On some properties of permutation tableaux Alexander Burstein Iowa - PowerPoint PPT Presentation

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Boundary Path Extend the 01/2 boundary to a SW lattice path ( n − k , 0) → (0 , k ) on n steps. Label the path steps 1 through n from NE to SW end. Label the rows and columns with the label of the corresponding step. Label each cell p with its row and column labels ( l r ( p ) , l c ( p )). Note that l c ( p ) > l r ( p ) for every cell p in the tableau. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Example Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Example 20 19 18 17 16 14 12 9 8 4 1 2 3 5 6 7 10 11 13 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Example 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths start at NW boundary Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths start at NW boundary salient points at 1s Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths start at NW boundary salient points at 1s 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths start at NW boundary salient points at 1s 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths start at NW boundary salient points at 1s 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations starting with a tableau T use SE paths start at NW boundary salient points at 1s 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = � � 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = � � 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 Φ is a bijection. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = � � 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 Φ is a bijection. How to find Φ − 1 ? Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = � � 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 Φ is a bijection. How to find Φ − 1 ? Steingr´ ımsson, Williams 2005: Reconstruction of T from π by columns from right to left. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = � � 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 Φ is a bijection. How to find Φ − 1 ? Steingr´ ımsson, Williams 2005: Reconstruction of T from π by columns from right to left. Reconstruction by rows from bottom to top is similar. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Row and Column Labeling Statistics on Tableaux and Permutations Tableaux and Permutations Essential 1s Open Problems Φ : Tableaux → Permutations π = Φ( T ) = � � 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 Φ is a bijection. How to find Φ − 1 ? Steingr´ ımsson, Williams 2005: Reconstruction of T from π by columns from right to left. Reconstruction by rows from bottom to top is similar. We will reconstruct T from π by columns from left to right (and by rows from top to bottom). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i 1 , . . . , i k be the row labels of 1s in the leftmost column. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i 1 , . . . , i k be the row labels of 1s in the leftmost column. Then π ( n ) < π ( i 1 ) < π ( i 2 ) < · · · < π ( i k ) = n ... Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i 1 , . . . , i k be the row labels of 1s in the leftmost column. Then π ( n ) < π ( i 1 ) < π ( i 2 ) < · · · < π ( i k ) = n ... ...and for j ∈ ( i r , i r +1 ), π ( j ) < π ( i r ). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i 1 , . . . , i k be the row labels of 1s in the leftmost column. Then π ( n ) < π ( i 1 ) < π ( i 2 ) < · · · < π ( i k ) = n ... ...and for j ∈ ( i r , i r +1 ), π ( j ) < π ( i r ). Hence, π ( i 1 ) , π ( i 2 ) , . . . , π ( i k ) = n are the LR-minima of the subsequence of π on letters greater than π ( n ). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Leftmost column We want to find the row labels of 1s in the leftmost column. Notice that any two SE paths in the tableau may intersect at most once – at their first common point. Let i 1 , . . . , i k be the row labels of 1s in the leftmost column. Then π ( n ) < π ( i 1 ) < π ( i 2 ) < · · · < π ( i k ) = n ... ...and for j ∈ ( i r , i r +1 ), π ( j ) < π ( i r ). Hence, π ( i 1 ) , π ( i 2 ) , . . . , π ( i k ) = n are the LR-minima of the subsequence of π on letters greater than π ( n ). If T ′ is T without the leftmost column, then Φ( T ) = Φ( T ′ )( i 1 i 2 . . . i k n ) . Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , c i contains b , Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , c i contains b , c j contains a < c , Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , c i contains b , c j contains a < c , a < b < c , Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , c i contains b , c j contains a < c , a < b < c , then c j also contains b . Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Column Decomposition The column decomposition of a permutation π is the (unique) representation of π as a product c n − k c n − k − 1 . . . c 1 of increasing cycles c i (1 ≤ i ≤ n − k ) such that maximal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , c i contains b , c j contains a < c , a < b < c , then c j also contains b . This condition is equivalent to the 1-hinge rule for tableaux. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 17 7 18 3 8 11 15 5 4 12 13 2 20 6 19 10 16 9 1 14 14 7 17 3 8 11 15 5 4 12 13 2 18 6 19 10 16 9 1 20 1 7 14 3 8 11 15 5 4 12 13 2 17 6 18 10 16 9 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 17 10 16 18 19 20 1 7 9 3 8 11 14 5 4 12 13 2 15 6 16 10 17 18 19 20 1 7 9 3 8 10 11 5 4 12 13 2 14 6 15 16 17 18 19 20 1 6 7 3 8 9 10 5 4 11 12 2 13 14 15 16 17 18 19 20 1 2 6 3 7 8 9 5 4 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 6 7 8 5 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 π = (3 4)(5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19) (1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition π = (3 4)(5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19)(1 3 13 20) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Column Decomposition π = (3 4)(5 6 7 8)(3 5 6 7 9)(2 3 5 6 7 10 11 12)(2 3 6 7 10 11 13 14)(6 7 13 15 16)(15 17)(3 7 13 15 18)(1 3 13 15 19)(1 3 13 20) 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Row Decomposition The row decomposition of a permutation π is the (unique) representation of π as a product c k c k − 1 . . . c 1 of decreasing cycles c i (1 ≤ i ≤ k ) such that minimal elements of c i ’s are distinct from one another and from other elements in c i ’s, and if i < j , c i contains b , c j contains c > a , c > b > a , then c j also contains b . This condition is also equivalent to the 1-hinge rule for tableaux. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Statistics on Tableaux and Permutations Essential 1s Open Problems Example of Row Decomposition π = (19 18 17 16 14 15)(20 19 18 16 14 13)(14 12 11)(14 12 10)(18 16 14 12 9 8 7)(16 14 12 9 8 6)(12 9 8 5)(20 19 18 14 12 9 4 3) (14 12 2)(20 19 1) 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Alignments and Crossings 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Alignments and Crossings 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Intersections: ee , nn , en ( e < n ) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Alignments and Crossings 20 19 18 17 16 14 12 9 8 4 • • 1 • • 2 • • • • • • • 3 • • • 5 • • • • • 6 • • • • • • 7 • • 10 • • 11 • • • • • 13 • • • • 15 Intersections: ee , nn , en ( e < n ) Non-intersections: ne ( n < e ) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Alignments and Crossings 20 19 18 17 16 14 12 9 8 4 • • 1 nn nn nn nn nn nn nn nn • • 2 ee ee en en en nn nn nn • • • • • • • 3 nn nn nn • • • 5 ee ee en en en en ne • • • • • 6 ee ee en en ne • • • • • • 7 ee ee nn ne • • 10 ee ee ee en en ne ne ne • • 11 ee ee ee en en ne ne ne • • • • • 13 en ne ne ne ne • • • • 15 ee ne ne ne ne ne Intersections: ee , nn , en ( e < n ) Non-intersections: ne ( n < e ) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Statistics on Permutation Tableaux Let π = Φ( T ). Define A EE ( π ) = |{ ( i , j ) | j < i ≤ π ( i ) < π ( j ) }| = # ee ( T ) A NN ( π ) = |{ ( i , j ) | π ( j ) < π ( i ) < i < j }| = # nn ( T ) A EN ( π ) = |{ ( i , j ) | j ≤ π ( j ) < π ( i ) < i }| = # en ( T ) A NE ( π ) = |{ ( i , j ) | π ( i ) < i < j ≤ π ( j ) }| = # ne ( T ) = #2s( T ) C EE ( π ) = |{ ( i , j ) | j < i ≤ π ( j ) < π ( i ) }| C NN ( π ) = |{ ( i , j ) | π ( i ) < π ( j ) < i < j }| It can be shown that C EE ( π ) + C NN ( π ) = #nontop 1s( T ) Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Steingr´ ımsson and Williams define a map Ψ : S n → S n that takes descents (des) and ascents (ndes) to weak excedances (wex) and deficiencies (nwex). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Steingr´ ımsson and Williams define a map Ψ : S n → S n that takes descents (des) and ascents (ndes) to weak excedances (wex) and deficiencies (nwex). They show that Ψ has the following properties. If π = Ψ( σ ), then des σ = wex π − 1 (31-2) σ = A EE ( π ) + A NN ( π ) � des σ � (21-3) σ + (3-21) σ − = A EN ( π ) 2 (2-31) σ = C EE ( π ) + C NN ( π ) � des σ � (1-32) σ + (32-1) σ − = A NE ( π ) 2 Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems This implies: A EN + A NN and C EE + C NN are equidistributed, A EN and A NE are equidistributed. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems This implies: A EN + A NN and C EE + C NN are equidistributed, A EN and A NE are equidistributed. To see the latter note that the map i ◦ r ◦ c (inverse of reversal of complement, or reflection across the antidiagonal of the permutation diagram) preserves wex , A EE , A NN , C EE , C NN , and exchanges A EN and A NE . Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems This implies: A EN + A NN and C EE + C NN are equidistributed, A EN and A NE are equidistributed. To see the latter note that the map i ◦ r ◦ c (inverse of reversal of complement, or reflection across the antidiagonal of the permutation diagram) preserves wex , A EE , A NN , C EE , C NN , and exchanges A EN and A NE . Question: Describe the equivalent (under Φ) of irc on tableaux directly. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Catalan tableaux Recall that | S n (2-31) | = | S n (2-3-1) | = C n , the n th Catalan number. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Catalan tableaux Recall that | S n (2-31) | = | S n (2-3-1) | = C n , the n th Catalan number. Hence, (SW, 2005) C n is the number of tableaux with no nontop 1s (i.e. with a single 1 per column). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Catalan tableaux Theorem If a tableau T has a single 1 per column, and π = Φ( T ) , then (the underlying sets of) the cycles of π form a noncrossing partition. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Catalan tableaux Theorem If a tableau T has a single 1 per column, and π = Φ( T ) , then (the underlying sets of) the cycles of π form a noncrossing partition. Proof. T has a single 1 per column, so no element of π may occur in more than one cycle of its row decomposition. Suppose π contains two cycles c i and c j and elements a > b > c > d such that a , c are in c i and b , d are in c j . Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Catalan tableaux Theorem If a tableau T has a single 1 per column, and π = Φ( T ) , then (the underlying sets of) the cycles of π form a noncrossing partition. Proof. T has a single 1 per column, so no element of π may occur in more than one cycle of its row decomposition. Suppose π contains two cycles c i and c j and elements a > b > c > d such that a , c are in c i and b , d are in c j . If i < j , then c j contains c . If i > j , then c i contains b . Neither is possible. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Catalan tableaux Theorem If a tableau T has a single 1 per column, and π = Φ( T ) , then (the underlying sets of) the cycles of π form a noncrossing partition. Proof. T has a single 1 per column, so no element of π may occur in more than one cycle of its row decomposition. Suppose π contains two cycles c i and c j and elements a > b > c > d such that a , c are in c i and b , d are in c j . If i < j , then c j contains c . If i > j , then c i contains b . Neither is possible. Thus, the c i ’s form a noncrossing partition. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Note that A EE ( T ) = A NN ( T ) = 0 for monotone T . Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Note that A EE ( T ) = A NN ( T ) = 0 for monotone T . Let π = Φ( T ) and let σ = Ψ − 1 ( π ). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Note that A EE ( T ) = A NN ( T ) = 0 for monotone T . Let π = Φ( T ) and let σ = Ψ − 1 ( π ). Then σ avoids 31-2 (i.e. 3-1-2). Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Note that A EE ( T ) = A NN ( T ) = 0 for monotone T . Let π = Φ( T ) and let σ = Ψ − 1 ( π ). Then σ avoids 31-2 (i.e. 3-1-2). Note that the subsequences of weak excedance values of π and deficiency values of π are increasing. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Note that A EE ( T ) = A NN ( T ) = 0 for monotone T . Let π = Φ( T ) and let σ = Ψ − 1 ( π ). Then σ avoids 31-2 (i.e. 3-1-2). Note that the subsequences of weak excedance values of π and deficiency values of π are increasing. Hence, π avoids 3-2-1. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Alignments and Crossings vs. Pattern Statistics Row and Column Decomposition Catalan tableaux Statistics on Tableaux and Permutations Monotone tableaux Essential 1s Open Problems Monotone tableaux A monotone tableau is one that has no 0 below or to the right of any 1. Note that A EE ( T ) = A NN ( T ) = 0 for monotone T . Let π = Φ( T ) and let σ = Ψ − 1 ( π ). Then σ avoids 31-2 (i.e. 3-1-2). Note that the subsequences of weak excedance values of π and deficiency values of π are increasing. Hence, π avoids 3-2-1. Theorem S n (31 - 2) Ψ → S n (3 - 2 - 1) Φ ← monotone tableaux of semiperimeter n. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Distribution Statistics on Tableaux and Permutations Bare Tableaux Essential 1s Open Problems Definition Essential 1s – Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Distribution Statistics on Tableaux and Permutations Bare Tableaux Essential 1s Open Problems Definition Essential 1s – leftmost 1s in their rows Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Distribution Statistics on Tableaux and Permutations Bare Tableaux Essential 1s Open Problems Definition Essential 1s – leftmost 1s in their rows or topmost 1s in their columns. Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Distribution Statistics on Tableaux and Permutations Bare Tableaux Essential 1s Open Problems Definition Essential 1s – leftmost 1s in their rows or topmost 1s in their columns. Doubly essential 1s – Alex Burstein On some properties of permutation tableaux

Outline Permutation Tableaux Row and Column Decomposition Distribution Statistics on Tableaux and Permutations Bare Tableaux Essential 1s Open Problems Definition Essential 1s – leftmost 1s in their rows or topmost 1s in their columns. Doubly essential 1s – leftmost 1s in their rows Alex Burstein On some properties of permutation tableaux