Some 1 n generalized grid classes are context-free Robert Brignall - PowerPoint PPT Presentation

Some 1 × n generalized grid classes are context-free Robert Brignall Jakub Sliaˇ can Permutation Patterns 2018 1 / 28

View permutations as drawings 9 8 7 6 ← → 635814972 5 4 3 2 1 1 2 3 4 5 6 7 8 9 2 / 28

Enumerating permutation classes Class Collection of permutations closed under containment (if π ∈ C , then all subpermutations σ ⊂ π are also in C ). Enumeration Determining the number of permutations of each length in C . 3 / 28

Context-free class Definition A class C is context-free if it coincides with the first component of the system of equations  S 1 = f 1 ( Z , S 1 , . . . , S r )    . . .   S r = f r ( Z , S 1 , . . . , S r )  where f i are constructors only involving +, × , and E = ∅ . 4 / 28

Context-free class: example S ⊖ S S = Z + + S ⊕ S S = Z + S ⊕ S + S ⊖ S S S ⊖ = Z + S ⊖ = Z + S ⊕ S S ⊕ S ⊕ = Z + S ⊖ S , S ⊖ S ⊕ = Z + S 5 / 28

Context-free classes are nice Many things are context-free, e.g. finitely many simples = ⇒ context-free Shades of niceness rational ⊂ algebraic ⊂ D -finite ⊂ D -algebraic ⊂ power series Theorem (Chomsky-Sch¨ utzenberger) A combinatorial class C that is context-free admits an algebraic generating function. 6 / 28

Grid classes Definition Permutation grid class is a permutation class. It consists of permutations that can be chopped up by vertical and horizontal lines into sub-permutations belonging to designated classes. 7 / 28

Grid classes Definition Permutation grid class is a permutation class. It consists of permutations that can be chopped up by vertical and horizontal lines into sub-permutations belonging to designated classes. � Av (12) � Av (21) belongs to . Av (12) Av (21)

Grid classes Definition Permutation grid class is a permutation class. It consists of permutations that can be chopped up by vertical and horizontal lines into sub-permutations belonging to designated classes. � Av (12) � Av (21) belongs to . Av (12) Av (21) 7 / 28

Example: where the trouble lies Av (321) Av (12) as witnessed by the middle two 2615743 is in partitions. No! Yes! Yes! No! 8 / 28

We can’t enumerate this C 11 C 12 C 13 C 1 m C 21 C 22 C 23 C 2 m . . . C 31 C 32 C 33 C 3 m . ... . . C n 1 C n 2 C n 3 C nm Even if C ij are permutation classes that we CAN enumerate 9 / 28

. . . or this M M M M M M M C M M . . . M M M M M . ... . . M M M M M monotone classes, C non-monotone class 10 / 28

. . . actually, not even this M M M M M M M M M M . . . M M M M M . ... . . M M M M M monotone classes But! we know their growth rates = (spectral radius) 2 of the row-column graph [Bev15a]. 11 / 28

. . . also . . . these have rational generating functions [AAB + 13]   M M M M M     M M M M M       . . .   M M M M M       Geom             .  ...  .   .     M M M M   12 / 28

. . . and . . . generating functions conjectured for monotone increasing strips [Bev15b] . . . 13 / 28

Today M 1 · · · M k M k +1 · · · M k + l C Theorem Let C be a context-free permutation class that admits a combinatorial specification which tracks both the right-most and the left-most points. Let M i be a sequence of n − 1 monotone permutation classes. Then M 1 | . . . |M k |C|M k +1 | . . . |M k + ℓ is a context-free permutation class that admits an algebraic generating function. 14 / 28

Leftmost gridlines Griddable → gridded Convention: Let π be a permutation from C 1 |C 2 . The gridline in π is chosen to be the left-most possible. I.e. if it was any further left, the sub-permutation to the right of it would not belong to the designated class C 2 . 15 / 28

Leftmost gridlines: example C| Av (21) 16 / 28

Gaps associated with points x y The gap associated with x is the space on the RHS below x and above the next point below it on the LHS. 17 / 28

What we want to do: example Enumerate Av (21 | 21 | 21). Append cells from left to right. 1. Start with a single increasing sequence on the LHS. 2. Now append stuff on the RHS. 3. Finally, append the third cell. 18 / 28

Tracking the rightmost point The rightmost point of C is critical. So pick the combinatorial specification of C that tracks the rightmost point. 19 / 28

Tracking the rightmost point The rightmost point of C is critical. So pick the combinatorial specification of C that tracks the rightmost point. S ∗ S ⊖ S ∗ = Z ∗ + + S ⊕ S ∗ S ⊖ S S ∗ = Z ∗ + S ⊕ S ∗ + S ∗ S ⊖ S = Z + + S ⊕ S S = Z + S ⊕ S + SS ⊖ S ⊖ = Z + S ⊕ S S S ⊖ = Z + S ⊕ = Z + SS ⊖ . S ⊕ S ⊖ S ⊕ = Z + S 19 / 28

Operators Consider Ω 1 , an operator that appends a single point on the right of a class T m = X 1 . . . X m (bottom to top). Ω 1 ( Z ) = Z ∗ Z Ω 1 ( Z ∗ ) = Z ∗ Z � Ω 1 ( X ∗ 1 )Ω 0 ( X 2 · · · X m ) if k = 1 Ω 1 ( T m ) = Ω 1 ( X 1 )Ω 0 ( X 2 · · · X m ) + Ω 0 ( X 1 )Ω 1 ( X 2 · · · X m ) , if k > 1. Z Z / Z ∗ Ω 1 Z ∗ 20 / 28

The beast operator Ω 11 is the most involved operator – placing a sequence on the RHS with designated bottom and top point. Z Z ∗ Z / Z ∗ Ω 11 M E Z 21 / 28

All operators We need the following information captured when appending sequences on the RHS. ◮ Ω 0 : Nothing appended on the RHS. ◮ Ω 1 : Single point appended on the RHS (leftmost & rightmost coincide) ◮ Ω ∞ : Possibly empty sequence by itself. ◮ Ω 10 : Point followed by a (possibly empty) sequence above. ◮ Ω 01 : Point preceded by a (possibly empty) sequence below. ◮ Ω 11 : Point followed by a (possibly empty) sequence followed by another point. 22 / 28

Apply Ω 11 to a class C = X 1 X 2 X ∗ 3 X 4 p p p X 4 X 4 X 4 X ∗ X ∗ X ∗ 3 3 3 X 2 X 2 X 2 X 1 X 1 X 1 Ω 11 ( X 1 ) X 2 X 3 X 4 Z X 1 Ω 11 ( X 2 ) X 3 X 4 Z X 1 X 2 Ω 11 ( X ∗ 3 ) X 4 Z p p p X 4 X 4 X 4 X ∗ X ∗ X ∗ 3 3 3 X 2 X 2 X 2 X 1 X 1 X 1 Ω 10 ( X 1 )Ω ∞ ( X 2 )Ω 01 ( X ∗ 3 ) X 4 Z Ω 10 ( X 1 )Ω ∞ ( X 2 X ∗ 3 )Ω 01 ( X 4 ) Z Ω 10 ( X 1 )Ω 01 ( X 2 ) X 3 X 4 Z p p p X 4 X 4 X 4 X ∗ X ∗ X ∗ 3 3 3 X 2 X 2 X 2 X 1 X 1 X 1 Ω 10 ( X 1 )Ω ∞ ( X 2 X ∗ 3 X 4 )Ω 01 ( Z ) X 1 Ω 10 ( X 2 )Ω 01 ( X ∗ 3 ) X 4 Z X 1 Ω 10 ( X 2 )Ω ∞ ( X ∗ 3 )Ω 01 ( X 4 ) Z 23 / 28

Apply Ω 11 to a class C = X 1 X 2 X ∗ 3 X 4 p p p X 4 X 4 X 4 X ∗ X ∗ X ∗ 3 3 3 X 2 X 2 X 2 X 1 X 1 X 1 X 1 Ω 10 ( X 2 )Ω ∞ ( X ∗ 3 X 4 )Ω 01 ( Z ) X 1 X 2 Ω 10 ( X ∗ 3 )Ω 01 ( X 4 ) Z X 1 X 2 Ω 10 ( X ∗ 3 )Ω ∞ ( X 4 )Ω 01 ( Z ) 24 / 28

Appending a monotone decreasing class v C 4 C ∗ x 3 C 2 C 1 q y Θ : ∗ �→ ∗ , q �→ p y p H 4 H 3 x H ∗ 2 H 1 v 25 / 28

Appending on the left z p C 4 C ∗ a 3 C ◦ b 2 C 1 w Φ : ∗ �→ ◦ z p C 4 C ∗ a 3 C ◦ b 2 C 1 w 26 / 28

Putting it all together Consider C| Av (21). ∗ ) + Ω 11 ( C ∗ ) F = E + M + Ω 1 ( C Z ◮ Either empty, or non-empty increasing, or non-empty C next to non-empty Av (21). ◮ Need phantom points, hence C . ◮ Need to track rightmost points only, so C ∗ . ◮ Need to remove the phantom point after we’re done, hence 1 / Z in the last term. In general more complicated, but same ideas. 27 / 28

Things to notice ◮ algorithmic approach → can be automated ◮ it’s constructive: can enumerate (provide g.f. for) every such 1 × n grid class ◮ rational? D-finite? ◮ n × m acyclic grid classes? ◮ etc. 28 / 28

M. H. Albert, M. D. Atkinson, M. Bouvel, N. Ruˇ skuc, and V. Vatter. Geometric grid classes of permutations. Transactions of the American Mathematical Society , 365(11):5859–5881, 2013. D. I. Bevan. Growth rates of permutation grid classes, tours on graphs, and the spectral radius. Transactions of the American Mathematical Society , 367(8):5863–5889, 2015. D. I. Bevan. On the growth of permutation classes . PhD thesis, The Open University, 2015. 28 / 28