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Inversions on Permutations Avoiding Consecutive Patterns Naiomi Cameron* 1 Kendra Killpatrick 2 12th International Permutation Patterns Conference 1 Lewis & Clark College 2 Pepperdine University July 11, 2014 Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 1 / 30
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Inversions on Permutations Avoiding ConsecutivePatterns

Naiomi Cameron* 1 Kendra Killpatrick2

12th International Permutation Patterns Conference

1Lewis & Clark College2Pepperdine University

July 11, 2014

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 1 / 30

1 Permutations and Inversions

2 Generalized Pattern Avoidance

3 Fibonacci Tableaux

4 Inversion Polynomials for Consecutive Pattern Avoiding Permutations

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 2 / 30

The Basics

A permutation π = π1π2 · · ·πn of length n will simply be any way to writethe numbers 1 through n in some order. We use Sn to denote the group ofall permutations of length n and |Sn| = n!.

Definition

Given a permutation π = π1π2 · · ·πn ∈ Sn, we define an inversion to be apair (i , j) such that i < j and πi > πj .

Definition

The inversion statistic, inv, is given by

inv(π) = the total number of inversions in π

Example

π = 328574619, inv(π) = 15

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 3 / 30

Classical Pattern Avoidance

Definition

Let π ∈ Sn. We say “π contains σ as a pattern” if π has a subsequencethat is order isomorphic to σ.

Definition

We say that π avoids σ as a pattern if π contains no subsequence orderisomorphic to σ.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 4 / 30

Classical vs. Generalized Pattern Avoidance

Classical: σ is written as a permutation of the numbers 1, 2, . . . , kand elements of the pattern need not appear as adjacent in π.

Generalized Pattern Avoidance: σ is written as a list of thenumbers 1, 2, . . . , k with dashes inserted between elements thatneed not appear as adjacent in π and no dashes otherwise.

Example

π = 45213 contains 3− 12 and 3− 1− 2 but avoids 312.

A classical pattern is a generalized pattern with all internal dashes.

A generalized pattern with no internal dashes is called a consecutivepattern.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 5 / 30

Example

Find all permutations in S4 that avoid 312 as a consecutive pattern.

1234 2134 3124 41231243 2143 3142 41321324 2314 3214 42131342 2341 3241 42311423 2413 3412 43121432 2431 3421 4321

Permutations in bold contain 312.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 6 / 30

Example

Find all permutations in S4 that avoid 312 as a consecutive pattern.

1234 2134 3124 41231243 2143 3142 41321324 2314 3214 42131342 2341 3241 42311423 2413 3412 43121432 2431 3421 4321

Permutations in bold contain 312.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 6 / 30

Wilf-equivalence

Definition

Let Π be a collection of generalized patterns. We let Avn(Π) denote theset of permutations in Sn that avoid every pattern in Π.

Example

|Av4(312)| = 16.

Definition

We say that two sets of generalized permutation patterns Π and Π′ areWilf equivalent if |Avn(Π)| = |Avn(Π′)|.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 7 / 30

st-Wilf Equivalence

Sagan and Savage [14] recently defined a q-analogue of Wilf equivalenceby considering any permutation statistic st from ]n≥0Sn → N, where N isthe set of nonnegative integers, and letting

F stn (Π; q) =

∑σ∈Avn(Π)

qst(σ).

For Π and Π′ subsets of permutations, they defined Π and Π′ to be st-Wilfequivalent if F st

n (Π; q) = F stn (Π′; q) for all n ≥ 0.

We use the same definition for Π and Π′ sets of generalized permutationpatterns.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 8 / 30

Dokos et al.[7] give a thorough investigation of st-Wilf equivalence forclassical patterns of length 3 for both the major index, maj, and theinversion statistic, inv.

Elizalde-Noy, Kitaev, Kitaev-Mansour, Aldred-Atkinson-McCaughan[1, 9, 10, 12, 13] collectively accomplished a comprehensiveenumeration of permutations avoiding multiple consecutive patternsof length three.

The main focus of the present work is the inversion statistic onpermutations avoiding sets of consecutive patterns of length three.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 9 / 30

Inversion Polynomial In(Π; q)

Definition

Let Π be a set of (generalized) patterns. The inversion polynomial onAvn(Π) is given by

In(Π; q) =∑

σ∈Avn(Π)

qinv(σ)

Note that In(Π; 1) = |Avn(Π)| .

Example

I4({3− 1− 2}; q) = 1 + 3q + 3q2 + 3q3 + 2q4 + q5 + q6

I4({312}; q) = 1 + 3q + 3q2 + 4q3 + 3q4 + q5 + q6

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 10 / 30

Fibonacci Tableaux

Recall that the number of ways to write a positive integer n as a sum of1’s and 2’s is a Fibonacci number.

Definition

A Fibonacci shape of size n is an ordered list of 1’s and 2’s which sumsto n. The Ferrers diagram for a Fibonacci shape is formed by replacingeach 1 with a single dot and each 2 with two dots.

Example

The Fibonacci shape 122121 has size 9 and Ferrers diagram:

• • •• • • • • •

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 11 / 30

Fibonacci Tableaux

Definition

A standard Fibonacci tableau for a Fibonacci shape µ of size n is afilling of the Ferrers diagram of µ with the numbers 1, 2, . . . , n so that thebottom row decreases from left to right and each column decreases frombottom to top.

Example

A standard Fibonacci tableau of shape µ = 122121 is

3 4 29 8 7 6 5 1

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 12 / 30

Fibonacci Tableaux

3 4 29 8 7 6 5 1

Definition

The column-reading word wc(T ) of a standard Fibonacci tableau T isobtained by reading the columns of T from right to left, bottom to top.

Example

For the tableau above, wc(T ) = 152674839.

Definition

The inversion number of a standard Fibonacci tableau T is defined asinv(T ) := inv(wc(T )).

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 13 / 30

Fibonacci Tableaux and Consecutive Pattern Avoidance

Observation

The set of column reading words for standard Fibonacci tableaux of size nis Avn(321, 312), the set of permutations of length n that avoid theconsecutive patterns 312 and 321.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 14 / 30

The Inversion Polynomial on Avn(321, 312)

Theorem (C-K, 2013)

Let Π = {321, 312}. Then I0(Π; q) = I1(Π; q) = 1 and for n ≥ 2,

In(Π; q) = In−1(Π; q) + (q + q2 + · · ·+ qn−1)In−2(Π; q).

Let ν be a Fibonacci shape and define

Iν(q) :=∑

T shape ν

qinv(T )

Note that ∑|ν|=n

Iν(q) =∑ν=1µ

Iν(q) +∑ν=2µ

Iν(q)

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 15 / 30

The Inversion Polynomial on Avn(321, 312)

I1µ(q) = Iµ(q) where |µ| = n − 1

inv

(• • •

n • • • • •

)= inv

(• • •• • • • •

)I2µ(q) =

(q + q2 + · · · q|µ|+1

)Iµ(q) where |µ| = n − 2.

inv

(k • • •n • • • • •

)= n − k + inv

(• • •• • • • •

)where k = 1, · · · , n − 1

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 16 / 30

The Inversion Polynomial on Avn(321, 312)

Hence, for Π = {321, 312}

In(Π; q) =∑|µ|=n−1

Iµ(q) +∑|µ|=n−2

(q + q2 + · · · q|µ|+1

)Iµ(q)

= In−1(Π; q) + (q + q2 + · · ·+ qn−1)In−2(Π; q)

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 17 / 30

Inversion Polynomials for Consecutive Pattern AvoidingPermutations

In our recent paper, we compute the inversion polynomials for all but oneset of permutations that simultaneously avoid a set of three or moreconsecutive patterns.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 18 / 30

Class 5 - Π = {321, 312, 213, 132}

Theorem

In(Π; q) = 1 + q2 + · · ·+ qn−1

In this case, Avn(Π) is in correspondence with standard Fibonacci tableauxthat have at most one column of height two which, if it exists, must bethe first column of the Fibonacci tableau.

If the Fibonacci tableau has all columns of height one then it correspondswith the permutation π = 123 · · · n which has an inversion number of 0.

If the Fibonacci tableau begins with a column of height two then the entryin the top row can be one of the numbers 1, 2, · · · , n − 2 (not n − 1 sincethe corresponding permutation must avoid 132). If k is the number in thetop row then the corresponding permutation has an inversion statistic ofn − k. Summing over all possible k gives the result.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 19 / 30

Class 7 - Π = {321, 312, 231}

Theorem

In(Π; q) = In−1(Π; q) +

bn/2c∑k=1

qk−1Ck−1(q) · In−2k+1(Π; q)

Avn(Π) is in one-to-one correspondence with standard Fibonacci tableauxfor which the elements in the top of each column must decrease from leftto right. (Note: the element in a column of height one is both in the toprow of its column and in the bottom row of its column.)

T =e c •

• d b a • •

wc(T ) = • • •abcde•. Since abc must avoid 231, c > a. Since cde mustalso avoid 231, e > c .

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 20 / 30

Let I jn(Π; q) denote the inversion polynomial for Fibonacci tableaux of sizen whose shape starts with a j, where j = 1, 2.

If the Fibonacci tableau begins with a column of height one, then theremust be an n in this column and n appears as the last element in the wordof the tableau. Removing this n will not change the inversion statistic forthis permutation and will give a tableau of size n − 1 with the givenrestrictions. Therefore,

I 1n (Π; q) = In−1(Π; q).

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 21 / 30

If the Fibonacci tableau begins with a column of height two, let k be thesmallest integer such that the first k − 1 columns have height two and thekth column has height one.

Since the elements in the top row of each column and the bottom row ofeach column must decrease from left to right, the numbers n,n − 1, n − 2, . . . n − 2k + 3 must be in the first k − 1 columns andn − 2k + 2 must be in the kth column (of height 1). The tableaux formedfrom the remaining columns to the right of column k corresponds to apermutation of size n − 2(k − 1)− 1 = n − 2k + 1.

For example, for n = 15 and k = 4 we have

T =13 11 10 6 5 115 14 12 9 8 7 4 3 2

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 22 / 30

T =13 11 10 | | 6 5 115 14 12 | 9 | 8 7 4 3 2

Since n − 2k + 2 is in column k , removal of this element does not changethe inversion statistic and also gives a tableau in the first k − 1 columnsthat corresponds (with relabeling) to a tableau of size 2(k − 1) with allcolumns of height two whose elements in the top row of each columndecrease from left to right.

13 11 1015 14 12

→ 4 2 16 5 3

→ 1 2 43 5 6

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 23 / 30

These tableaux are counted by the Catalan number Ck−1 and the inversionpolynomial on such tableau is given by qk−1Ck−1(q) where Ck−1(q) arethe q-Catalan polynomials. This gives us the inversion polynomial

qk−1Ck−1(Π; q)In−2k+1)(Π; q),

and we have

I 2n (Π; q) =

bn/2c∑k=1

qk−1Ck−1(q) · In−2k+1(Π; q).

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 24 / 30

Thus, with initial conditions

I 12 (Π; q) = 1 and I 2

2 (Π; q) = I 23 (Π; q) = q,

we have

In(Π; q) = In−1(Π; q) +

bn/2c∑k=1

qk−1Ck−1(q) · In−2k+1(Π; q)

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 25 / 30

Further Research

We have determined the inversion polynomial for (almost) all Π whereΠ is a collection of three or more consecutive patterns. We continueto work on the remaining class of three consecutive patterns and theremaining classes of two consecutive patterns.

If {321, 132} is not a subset of Π, we use a more general notion calleda strip tableaux to model the permutations in Avn(Π).

We are also working on mixed patterns (some consecutive letters andsome not) and on how we can use our strip shape model to countthese classes.

Killpatrick’s summer undergraduate research students areinvestigating consecutive patterns of length four.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 26 / 30

Acknowledgments

Thanks for your attention!

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 27 / 30

References I

R. Aldred, M. Atkinson, D. McCaughan, Avoiding consecutive patternsin permutations, Adv. in Appl. Math. 45 (2010), no. 3, 449461

E. Babson, E. Steingrimsson. Generalized permutation patterns and aclassification of the Mahonian statistics. Sem. Lothar. Combin. 44(2000), Art. B44b, 18 pp. (electronic).

N. Cameron, K. Killpatrick, Symmetry and log-concavity results forstatistics on Fibonacci tableaux, Ann. Comb. 17 (2013), 603-618.

Carlitz, L. and Riordan, J. Two element lattice permutation numbersand their q-generalization. Duke J. Math 31 (1964), 371-388.

S. Cheng, S. Elizalde, A. Kasraoui, B. Sagan, Inversion polynomialsfor 321-avoiding permutations, Discrete Math. 313 (2013), no. 22,25522565.

A. Claesson, Generalized Pattern Avoidance, European J. Combin. 22(2001), no. 7, 961-971.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 28 / 30

References II

T. Dokos, T. Dwyer, B. Johnson, B. Sagan, K. Selsor, Permutationpatterns and statistics, Discrete Math. 312 (2012), no. 18, 2760-2775.

S. Elizalde, Asymptotic enumeration of permutations avoidinggeneralized patterns, Advances in Applied Mathematics 36 (2006),138-155.

S. Elizalde, M. Noy, Clusters, generating functions and asymptotics forconsecutive patterns in permutations, Adv. in Appl. Math. 49 (2012),351-374.

S. Elizalde, M. Noy, Consecutive Patterns in Permutations, Adv. inAppl. Math. 30 (2003), 110-125.

Killpatrick, K., On the parity of certain coefficients for a q-analogue ofthe Catalan numbers, Electron. J. Combin. 16 (2009), #R00.

S. Kitaev, Multi-avoidance of generalised patterns. Discrete Math. 260(2003), no. 1-3, 89100.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 29 / 30

References III

S. Kitaev, T. Mansour, Simultaneous avoidance of generalizedpatterns, Ars Combinatoria 75 (2005), 267-288.

B. Sagan, C. Savage, Mahonian pairs, J. Combin. Theory Ser. A 119(2012), no. 3, 526-545.

R. Stanley, The Fibonacci Lattice, Fibonacci Quart. 13 (1975),215-232.

Cameron, Killpatrick (PPC2014) Inversions on Permutations Avoiding Consecutive Patterns July 11, 2014 30 / 30


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