Relations between the shape of a permutation and the shape of the - PDF document

Relations between the shape of a permutation and the shape of the base poset derived from the corresponding Lehmer codes 2013 July 5 at Permutation Pattens 2013 Masaya Tomie Morioka University tomie@morioka-u.ac.jp

1. Lehmer Codes and Weak Bruhat Order 2. Denoncourt’s Work 3. Relations Between ω and M ω 4. Relations Between ∆( ω ) and M ω

1. Lehmer Codes and Weak Bruhat Order ω = ω (1) ω (2) · · · ω ( n ) ∈ S n

1. Lehmer Codes and Weak Bruhat Order ω = ω (1) ω (2) · · · ω ( n ) ∈ S n c ( ω ) = ( c 1 ( ω ) , c 2 ( ω ) , · · · , c n ( ω )) Lehmer Code

1. Lehmer Codes and Weak Bruhat Order ω = ω (1) ω (2) · · · ω ( n ) ∈ S n c ( ω ) = ( c 1 ( ω ) , c 2 ( ω ) , · · · , c n ( ω )) Lehmer Code ⇐ ⇒ c 1 ( ω ) : the number of i ( ≥ 1) such that ω (1) > ω ( i ) c 2 ( ω ) : the number of i ( ≥ 2) such that ω (2) > ω ( i ) . . . c n ( ω ) : the number of i ( ≥ n ) such that ω ( n ) > ω ( i )

Example ω = 423615

Example ω = 423615 • c 1 = 3 4 23 6 1 5

Example ω = 423615 • c 1 = 3 4 23 6 1 5 • c 2 = 1 4236 1 5

Example ω = 423615 • c 1 = 3 4 23 6 1 5 • c 2 = 1 4236 1 5 • c 3 = 1 4236 1 5

Example ω = 423615 • c 1 = 3 4 23 6 1 5 • c 2 = 1 4236 1 5 • c 3 = 1 4236 1 5 • c 4 = 2 4236 15 • c 5 = 0 423615 • c 6 = 0 423615 c ( ω ) = (3 , 1 , 1 , 2 , 0 , 0)

Example ω = 423615 • c 1 = 3 4 23 6 1 5 • c 2 = 1 4236 1 5 • c 3 = 1 4236 1 5 • c 4 = 2 4236 15 • c 5 = 0 423615 • c 6 = 0 423615 c ( ω ) = (3 , 1 , 1 , 2 , 0 , 0) Endow a product order on Lehmer Codes

ω = ω (1) ω (2) · · · ω ( n ) ∈ S n Inv( ω ) = { ( i, j ) | i < j, ω ( i ) > ω ( j ) }

ω = ω (1) ω (2) · · · ω ( n ) ∈ S n Inv( ω ) = { ( i, j ) | i < j, ω ( i ) > ω ( j ) } Definition (Weak Bruhat Order) For ω, τ ∈ S n ω ≤ τ ⇐ ⇒ Inv( ω ) ⊂ Inv( τ )

ω = ω (1) ω (2) · · · ω ( n ) ∈ S n Inv( ω ) = { ( i, j ) | i < j, ω ( i ) > ω ( j ) } Definition (Weak Bruhat Order) For ω, τ ∈ S n ω ≤ τ ⇐ ⇒ Inv( ω ) ⊂ Inv( τ ) 321 312 231 132 213 123

3. Denoncourt’s Work H. Denoncourt, A refinement of weak order intervals into dis- tributive lattices, arXiv:1102.2689.

3. Denoncourt’s Work H. Denoncourt, A refinement of weak order intervals into dis- tributive lattices, arXiv:1102.2689. ω ∈ S n c ( ω ) : The corresponding Lehmer code

3. Denoncourt’s Work H. Denoncourt, A refinement of weak order intervals into dis- tributive lattices, arXiv:1102.2689. ω ∈ S n c ( ω ) : The corresponding Lehmer code Λ ω := [ e, ω ] Interval in Weak Bruhat Order

3. Denoncourt’s Work H. Denoncourt, A refinement of weak order intervals into dis- tributive lattices, arXiv:1102.2689. ω ∈ S n c ( ω ) : The corresponding Lehmer code Λ ω := [ e, ω ] Interval in Weak Bruhat Order Theorem (Denoncourt 2011) 1. c is an order preserving map 2. c (Λ ω ) is a distributive lattice in N n

Definition (Denoncourt 2011) For ω ∈ S n , i with c i ( ω ) ̸ = 0 and 1 ≤ x ≤ c i ( ω ) , define m i,x ( ω ) ∈ N n s.t.

Definition (Denoncourt 2011) For ω ∈ S n , i with c i ( ω ) ̸ = 0 and 1 ≤ x ≤ c i ( ω ) , define m i,x ( ω ) ∈ N n s.t. 1. j -th ( j < i ) coordinate of m i,x ( ω ) is 0

Definition (Denoncourt 2011) For ω ∈ S n , i with c i ( ω ) ̸ = 0 and 1 ≤ x ≤ c i ( ω ) , define m i,x ( ω ) ∈ N n s.t. 1. j -th ( j < i ) coordinate of m i,x ( ω ) is 0 2. i -th coordinate of m i,x ( ω ) is x

Definition (Denoncourt 2011) For ω ∈ S n , i with c i ( ω ) ̸ = 0 and 1 ≤ x ≤ c i ( ω ) , define m i,x ( ω ) ∈ N n s.t. 1. j -th ( j < i ) coordinate of m i,x ( ω ) is 0 2. i -th coordinate of m i,x ( ω ) is x 3. For j > i with ω ( i ) > ω ( j ) , j -th coordinate of m i,x ( ω ) is 0

Definition (Denoncourt 2011) For ω ∈ S n , i with c i ( ω ) ̸ = 0 and 1 ≤ x ≤ c i ( ω ) , define m i,x ( ω ) ∈ N n s.t. 1. j -th ( j < i ) coordinate of m i,x ( ω ) is 0 2. i -th coordinate of m i,x ( ω ) is x 3. For j > i with ω ( i ) > ω ( j ) , j -th coordinate of m i,x ( ω ) is 0 4. For j > i with ω ( i ) < ω ( j ) , j -th coordinate of m i,x ( ω ) is max { 0 , x − c i,j ( x ) } where c i,j is the number of i ≤ k ≤ j s.t. ω ( i ) > ω ( k ) .

We denote the join-irreducibles by M ω .

We denote the join-irreducibles by M ω . Theorem (Denoncourt 2011) The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) }

We denote the join-irreducibles by M ω . Theorem (Denoncourt 2011) The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) } Example ω = 5371642

We denote the join-irreducibles by M ω . Theorem (Denoncourt 2011) The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) } Example ω = 5371642 c ( ω ) = (4 , 2 , 4 , 0 , 2 , 1 , 0)

We denote the join-irreducibles by M ω . Theorem (Denoncourt 2011) The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) } Example ω = 5371642 c ( ω ) = (4 , 2 , 4 , 0 , 2 , 1 , 0) m 1 , 4 = (4 , , , , , , )

We denote the join-irreducibles by M ω . Theorem (Denoncourt 2011) The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) } Example ω = 5371642 c ( ω ) = (4 , 2 , 4 , 0 , 2 , 1 , 0) m 1 , 4 = (4 , , , , , , ) m 1 , 4 = (4 , 0 , , 0 , , 0 , 0 ) 5 3 7 1 6 42

We denote the join-irreducibles by M ω . Theorem (Denoncourt 2011) The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) } Example ω = 5371642 c ( ω ) = (4 , 2 , 4 , 0 , 2 , 1 , 0) m 1 , 4 = (4 , , , , , , ) m 1 , 4 = (4 , 0 , , 0 , , 0 , 0 ) 5 3 7 1 6 42 m 1 , 4 = (4 , 0 , 3 , 0 , , 0 , 0) 53 7 1642

We denote the join-irreducibles by M ω . Theorem The join-irreducibles of c (Λ ω ) are M ω = { m i,x ∈ N n | c i ( ω ) ̸ = 0 , 1 ≤ x ≤ c i ( ω ) } Example ω = 5371642 c ( ω ) = (4 , 2 , 4 , 0 , 2 , 1 , 0) m 1 , 4 = (4 , , , , , , ) m 1 , 4 = (4 , 0 , , 0 , , 0 , 0 ) 5 3 7 1 6 42 m 1 , 4 = (4 , 0 , 3 , 0 , , 0 , 0) 53 7 1642 m 1 , 4 = (4 , 0 , 3 , 0 , 2 , 0 , 0) 5371 6 42

m 1 , 4 = (4 , 0 , 3 , 0 , 2 , 0 , 0) m 1 , 3 = (3 , 0 , 2 , 0 , 1 , 0 , 0) m 1 , 2 = (2 , 0 , 1 , 0 , 0 , 0 , 0) m 1 , 1 = (1 , 0 , 0 , 0 , 0 , 0 , 0) m 2 , 2 = (0 , 2 , 2 , 0 , 1 , 1 , 0) m 2 , 1 = (0 , 1 , 1 , 0 , 0 , 0 , 0) m 3 , 4 = (0 , 0 , 4 , 0 , 0 , 0 , 0) m 3 , 3 = (0 , 0 , 3 , 0 , 0 , 0 , 0) m 3 , 2 = (0 , 0 , 2 , 0 , 0 , 0 , 0) m 3 , 1 = (0 , 0 , 1 , 0 , 0 , 0 , 0) m 5 , 2 = (0 , 0 , 0 , 0 , 2 , 0 , 0) m 5 , 1 = (0 , 0 , 0 , 0 , 1 , 0 , 0) m 6 , 1 = (0 , 0 , 0 , 0 , 0 , 1 , 0)

m 14 m 34 m 13 m 33 m 12 m 22 m 32 m 52 m 11 m 21 m 51 m 31 m 61

3. Relations Between ω and M ω

3. Relations Between ω and M ω Lemma ω is a 231-avoiding permutation = ⇒ M ω is disjoint union of chains.

3. Relations Between ω and M ω Lemma ω is a 231-avoiding permutation = ⇒ M ω is disjoint union of chains. M 4231 is disjoint union of length 2 and 1 chains.

3. Relations Between ω and M ω Lemma ω is a 231-avoiding permutation = ⇒ M ω is disjoint union of chains. M 4231 is disjoint union of length 2 and 1 chains. Definition P, Q Posets P is called to be Q free poset iff there are no subposets R ⊂ P s.t. R ≃ Q .

3. Relations Between ω and M ω Lemma ω is a 231-avoiding permutation = ⇒ M ω is disjoint union of chains. M 4231 is disjoint union of length 2 and 1 chains. Definition P, Q Posets P is called to be Q free poset iff there are no subposets R ⊂ P s.t. R ≃ Q . 1. (2 + 2)–(3 + 1)-free poset is enumerated by Catalan number. 2. A (2 + 2)-free poset is a interval order.

3. Relations Between ω and M ω Lemma ω is a 231-avoiding permutation = ⇒ M ω is disjoint union of chains. M 4231 is disjoint union of length 2 and 1 chains. Definition P, Q Posets P is called to be Q free poset iff there are no subposets R ⊂ P s.t. R ≃ Q . 1. (2 + 2)–(3 + 1)-free poset is enumerated by Catalan number. 2. A (2 + 2)-free poset is a interval order. A poset P is B 2 -free iff P has no 4 elements isomorphic to Boolean algebra of rank 2.

Theorem (T. 2011) ω is a 3412-3421-avoiding permutation ⇐ ⇒ M ω is a B 2 free poset. M. Tomie, A relation between the shape of a permuta- tion and the shape of the base poset derived from the Lehmer codes, arXiv:1111.3094.

Theorem (T. 2011) ω is a 3412-3421-avoiding permutation ⇐ ⇒ M ω is a B 2 free poset. M. Tomie, A relation between the shape of a permuta- tion and the shape of the base poset derived from the Lehmer codes, arXiv:1111.3094. 3412-3421-avoiding permutation is called Schr¨ oder permutation

Theorem (T. 2011) ω is a 3412-3421-avoiding permutation ⇐ ⇒ M ω is a B 2 free poset. M. Tomie, A relation between the shape of a permuta- tion and the shape of the base poset derived from the Lehmer codes, arXiv:1111.3094. 3412-3421-avoiding permutation is called Schr¨ oder permutation Example ω = 315462 m 12 m 32 m 11 m 41 m 51 m 31

Consider the number of components of M ω