Enumerating Anchored Permutations with Bounded Gaps Maria Monks Gillespie, CSU Ken G. Monks, University of Scranton Ken M. Monks, Front Range Community College Rocky Mountain Algebraic Combinatorics Seminar Nov 15, 2019
Classical stair climbing problems How many ways can you climb a staircase with n stairs, taking either 1 or 2 stairs at a time at each step?
Classical stair climbing problems How many ways can you climb a staircase with n stairs, taking either 1 or 2 stairs at a time at each step? Fibonacci: If a n is total, a 0 “ 0, a 1 “ 1, a n “ a n ´ 1 ` a n ´ 2 for n ě 2.
Classical stair climbing problems How many ways can you climb a staircase with n stairs, taking either 1 or 2 stairs at a time at each step? Fibonacci: If a n is total, a 0 “ 0, a 1 “ 1, a n “ a n ´ 1 ` a n ´ 2 for n ě 2. Take up to k stairs at a time? If b n is number for n stairs: b n “ b n ´ 1 ` b n ´ 2 ` ¨ ¨ ¨ ` b n ´ k
Forwards and backwards steps § Take steps of at most k stairs up or down, stepping on every stair exactly once, starting at stair 1 and ending at stair n ? 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5
Forwards and backwards steps § Take steps of at most k stairs up or down, stepping on every stair exactly once, starting at stair 1 and ending at stair n ? § Permutation π “ π 1 , . . . , π n of stairs 1 , . . . , n is anchored if π 1 “ 1 and π n “ n . § k -bounded if | π i ´ π i ` 1 | ď k for all i . 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5
Forwards and backwards steps § Take steps of at most k stairs up or down, stepping on every stair exactly once, starting at stair 1 and ending at stair n ? § Permutation π “ π 1 , . . . , π n of stairs 1 , . . . , n is anchored if π 1 “ 1 and π n “ n . § k -bounded if | π i ´ π i ` 1 | ď k for all i . § Let F p k q be number of k -bounded anchored permutations of n n . Recursion for F p k q ? n 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5
The case k “ 2 § F p 2 q is number of ways to climb stairs with steps ˘ 1 , ˘ 2. n 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
The case k “ 2 § F p 2 q is number of ways to climb stairs with steps ˘ 1 , ˘ 2. n § A step of ` 2 must be followed by ´ 1, ` 2, return to diagonal 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
The case k “ 2 § F p 2 q is number of ways to climb stairs with steps ˘ 1 , ˘ 2. n § A step of ` 2 must be followed by ´ 1, ` 2, return to diagonal § Recursion: F p 2 q “ F p 2 q n ´ 1 ` F p 2 q n n ´ 3 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
The case k “ 3 Define F n “ F p 3 q n . § Using generating functions: F n “ 2 F n ´ 1 ´ F n ´ 2 ` 2 F n ´ 3 ` F n ´ 4 ` F n ´ 5 ´ F n ´ 7 ´ F n ´ 8
The case k “ 3 Define F n “ F p 3 q n . § Using generating functions: F n “ 2 F n ´ 1 ´ F n ´ 2 ` 2 F n ´ 3 ` F n ´ 4 ` F n ´ 5 ´ F n ´ 7 ´ F n ´ 8 § Solved conjecture listed on OEIS! Screenshot from 2018:
The case k “ 3: Proof § Joker : 31425, or any vertical translation thereof appearing consecutively 6 5 4 3 2 1 1 2 3 4 5 6
The case k “ 3: Proof § Joker : 31425, or any vertical translation thereof appearing consecutively 10 9 8 6 7 5 6 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 § Lemma: (G.,M.,M.) At p i , i q after permutation of 1 , 2 , . . . , i : up-step of ` 3 must be followed either by a Joker or by a Cascading 3 -pattern .
The case k “ 3: Proof § F n “ # 3-bounded anchored permutations of length n § G n “ # 3-bounded permutations π with π 1 “ 1 or π 1 “ 2 and π n “ n § H n “ # 3-bounded permutations π with π 1 “ 3 and π n “ n that do not begin with the Joker
The case k “ 3: Proof § F n “ # 3-bounded anchored permutations of length n § G n “ # 3-bounded permutations π with π 1 “ 1 or π 1 “ 2 and π n “ n § H n “ # 3-bounded permutations π with π 1 “ 3 and π n “ n that do not begin with the Joker § System of recursions: 1. F n “ G n ´ 1 ` H n ´ 1 ` F n ´ 5 2. G n “ F n ` G n ´ 2 ` F n ´ 3 ` G n ´ 4 ` H n ´ 2 3. H n “ F n ´ 3 ` G n ´ 3 ` F n ´ 4 ` G n ´ 5 ` H n ´ 3
The case k “ 3: Proof § F n “ # 3-bounded anchored permutations of length n § G n “ # 3-bounded permutations π with π 1 “ 1 or π 1 “ 2 and π n “ n § H n “ # 3-bounded permutations π with π 1 “ 3 and π n “ n that do not begin with the Joker § System of recursions: 1. F n “ G n ´ 1 ` H n ´ 1 ` F n ´ 5 2. G n “ F n ` G n ´ 2 ` F n ´ 3 ` G n ´ 4 ` H n ´ 2 3. H n “ F n ´ 3 ` G n ´ 3 ` F n ´ 4 ` G n ´ 5 ` H n ´ 3 § Set F p x q , G p x q , H p x q to be generating functions of F n , G n , H n . Solve system of three equations: x ´ x 2 ´ x 4 F p x q “ 1 ´ 2 x ` x 2 ´ 2 x 3 ´ x 4 ´ x 5 ` x 7 ` x 8 Recursion follows. QED
k ě 4? § Much more difficult! § Must a finite-depth linear recurrence relation always exist?
k ě 4? § Much more difficult! § Must a finite-depth linear recurrence relation always exist? § (G.,M.,M.) Answer: YES!
k ě 4? § Much more difficult! § Must a finite-depth linear recurrence relation always exist? § (G.,M.,M.) Answer: YES! § Transfer-matrix method: to show gen. function is rational § Finite directed graph: p V , E q where E Ď V ˆ V # 1 p i , j q P E § Adjacency matrix: For i , j P V , define A ij “ 0 p i , j q R E
k ě 4? § Much more difficult! § Must a finite-depth linear recurrence relation always exist? § (G.,M.,M.) Answer: YES! § Transfer-matrix method: to show gen. function is rational § Finite directed graph: p V , E q where E Ď V ˆ V # 1 p i , j q P E § Adjacency matrix: For i , j P V , define A ij “ 0 p i , j q R E Theorem (Transfer-matrix.) Let p ij p n q “ # directed paths from i to j of length n. Then p ij p n q x n “ p´ 1 q i ` j det p I ´ xA ; j , i q 8 ÿ P C p x q det p I ´ xA q n “ 0 where det p B ; j , i q is the minor with row j, column i deleted.
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k § Consecutive k -pattern: of π is a permutation of 1 , 2 , . . . , k whose relative order matches π i ` 1 , π i ` 2 , . . . , π i ` k for some i Example π “ 51432 has consecutive 3-patterns:
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k § Consecutive k -pattern: of π is a permutation of 1 , 2 , . . . , k whose relative order matches π i ` 1 , π i ` 2 , . . . , π i ` k for some i Example π “ 51432 has consecutive 3-patterns: 312,
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k § Consecutive k -pattern: of π is a permutation of 1 , 2 , . . . , k whose relative order matches π i ` 1 , π i ` 2 , . . . , π i ` k for some i Example π “ 51432 has consecutive 3-patterns: 312, 132,
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k § Consecutive k -pattern: of π is a permutation of 1 , 2 , . . . , k whose relative order matches π i ` 1 , π i ` 2 , . . . , π i ` k for some i Example π “ 51432 has consecutive 3-patterns: 312, 132, 321
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k § Consecutive k -pattern: of π is a permutation of 1 , 2 , . . . , k whose relative order matches π i ` 1 , π i ` 2 , . . . , π i ` k for some i § Pattern graph P k : nodes are k -patterns, edge τ Ñ σ iff pattern of τ 2 , . . . , τ k matches pattern of σ 1 , . . . , σ k ´ 1 Example π “ 51432 has consecutive 3-patterns: 312, 132, 321 Path of 51432 in P 3 is 312 Ñ 132 Ñ 321
Transfer-matrix method for non-anchored case § Avgustinovich and Kitaev: k -bounded permutations (not nec. anchored) have rational generating functions for all k § Consecutive k -pattern: of π is a permutation of 1 , 2 , . . . , k whose relative order matches π i ` 1 , π i ` 2 , . . . , π i ` k for some i § Pattern graph P k : nodes are k -patterns, edge τ Ñ σ iff pattern of τ 2 , . . . , τ k matches pattern of σ 1 , . . . , σ k ´ 1 § k -determined permutation: determined by its path of consecutive patterns in P k Example π “ 51432 has consecutive 3-patterns: 312, 132, 321 Path of 51432 in P 3 is 312 Ñ 132 Ñ 321 Path of 52431 in P 3 is 312 Ñ 132 Ñ 321, so not 3-determined
Transfer-matrix method for non-anchored case Theorem (Avgustinovich, Kitaev, 2008) For any permutation π : π is p k ` 1 q -determined ô π ´ 1 is k-bounded ô π avoids all k-prohibited patterns of length at most 2 k ` 1 A k -prohibited pattern is of the form xX p x ` 1 q or p x ` 1 q Xx where | X | ě k . Definition P 2 k ` 1 , k is the subgraph of P 2 k ` 1 on nodes that do not contain a k -prohibited pattern.
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