Pattern Popularity in Γ1 − Non Deranged Permutations: An Algebraic and Algorithmic Approach

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Department of Mathematics, Usmanu Danfodiyo University Sokoto, P.M.B. 2346, Sokoto, Nigeria

Corresponding author: K. O. Aremu, [email protected]

ABSTRACT: Given a permutation pattern say

1... k Sk

  

  and permutation

  

 ₁... _nS_n, we say that



contains the pattern



if there exist

1

1   i ... ik n such thatred( i1... ik) . Each subsequence in



is known as an occurrence of the pattern



. Conversely, if there exist no occurrence of



in



, then we say that the permutation



avoids the pattern



. The popularity of a pattern



is the total number of copies of



within all permutations of a set. In this work, we address popularity of length-3 patterns in

1

  non deranged permutations in two approaches; algebraically and algorithmically. We first establish algebraically that pattern



₁ is the most popular and

pattern



₃,



₄ and



₅ are equipopular in 1

p



G . We further

provide efficient algorithms that also report same results on popularity and equipopularity of patterns of length-3 in

1

p



G as obtained by the algebraic approach.

KEYWORDS: Permutation, Pattern, Popularity, Equipopularity, Sorting.

1. INTRODUCTION

Permutation is a mathematical concept, which appears in various mathematical and practical problems. For example, sorting, ordering, matching and so on can be described as permutation problem. Permutation did not simply spring into existence, but rather, it is the culmination of a long period of mathematical investigation. Patterns in permutations have been studied over a long period of time; it has been an active field of research with over hundreds of articles published in the past few decades. There are several notable survey papers on this subject, a few includes ([Zha14, Rud13, Ste10]).

There are several notions of patterns in different combinatorial objects, e.g. permutations, graphs, matrices, words, compositions, etc. Permutation patterns include; classical patterns, barred patterns, vincular patterns, bivincular patterns and partial ordered patterns. The classical patterns was introduced by Knuth in 1968 but was much studied by Simon and Schmist in 1985. The notion was

further extended to word by Burstein in 1998 and later generalised for word by Burstein & Mansour in 2002. The Barred pattern was introduced for permutation by West in 1990. In 2002, the Vincular pattern (Generalized patterns) was introduced for permutation by Babson and Steingrimsson. This pattern (Vincular) was extended to word by Bustein & Mansour in 2003. The Bivincular pattern was introduced for permutations by Bousquet-Melou, Claesson, Dukes, and Kitaev in 2010. Kitaev introduced partially ordered patterns for permutations in 2003. The pattern (bivincular) was later extended to word by Kitaev & Mansour. [Kit11]

With no doubt, patterns in permutations have been well studied for over a century. As seems to be the case, these patterns were studied on permutations arbitrarily. The Symmetric group S_n is the set of all permutations of a set of cardinality n. There are several types of other smaller permutation groups (Subgroup ofS_n) of set, a notable one among them is the alternating groupA_n. Recently, [Are16] proposed a permutation group called the ₁- non deranged permutation group denoted as 1

p



G . A permutation in this group is expressed as a sequence of 1 and numbers of integer modulo p where p is a prime and greater than or equal to 5. More precisely, this work will be centred on the ₁- non deranged permutations as the notion of pattern popularity has never been studied on this set of permutations. In this present work, a review of length-3 classical patterns in 1

p



G is presented. 1

p



G is examined for certain properties (namely, containment and avoidance). In truth, it would be more correct to say that the approach of this research is both algebraic and algorithmic techniques, which accidentally obtain interesting results.

A permutation



on the set { ,



₁ ,



_n} is a sequence of distinct letters



(1),



( )n such that

1

( ) { ,i , _n}

1, 2, n

a a a



 is of the form

(1 ) (1 2 ) (1 ( 1) )

i i mp i mp p i mp

      where n

= p and p is a prime greater than or equal to 5. [Are17]. An occurrence of a pattern



in a permutation



_{is a subsequence in}



_{whose letters} are in the same relative order as those in



. So, a



- avoiding permutation is a permutation that does not contain the pattern



. The popularity P_S( )



of a pattern



in a set Sof permutations [Rud13]

( ) ( , )

S

S

P f



  







2. EQUIPOPULARITY OF PATTERNS IN ₁-

NON DERANGED PERMUTATIONS

When two patterns occur equally in a set of permutation, we say that these patterns are equipopular. In particular, this subsection addresses length-3 patterns that are equally popular in 1

p



G . To prove some of the results in this section, we introduce some new notions on permutations. Definition 2.1

A permutation

  

 (1) (2)... ( )



n is increasing if ( )i

 and



(i1)are two successive element such that



( )i 



(i1) and decreasing if

( )i (i 1)







 . Let



and be two non-negative integers such that



is the number of times of all increment and  number of times of decrement in



, we then define



as a

 

 sequence: i. completely increasing if  = 0 ii. completely decreasing if



= 0

iii. equally increasing and decreasing if    iv. almost increasing if   

v. almost decreasing if  

Example 2.1 Consider the permutations



₁ = 1234567,



₂= 1357246 and



₄ = 1526374. Then,

1



is a completely increasing sequence,



₂ is an almost increasing sequence and



₄ is an equally increasing and decreasing sequence.

Definition 2.2. A permutation



is said to contain a break in sequence if there exists any two successive elements of



such that

|



( )i 



(i 1) | 1

Remark 2.2. For simplicity, we let



₁= 123,



₂= 132,

3



= 213,



₄= 231,



₅= 312,



₆= 321 be all length-3

patterns. By the above definition, we see that



₃,



₄,



₅, have breaks while



₁,



₂,



₆ have no break.

Lemma 2.3. Let 1

i p



_ 

G . Then every



_i eand

1

i p

  _ has a break between



_i( )j and



_i(j1).

Proof. Since every 1

i p



_ 

G is an arithmetic progression of modulo p,



_i( )j and



_i(j1) will not be in the equivalence classes [j] and [j + 1] modulo p respectively, and hence,

|



_i( )j 



_i(j 1) | 1



Definition 2.3. A pair in a permutation



is a subsequence ( ( ) ( ))

 

i j of



, such that i < j. Lemma 2.4. Let



_i, _j 1

p



G and P[



_i]denote the set of pairs of



_i. Then the following hold after deleting 1, the fix point of any



_i:

i. 1

[ _i] [ _i ]

P P  

ii. ( [P



_i]P[



_i1])( [P



_j]P[



_j1])  iii. | [P



_i]P[



_i1] | | [ P



_j]P[



_j1] | Example 2.5. Let P[ ] { :



 P i_i  } be the set of pairs of



. The length-3 containments of



are given as { (1)P_i P_j:P_i(1)P_j(1),i j}

Remark 2.6. Every containment of a pattern in a permutation can be expressed as subsequences of increasing and/or decreasing pairs. Equivalently, pairs of a permutation can be concatenated to form any length k containment.

The next result shows the length-3 patterns that are equipopular in 1

p



G which is also validated by an algorithm.

Theorem 2.7. Let



_i be a permutation in 1

p



G and and 1  i 6) be a pattern of length-3. If

1 ( )_i

P 



G is the popularity of a pattern



iin

1

p



G , then

1( 3) 1( 4) 1( 5).

p p p

P_G   P_G  P_G 

Proof. We want to show that

1( 3) 1( 4) 1( 5)

p p p

P_G   P_G   P_G  m such that

. The pattern



₁and



₆ have no breaks and thus their pairs are completely increasing and completely decreasing subsequences respectively. Therefore, by Lemma 2.3, 1( 6) .

p

P_G   m Also by Lemma 2.3 and

Remark 2.6, 1( )1 ,

p

1,





₃,



₄ and



₅ avoid_p1. Hence, since each of

3,





4 and



5 has a break,

1( 3) 1( 4) 1( 5)

p p p

P   P   P  m

G G G holds by

definition 2.3 and remark 2.6.

 Remark 2.8. Suppose



_A { ,

     

_{1 2}, ₃, ₄, ₅, ₆}is the set of all length-3 patterns, the pattern



₃,



₄ and

5



are equipopular in 1

p



G . We write

3 4 5

{ , , }

E





  

to be set of equipopular patterns which is subset of



_A. For convenience, we represent the equipopular patterns with



_E throughout the rest of this work.

Definition 2.4. A permutation



contains another permutation



as a pattern (denoted

 

) if the plot of



contains a subset which is equivalent to the plot of



. The number of occurrences of



in



(denotedV( )



) is the number of such subsets.

Algorithm 1: Algorithm to show that the pattern



₃,



₄

and



₅ are equally popular in 1

p



G

Input: Input any



 1

p



G i nt o an array

Output: Number of occurrence of the pattern



₃,



₄ and



₅

1: Begin

2: Set integer c = 0 3: for i = 1 to n − 1

4: for j = i + 1 to n − 2

5: for k = j + 1 to n

6: Case I: τ3

7: if a[k] > a[i]

8: if a[i] > a[j]

9: Case II: τ4

10: if a[i] > a[k]

11: if a[j] > a[i]

12: Case III: τ5

13: ifa[i] > a[k]

14: if a[k] > a[j]

15: Increment c by 1 16: end if

17: end if 18: end for 19: end for 20: end for 21: End

The algorithm above helps to show that for every 1_,

i p



_ 

G

3( i) 4( i) 5( i)

V_  V_  V_ 

Implementing the algorithm validates Theorem 2.7

Table 1: Popularity of



₃,



₄and



₅ in 1

p



G

1

p



G



₃



₄



₅

5 2 2 2

7 16 16 16

11 168 168 168

13 382 382 382

17 1320 1320 1320

19 2184 2184 2184

23 5096 5096 5096

3. POPULARITY OF PATTERNS IN ₁- NON DERANGED PERMUTATION

The popularity of a pattern



is the total number of copies of \tau within all permutation of a set. [Rud13]. We begin this section, with some basic results on which the main result of this section is built.

Lemma 3.1. Every



₁ and 1 1

p p





 G , are completely increasing and almost decreasing sequences.

Proposition 3.2. Suppose 1

i p



_ 

G such that i = 1, p - 1 and if



_E represents the equipopular patterns

then ( ) 0

E i

V_  

Proof. Since every 1

i p



_ 

G such that i = 1, p - 1 are completely increasing and almost decreasing sequences respectively, therefore, a subsequence that is an almost decreasing sequence does not exist in a completely increasing and almost decreasing sequence respectively. Thus, ( ) 0

E i

V  

1, 1

i p

   . Hence the result.

 Lemma 3.3. Let _j and



_kbe two patterns such that

j

 has fixed at the first element,



_khas no fixed and both do not avoid 1

i p



_ 

G . Then ( ) ( )

j i k i

V_



V_



Theorem 3.4. Let 1

p



G be a set ₁- non deranged permutations, then the pattern



₁ is the most popular pattern in 1

p



G .

Proof. To prove that the pattern



₁ is the most

popular pattern of



_Ain 1

p



G , we only need to show that 1( )1

p

P_G  is greater than any other 1( )

p i

P_G  . First,

we show that 1( )1 1( )

p p E

P_G   P_G  and

1( 2) 1( )

p p E

P_G   P_G  . Since the first entry of every

1

i p



_ 

occurrences of the equipopular patterns in 1

p



G

compared to



₁ and



₂ by Lemma 3.3. Thus,

1( )1

p

P_G  and 1( 2)

p

P_G  strictly greater than 1( )

p E

P_G  .

Since



₁ and _p1 avoid pattern



_E by proposition 3.2, therefore, 1( 6) 1( )

p p E

P_G  P_G  . Next we show

that 1( 2) 1( 6)

p p

P_G  P_G  . Since pattern



2 fixes 1

then this holds by Lemma 3.3. Note again that 1( )1 1( 2)

p p

P_G   P_G  because the pattern



₁ and

every



₁are completely increasing sequences and have no break. Therefore, pattern



₁ is the most popular pattern in 1

p



G . Hence the result.

 Theorem 3.5. Let 1

p



G be a set of ₁- non deranged permutations, and



_E is the equipopular pattern. Then



_Eis the least popular pattern in 1_.

p



G

Proof. It is immediate from the proof of theorem 3.4

and lemma 3.3.

 Lemma 3.6. Suppose 1

i p



_ 

G then the popularity of the patterns forms a chain

1( ) 1( 6) 1( 2) 1( )1

p E p p p

P_G   P_G   P_G   P_G 

Proof. Since the first entry of the pattern



₂ fixes 1,

then by Lemma 3.3 1( 2) 1( 6)

p p

P_G  P_G  . Also since

1



is the most popular by theorem 3.4 and



_Eis the least popular by theorem 3.5 in 1

p



G , hence the result. In what follows, we provide an algorithm that analyses all length-3 patterns in 1

p



G where

5 p 23.

Algorithm 2: Algorithm to show that the number of

occurrence of



_A in 1

p



G

Input: Input any 1

i p



_ 

G into an array Output: Number of occurrence of



_A in



_i 1: Begin

2: Set integer c = 0 3: for i = 1 to n − 1

4: for j = i + 1 to n − 2

5: for k = j + 1 to n

6: Case I: τ3

7: if a[k] > a[i]

8: if a[i] > a[j]

9: Case II: τ4

10: ifa[i] > a[k]

11: if a[j] > a[i]

12: Case III: τ5

13: ifa[i] > a[k]

14: if a[k] > a[j]

15: Case IV: τ1

16: if a[j] < a[k]

17: if a[i] < a[j]

18: Case V: τ2

19: if a[j] > a[k]

20: if a[k] > a[i]

21: Case VI: τ6

22: ifa[i] > a[j]

23: if a[j] > a[k]

24: Increment c by 1

25: end if

26: end if 27: end for 28: end for 29: end for 30: End

Hereafter, we implemented the above algorithm and observed that the pattern



₁ is the most popular pattern and



_Eis the least popular patterns in 1

p



G .

The algorithm validates theorem 3.4, theorem 3.5 and lemma 3.6.

Table 2: Pattern popularity in 1

p



G for prim size 5 to 23

1

p



G



₁



₂



₆



₄



₅



₃

5 16 14 4 2 2 2

7 73 61 28 16 16 16 11 489 393 264 168 168 168 13 952 778 556 382 382 382 17 2800 2280 1840 1320 1320 1320 19 4353 3561 2976 2184 2184 2184 23 9289 7637 6748 5096 5096 5096

Lemma 3.7. Let 1

i Gp



_ 

. Then



_i and 1

i

 _have equal number of increasing pairs and equivalently, equal number of decreasing pairs.

Proof . Let

1

1 2 ...

.

(1) (2) ... ( ) p

p

p











 

_{ }

  G

The inverse of



is given as

1

(1) (2) ... ( )

,

1 2 ... p

p

p







_

 



 

whenever



(1)



(2) ...



( )p . From above we have that



( )j comes after ( ) i as an image in the inverse permutation whenever

( ( ))j ( ( ))i

 



 

Since 1

[ _i] [ _i ]

P P   by lemma 2.4(i),

1 1

( [P



_i]P[



_i ])( [P



_j]P[



_j ])  by lemma 2.4(ii) and by

( ( ))j ( ))i

 





the result follows.

 Proposition 3.8. Suppose 1

i p



_ 

G then,

1 ( _i) ( _i )

V  V 

Proof. Since every containment of any pattern is a

concatenation of increasing and/or decreasing pairs by remark 2.6, and also



_i and _i1 have equal number of increasing pairs and equivalently, equal number of decreasing pairs by lemma 2.4, then the proposition follows.

 Proposition 3.9. Suppose 1

i p



_ 

G ,

1( p 1) 0

V_  _ 

Proof. Every 1

1

p p





 G is an almost decreasing sequence and the pattern



₁ is a completely increasing sequence. Thus, there does not exist a subsequence that is almost increasing in an almost decreasing sequence. Therefore,

1( p 1) 0.

V   

Hence the result.

 Below is an algorithm that validates proposition 3.9.

Algorithm 3: Algorithm to show that the involution of

any Γ1 non-deranged permutation avoids



1

Input: Input a n y 1 1

p p





 G i n t o a n a r ra y.

Output: Zero 1: Begin

2: Set integer c = 0 3: for i = 1 to n − 1

4: for j = i + 1 to n − 2

5: for k = j + 1 to n

6: if a[j] < a[k]

7: if a[i] < a[j]

8: Increment c by 1 9: end if

10: end if 11: end for 12: end for 13: end for 14: End

Proposition 3.10. Suppose 1

i p



_ 

G then,

2( 1) 0

V_  

Proof. Given that every 1

1 p



_ 

G is a completely increasing sequence and the pattern



₂is an almost increasing sequence. Then, there does not exist an almost increasing subsequence in a completely increasing sequence. Thus,

2( 1) 0

V   . Hence the

result.

 Below is an algorithm that validates proposition 3.10.

Algorithm 4: Algorithm to show every 1

1 p



_ 

G

avoids



₂

Input: Input any 1

1 p



_ 

G into an array a Output: Zero

1: Begin

2: Set integer c = 0 3: for i = 1 to n − 1

4: for j = i + 1 to n − 2

5: for k = j + 1 to n

6: if a[j] > a[k]

7: if a[k] > a[i]

8: Increment c by 1

9: end for 10: end for

11: end for 12: end if 13: end if 14:End

A permutation ω is sortable through a system of stacks and queues if an identity permutation, 12...n could be obtained by passing ω through the system of stacks and queues exactly once, making very careful decisions in such away the order of the permutation is not made to be more disorderly. Theorem 3.11. [Knu73] A permutation is sortable through a single stack if and only if it avoids the pattern 231.

Lemma 3.12. Suppose 1

i p



_ 

G such that i = 1, p - 1. Then



₁and _p1are stack sortable.

Proof. To show that every



₁and _p1 in 1

p



G are

pattern



₄. Since every



₁ and 1 1

p p





 G are completely increasing and almost decreasing sequences. Then there cannot exist a subsequence that is an almost increasing sequence in a completely increasing and almost decreasing sequence respectively. Thus,

4( i)

V_  will always be zero for i = 1 and p - 1. Hence, by Theorem 3.11, every



₁ and _p1 in 1

p



G are stack sortable. Hence the result.  We end this section by explicitly providing a summary of tables of the number of occurrence



_E for 5 p 23 in 1

p



G (Tables 3 to 6).

Table 3: Number of occurrence of pattern



₁ in ₁-

non deranged permutations for prime size 5 to 23

1



5

G G₇ G₁₁ G₁₃ G₁₇ G₁₉ G₂₃

1



_{10 35 165 286 680 969 1771}

2



_{3 13 70 125 308 444 825}

3



_{3 6 55 76 235 290 623}

4



_{0 13 55 50 140 309 561}

5



_{6 15 61 155 309 426}

6



_{0 70 21 235 119 561}

7



_{22 125 155 221 330}

8



_{22 61 36 164 623}

9



_{15 76 308 45 265}

10



_{0 50 121 444 330}

11



_{21 65 221 66}

12



_{0 121 164 825}

13



_{140 290 353}

14



_{65 100 426}

15



_{36 100 140}

16



_{0 119 353}

17



_{45 170}

18



_{0 265}

19



₁₇₀

20



₁₄₀

21



₆₆

22



₀

TOTAL 16 73 489 952 2800 4353 9289

Table 4: Number of occurrence of



₂ in ₁ - non

deranged permutation of prime size 5 to 23

2



5

G G₇ G₁₁ G₁₃ G₁₇ G₁₉ G₂₃

1



0 0 0 0 0 0 0

2



4 10 35 56 120 165 286

3



4 13 40 70 140 203 336

4



6 10 40 76 160 200 350

5



13 50 74 158 200 375

6



15 35 80 140 230 350

7



49 56 158 218 389

8



49 74 168 227 336

9



50 70 120 228 396

10



45 76 164 165 389

11



80 170 218 385

12



66 164 227 286

13



160 203 386

14



170 233 375

15



168 233 399

16



120 230 386

17



228 401

18



153 396

19



401

20



399

21



385

22



231

Table 5: Number of occurrence of



_E in ₁ - non

deranged permutation of prime size 5 to 23

E



5

G G₇ G₁₁ G₁₃ G₁₇ G₁₉ G₂₃

1



_{0 0 0 0 0 0 0}

2



_{1 4 20 35 84 120 220}

3



_{1 4 22 40 95 140 252}

4



_{0 4 22 40 100 140 260}

5



_{4 20 41 101 140 270}

6



_{0 20 35 95 140 260}

7



_{22 35 101 146 272}

8



_{22 41 84 146 252}

9



_{20 40 84 120 270}

10



_{0 40 101 120 272}

11



_{35 95 146 220}

12



_{0 101 146 220}

13



_{100 140 272}

14



_{95 140 270}

15



_{84 140 252}

16



_{0 140 272}

17



_{120 260}

18



_{0 270}

19



₂₆₀

20



₂₅₂

21



₂₂₀

22



₀

TOTAL 2 0 2 382 1320 1702 5096

Table 6: Number of occurrence of



₆ in ₁ - non

deranged permutation of prime size 5 to 23

6



G5 G7 G11 G13 G17 G19 G23

1



_{0 0 0 0 0 0 0}

2



_{0 0 0 0 0 0 0}

3



_{0 4 4 20 20 56 56}

4



_{4 0 4 40 80 40 80}

5



_{4 40 28 64 40 160}

6



_{20 0 80 20 200 80}

7



_{28 0 64 92 236}

8



_{28 28 224 140 56}

9



_{40 20 0 336 300}

10



_{120 40 92 0 236}

11



_{80 160 92 660}

12



_{220 92 140 0}

13



_{80 56 216}

14



_{160 216 160}

15



_{224 216 476}

16



_{560 200 216}

17



_{336 420}

18



_{816 300}

19



₄₂₀

20



₄₇₆

21



₆₆₀

22



₁₅₄₀

4. CONCLUSION

In this research, we introduced algorithms to examine all length-3 classical patterns in 1

p



G .

Implementing the algorithms, we have shown that pattern



₃,



₄ and



₅ are equipopular. We have experimentally determined the popularity of all patterns of length-3 in 1

p



G by computing the occurrence of all patterns of length-3 in ₁- non deranged permutation. We computed for prime size 5 to 23 and observed that pattern



₁is the most popular while the least popular are the equipopular patterns in 1

p



G . The algorithm further shows that for

any 1

i p



_ 

G , the number of occurrence of a pattern in



_i will be equal to the number of occurrence of the same pattern in its inverse. Based on the facts obtained after implementing the algorithm, it is sufficient to conclude that the entire results coincide algebraically.

REFRENCES

[AEA17] Aremu K. O., Ejima O., Abdullahi M. S. - On the fuzzy ₁-non deranged

permutation group 1

p



G , Asian Journal of Mathematics and Computer Research, 18(4) 2017, pp. 152-157.

[AOA16] Aminu I. A., Ojonugwa E., Aremu K. O. - On the Representations of ₁- Non

deranged Permutation Group G_p ,

Advances in Pure Mathematics, 2016, 6, (09), pp. 608-614.

[Kit11] Kitaev S. - Patterns in permutations

and words. Springer Science &

Business Media, 2011.

[Knu73] Knuth D. E. - Fundamental algorithms: the art of computer

programming. 1973.

[Rud13] Rudolph K. - Pattern popularity in

132-avoiding permutations, The

Electronic Journal of Combinatorics 20(1). 2013.

[Ste10] Steingrımsson E. - Generalized

permutation patterns-a short survey,

Permutation patterns Steve Linton, Nik Ruškuc and Vincent Vatter, Ed. Cambridge University Press 2010, 376, pp. 137-152.

[Zha14] Zhao A. F. - Pattern Popularity in

Multiply Restricted Permutations,