 4 1 2 3 7

5 6 4 6 5 4 1 2 3 7 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 5 4 1 2 3 7

5 6 4 6 5 4 1 2 3 7 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 3 alone in its column 5 4 7 greater than 4 1 2 3 7

5 6 4 6 5 4 1 2 3 7 1 2 3 7 Algebraic constructions on set partitions 22/03/2007 13 de 16

Bell order Cover relation

6 5 4 1 2 3 7

5 6 6 4 6 5 4 5 4 7 1 2 3 7 1 2 3 7 1 2 3 Algebraic constructions on set partitions 22/03/2007 14 de 16

Poset and Hopf algebra Intervals & Product

{31 | 2} | {1 | 32} = {31 | 2 | 4 | 65}

3 3 3 6 1 2 1 2 = 1 2 4 5 Algebraic constructions on set partitions 22/03/2007 14 de 16

Poset and Hopf algebra Intervals & Product

{31 | 2} | {1 | 32} = {31 | 2 | 4 | 65}

3 3 3 6 1 2 1 2 = 1 2 4 5 {31 | 2} ⊔ {1 | 32} = {431 | 652} 4 6 3 3 3 5 = 1 2 F 1 2 1 2 Algebraic constructions on set partitions 22/03/2007 14 de 16

Poset and Hopf algebra Intervals & Product

Theorem For every set partitions α, β,

P P P α × β = X δ . (α|β)≤δ≤(α⊔β) Algebraic constructions on set partitions 22/03/2007 14 de 16

Poset and Hopf algebra Intervals & Product {1 | 2 | 43}

{1 | 42 | 3}

{41 | 2 | 3}{1 | 432} = {1 | 2} × {21}

{41 | 32}

{431 | 2} Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863 <

1 7 9 5 2 4 8 6 V 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

1 7 9 ◦ 5 5 < 7 2 4 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

1 7 9 5 ◦ 2 4 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

1 7 ◦ 5 9 2 4 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 1 5 9 ◦ 2 4 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 1 5 9 2 ◦ 4 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 1 5 9 2 4◦ 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 1 5 9 ◦ 2 4 8 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 5 9 1 2 4 8 ◦ 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 5 9 1 2 4 8 ◦ 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 5 1 2 9 8 ◦ 4 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 1 5 9 8 ◦ 2 4 6 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 5 9 8 1 2 4 6 ◦ 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 5 9 8 1 2 4 6 ◦ 3 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 5 9 1 2 4 8 ◦ 3 6 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 9 1 5 4 8 ◦ 2 3 6 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 9 5 4 8 S(σ) = 1 2 3 6 Algebraic constructions on set partitions 22/03/2007 15 de 16

Scrolling Jeu de taquin-like

σ = 179524863

7 9 5 4 8 S(σ)=1 2 3 6 = P (σ) Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 4 4 4 3 1 2 3 +1 2 3 + 1 2

4 4 3 4 3 1 2 +1 2 3 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 1 4 1 2 4 1 4 3 3 +2 3 + 2

4 4 1 3 4 3 2 +1 2 3 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 1 4 1 2 4 1 4 3 3 +2 3 + 2

4 4 1 3 4 3 2 +1 2 3 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 1 1 2 1 ◦ +◦ 2 + 2

1 ◦ 2 +1 2 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

1 2 +1 2 + 1 2

1 2 +1 2 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 1 4 1 2 4 1 4 3 3 +2 3 + 2

4 4 1 3 4 3 2 +1 2 3 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 1 4 1 2 4 1 4 3 3 +2 3 + 2

4 4 1 3 4 3 2 +1 2 3 + 1 2 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

= 4 4 4 3 3 +◦ 3 +

4 4 ◦ 3 4 3 +◦ 3 + Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

4 4 4 3 +3 + 3

4 4 4 3 +3 + 3 Algebraic constructions on set partitions 22/03/2007 16 de 16

Scrolling and Hopf algebra

2 1 2 × 1

2 2 2 1 +1 + 1