Automated Enumeration of Pattern Avoiding Permutations

Transcription

1 of Pattern Valparaiso University Pacific Workshop on Pattern March 31, 2011

2 Goal Find algorithmic ways to study S n (Q). Subgoal Find enumeration techniques that apply to a wide variety of pattern sets. Why? Produce exact enumeration results. Generate data to form further conjectures.

3 Theorem (Albert and Atkinson 2005) If S n (Q) contains finitely many simple permutations, then has an algebraic generating function.

4 Associates to each permutation a word describing how that permutation evolved. If the insertion encoding for S n (Q) forms a regular language then S n (Q) has a rational generating function that can be routinely computed. Theorem (Albert, Linton, and Ruškuc 2005) The insertion encoding for S n (Q) forms a regular language if and only if S n (Q) contains only finitely many vertical alternations.

5 Generating Tree Example

6 Generating Tree Example {123, 231}

7 Generating Tree Example {123, 231}

8 Generating Tree Example {123, 231}

9 Generating Tree Example {123, 231}

10 Generating Tree Example {123, 231} 1 *1*

11 Generating Tree Example {123, 231} 1 *1*

12 Generating Tree Example {123, 231} 1 *1* *1*2

13 Generating Tree Example {123, 231} 1 *1* *1*2

14 Generating Tree Example {123, 231} 1 *1* *1*2 4312

15 Generating Tree Example {123, 231} 1 *1* *1*2 *312

16 Rules: 1 1, , Generating Tree Example {123, 231} 1 *1* *1*2 *312

17 Generating Tree Example {123, 231} Rules: 1 1, , Transfer Matrix

18 Utility of Focuses on placement of smallest letters in a permutation. Easy to compute S n (Q) from transfer matrix for specific n. Easy to compute generating function from transfer matrix. Can be completely automated by computer. Theorem (Vatter 2006) S n (Q) corresponds to a finitely labeled generating tree if and only if Q contains both a child of an increasing permutation and a child of a decreasing permutation.

19 Scheme Definition/Notation Definition (informal) An enumeration scheme is an encoding for a family of recurrence relations enumerating members of a family of sets. Definition: prefix pattern S n (Q)[p] = {π S n (Q) π 1 π p p} Note S n (Q) = S n (Q)[1] = S n (Q)[12] S n (Q)[21] =

20 Scheme Notation Definition: prefix pattern with specified letters Note S n (Q)[p; w] = {π S n (Q)[p] π 1 π p = w} Sn (Q)[1] = n Sn (Q)[1; i] i=1

21 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[21]...

22 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[21]...

23 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[21]... So Sn ({123})[21; ij] = Sn 1 ({123})[1; j]

24 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[12]...

25 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[12]...

26 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[12]...

27 Scheme Example {123} S n ({123}) = S n ({123})[1] = S n ({123})[12] S n ({123})[21] Consider a member of S n ({123})[12]... So S n ({123})[12; ij] { 0 j < n = S n 1 ({123})[1; i] j = n

28 Scheme Example {123} Summary: Sn ({123}) = S n ({123})[1] = Sn ({123})[1; i] = n S n ({123})[1; i] i=1 i 1 S n ({123})[21; ij] j=1 + n j=i+1 S n ({123})[12; ij] Sn ({123})[21; ij] = Sn 1 ({123})[1; j] S n ({123})[12; ij] { 0 j < n = S n 1 ({123})[1; i] j = n

29 Utility of Focus on initial few letters in a permutation. Work for many more pattern sets (including singletons). Can be completely automated by computer. Can be transformed into a functional equation satisfied by generating function. (Baxter 2011) Can be used to q-count according to inversion number, and according to the number of occurrences of any consecutive pattern(s). (Baxter 2011) Extend to many other types of pattern avoidance problems.

30 Extensions Words avoiding permutations (Pudwell 2008) Instead of S n (Q) consider words with a 1 1 s,..., a n n s avoiding Q. avoiding barred patterns (Pudwell 2008) e.g avoiding dashed patterns (Baxter, Pudwell 2011) e.g

31 1 We have algorithmic techniques to deal with barred patterns and dashed patterns (i.e. lots of data). Can we characterize when they are equivalent and when they are not? Examples: S n ({25 314}) = S n ({2 41 3}) There is no dashed pattern equivalent to Sn ({3 5241}) = Sn ({31 4 2}) but S n ({3 5241}) S n ({31 4 2})

32 1 We have algorithmic techniques to deal with barred patterns and dashed patterns (i.e. lots of data). Can we characterize when they are equivalent and when they are not? Examples: S n ({25 314}) = S n ({2 41 3}) There is no dashed pattern equivalent to Sn ({3 5241}) = Sn ({31 4 2}) but S n ({3 5241}) S n ({31 4 2}) 2 Can we find algorithmic techniques to enumerate permutations avoiding bivincular patterns? 3 Can we find algorithmic techniques to study other types of patterns that appear in this workshop?

33 Thank You!