Corners in TreeLike Tableaux Pawe l Hitczenko & Amanda Lohss - - PowerPoint PPT Presentation

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Corners in Tree–Like Tableaux

Pawe l Hitczenko & Amanda Lohss September 27, 2016

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Tree–Like Tableaux Corners in Tree–Like Tableaux Occupied Corners in Tree–Like Tableaux Diagonal Boxes in Symmetric Tree–Like Tableaux

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Tree–Like Tableaux (Aval, Boussicault, Nadeau 2011)

• Figure: A tree–like

tableaux of size 13. There are n! tableaux

• f size n.

Definition A tree-like tableaux of size n is a Ferrers diagrams of half-perimeter n + 1 such that,

1 The box in the first column and first

row is pointed.

2 Either all boxes to the left of a

pointed box is empty or all boxes above are empty.

3 Every row and every column contains

at least one point.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Symmetric Tableaux

Insertion Procedure

Definition: The ”special point” is the right-most point among those that occur at the bottom of a column. Add a pointed column for each north step and a pointed row for each west step. If the step is below the special point, add a ribbon.

• •
• •
• •
• Pawe

l Hitczenko and Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Natural Tree Structure

• •
• •
• There is a bijection between tree–like tableaux and permutations

which “preserves trees”.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

The PASEP

Tree-like tableaux provide a combinatorial formula for the partially asymmetric simple exclusion process (PASEP), an important particle model with applications in physics, biology and biochemistry.

α u q β

Figure: An example of the PASEP as defined by a Markov chain of size 8.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Connection with the PASEP

• • ◦ ◦ • • • ◦ ◦ • • ◦

Figure: A tree–like tableau and its associated state of the PASEP as represented by a Markov chain of size 12. Tree-like tableaux provide a combinatorial formula for the PASEP.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Main Objective: Laborde Zubieta 2015

Conjecture: The number of corners in tree–like tableaux of size n is n! × n+4

6 .

Conjecture: The number of corners in symmetric tree–like tableaux

• f size 2n + 1 is 2n × n! × 4n+13

12

.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Permutation Tableaux

1 0 1 1 1 1 1 1 1 1

Figure: A permutation tableaux of size 12. Definition A permutation tableaux of size n is a Ferrers diagram of half-perimeter n such that

1

There is at least one 1 in every column.

2

There is no 0 with a 1 above it and a 1 to the left of it simultaneously.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

The Bijection: (Aval, Boussicault, Nadeau 2011)

0 0 1 1 1 1 ← →

• •
• Figure: An example of the bijection

c(Tn) = c(Pn) + |{P ∈ Pn : Mn(P) = S}| = c(Pn) + (n − 1)! Theorem (Hitczenko, L.) For permutation tableaux of size n, EnCn = n + 4 6 − 1 n .

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Symmetric Tree–Like Tableaux

• •
• Figure: A symmetric tree–like tableaux of size 9. There are 2n · n!

tableaux of size n.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Type-B Permutation Tableaux

1 0 0 1 1 0 1

Figure: A type-B permutation tableaux

• f size 6.

Definition A type-B permutation tableaux of size n is a shifted Ferrers diagram of half-perimeter n such that,

1

The rules of permutation tableaux are satisfied.

2

If there is a 0 on the diagonal, it is a 0-row.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

The Bijection: (Aval, Boussicault, Nadeau 2011)

1 1 1 0 0 ← →

• •

Figure: An example of the bijection between type–B permutation tableaux and symmetric tree–like tableaux.

1

Add a column and a root point then point unrestricted rows.

2

Replace all 0R’s with points (except on 0-rows).

3

Replace all non-diagonal 1T’s with points.

4

Make symmetric.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Symmetric Tableaux

The Bijection

1 1 0 1

• •
• •

← →

Figure: Transformation of the Shape

Key Relationship: c(T sym

2n+1)

= 2c(Bn) + 2|{B ∈ Bn : Mn(B) = S}| + |{B ∈ Bn : M1(B) = W }| = c(Bn) + 2n(n − 1)! + 2n−1n!,

Pawe l Hitczenko and Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Technique for proof: Corteel and Hitczenko 2007

If Un−1 is the number of unrestricted rows, then there are 2Un−1+1 extensions. P(Un = Un−1 + 1|Fn−1) = 2 2Un−1+1 = 1 2Un−1 . P(Un = k|Fn−1) = 1 2Un−1+1 Un−1 k − 1

• +

Un−1 k − 1

• =

1 2Un−1 Un−1 k − 1

• .

Therefore, L(Un|Fn−1) = 1 + Bin(Un−1, 1/2).

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Relationship Between Measures

For all B ∈ Bn−1, Pn(B) = 2Un−1(B)+1 |Bn| = 2Un−1(B)+1 |Bn−1| |Bn| Pn−1(B) Therefore, for any random variable X on Bn−1, En(X) = 2|Bn−1| |Bn| En−1(2Un−1(Bn−1)X)

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Simple Illustration

Proposition (known but new proof: Hitczenko, L.) For all n ≥ 0, |Bn| = 2nn!. Proof can be deduced using the following: L(Un|Fn−1) = 1 + Bin(Un−1, 1/2). En(X) = 2|Bn−1| |Bn| En−1(2Un−1(Bn−1)X) E

• aBin(m)

= a + 1 2

• Pawe

l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Corners in Type-B Permutation Tableaux

Theorem (Hitczenko, L.)

For type-B permutation tableaux of size n we have EnCn = 4n + 7 24 − 1 2n.

Proof. En n−1

• k=1

IMk=S,Mk+1=W

• =

n−1

• k=1

En

• IMk=S,Mk+1=W
• Pawe

l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Occupied Corners: Laborde Zubieta 2015

The generating polynomial Pn(x) :=

T∈Tn

xoc(T) satisfies the following recurrence:

• P

′

n(x) = nPn−1(x) + 2(1 − x)P

′

n−1(x)

P0(x) = 1 Theorem (Hitczenko, L.) As n → ∞, the limiting distribution of the number of occupied corners in a random tree–like tableau of size n is Pois(1).

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Proof.

Theorem (Hitczenko, L.) Let Pn(x) = m

k=0 an,kxk satisfy,

P

′

n(x) = fn(x)Pn−1(x) + gn(x)P

′

n−1(x)

where gn(1) = 0, g ′

n(x) = gn = o(fn), f ′ n (x) = 0 and

fn Pn−1(1) Pn(1) → c > 0, as n → ∞. If a sequence of random variables Xn is defined by P(Xn = k) = an,k Pn(1) = an,k

• j an,j ,

then Xn

d

→ Pois(c) as n → ∞.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Diagonal Boxes: Aval, Boussicault, Nadeau 2011

• •
• Figure: A symmetric tree–like tableaux with three diagonal boxes.

If B(n, k) is the number of symmetric tree–like tableaux of size 2n + 1 with k diagonal cells, then Bn(x) = n+1

k=1 B(n, k)xk satisfies the following recurrence,

• Bn(x) = nx(x + 1)Bn−1(x) + x(1 − x2)B′

n−1(x),

B0(x) = x.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Theorem (Hitczenko, L.) Let Dn be the number of diagonal boxes in a random symmetric tree–like tableau of size 2n + 1. As n → ∞, Dn − 3(n + 1)/4

• 7(n + 1)/48

d

− → N(0, 1).

• Proof. Notice that

P(Dn = k) = B(n, k)

• k≥0 B(n, k) = B(n, k)

Bn(1) , and then the conclusion will follow if the variance of Dn → ∞ as n → ∞ and Bn(x) has non-positive real roots.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Proposition (Hitczenko, L.) The variance of the number of diagonal cells in a random symmetric tree–like tableaux of size 2n + 1 is, var(Dn) = 7(n + 1) 48 .

• Proof. Straightforward computation of

var(Dn) = E(Dn)2 − E2Dn + EDn.

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Proposition (Hitczenko, L.) For all n ≥ 0, the polynomial Bn(x) a) has degree n + 1 with all coefficients non-negative, and b) all roots real and in the interval [−1, 0].

• Proof. Induction and Rolle’s Theorem.

Recall:

• Bn+1(x) = (n + 1)x(x + 1)Bn(x) + x(1 − x2)B′

n(x),

B0(x) = x. Key Step: Bn+1(x) = x(1 − x2) Kn(x) d dx [Kn(x)Bn(x)]

Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Future Directions

1 Asymptotic distribution of corners Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Future Directions

1 Asymptotic distribution of corners 2 Type-B permutation tableaux Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux

Tree–Like Tableaux Corners Occupied Corners Diagonal Boxes

Future Directions

1 Asymptotic distribution of corners 2 Type-B permutation tableaux 3 Problems significant to PASEP Pawe l Hitczenko & Amanda Lohss Corners in Tree–Like Tableaux