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More Transformations of Integer Sequences

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Further Transformations of Integer Sequences

This page was created by Christian G. Bower and is a sub-page of the On-Line Encyclopedia of Integer Sequences, maintained by N. J. A. Sloane.

File:BluePinLeft.gif Keywords: AFJ, AFK, AGJ, AGK, AIJ, BFJ, BFK, BGJ, BGK, BHJ, BHK, BIJ, BIK, CFJ, CFK, CGJ, CGK, CHJ, CHK, CIJ, CIK, DFJ, DFK, DGJ, DGK, DHJ, DHK, DIJ, DIK, EFJ, EFK, EGJ.

Table of Contents

  1. Definition.
  2. Algorithms.
  3. Catalogue of Sequences.
  4. Back to Integer Sequences.

Part 1: Definition

This is a generalization of transforms that count the ways objects can be partitioned.

Say we have boxes of different colors and sizes.

The sequence {an;n>=1} represents the number of colors a box holding n balls can be. The transformed sequence {bn;n>=1} represents the number of ways we can have a collection of boxes so that the total number of balls is n, subject to the following rules.

The boxes are ordered in one of the following ways:

A. Linear (ordered)
The boxes are in a line from beginning to end.

B. Linear with turning over (reversible)
The boxes are in a line that can be read in either direction.

C. Circular (necklace)
The boxes are in a circle.

D. Circular with turning over (bracelet)
The boxes are in a circle that can be read in either direction.

E. None (unordered)
The order of the boxes is not important.

One of the following distinctness rules applies:

F. Size
No two boxes are the same size.

G. Element
No two boxes are the same size and color.

H. Identity
Any two boxes can be distinguished by size, color and position.

I. None (indistinct)
No restriction.

Distinctness H (identity) has different implications depending on the chosen order.

  • If order A is chosen, distinctness H is the same as distinctness I.
  • If order B is chosen, the boxes cannot form a palindrome of length greater than one.
    Red 1 Blue 2 Red 1 is not allowed.
  • If order C is chosen, the sequence of boxes is aperiodic. It cannot be a repitition of a shorter subsequence.
    Red 1 Blue 2 Red 1 Blue 2 is not allowed.
  • If order D is chosen, the boxes are aperiodic and cannot be a palindrome of length greater than two.
  • If order E is chosen, distinctness H is the same as distinctness G.
One of the following labelling rules applies:

J. Labeled
The balls in the boxes are labeled.

K. Unlabeled
The balls in the boxes are not labeled.

Each transform is identified by a 3 letter code, e.g. BGJ to represent linear order with turning over, each object distinct, labeled.
An X is a wild card as in CXK, unlabeled necklace transforms.

AIK is the transform INVERT.
EGK is the transform WEIGH.
EIJ is the transform EXP.
EIK is the transform EULER.

There are 5×4×2=40 of these transforms.

However, the AHX and EHX transforms are redundant, leaving 36. Four of them are named. As far as I know, the other 32 are not. The new and old sequence listed illustrate the 32 new transforms.

Terminology:

  • XXXk means the transform XXX with exactly k boxes.
    These are denoted by XXX[k] in the On-Line Encyclopedia of Integer Sequences.
    AIK2 is the transform CONV.
  • Bracelet means necklace that can be turned over. More information about necklaces.
  • Compound windmill is a rooted planar tree where the sub-rooted tree extending from a node can be rotated independently of the rest of the tree. Much like some children's toys or carnival rides. Compound windmills can be dyslexic.
  • Dyslexic planar tree is a planar tree where each sub-rooted tree extending from a node can be read from left to right or right to left. It can be thought of as viewed by an observer who does not know left from right or as sub-rooted trees that can be turned around independent of the rest of the tree.
  • Eigensequence means a sequence that is stable under a given transform or is modified in some simple way. Eigensequences are covered in detail in: M. Bernstein & N. J. A. Sloane, Some canonical sequences of integers, Linear Algebra and its Applications, 226-228 (1995), 57-72.A000081, rooted trees, 1,1,2,4,9,20,48,115... is an eigensequence of the transform EULER. because the transformed sequence, 1,2,4,9,20,48,115,286,..., is the original sequence shifted left one place.
  • Identity bracelet means bracelet where every bead is distinguished by position and color, i.e. a bracelet generated by the transform DHK.

Part 2: Algorithms

an is the input sequence.

bn is the output sequence.

A(x) is the generating function of an.

B(x) is the generating function of bn.

(XXX a)n = sum{k=1 to n} (XXXk a)n

MÖBIUS · XXX refers to the Möbius transform of the sequence transformed by XXX. Similarly for MÖBIUS-1 · XXX. However, (MÖBIUS · XXX)k and (MÖBIUS-1 · XXX)k are defined as follows:

(MÖBIUS · XXX)kan = sum{d|k and d|n} (µ(d) × XXXk/dan/d)

(MÖBIUS-1 · XXX)kan = sum{d|k and d|n} (XXXk/dan/d)

AIK = INVERT
B(x) = A(x) / (1-A(x))

AIKk
B(x) = A(x)k

LPALk (Linear palindrome)
If n, k even: bn = (AIKk/2a)n/2
If n odd, k even: bn = 0
If n even, k odd: bn = sum{i>0 and i<n/2} (a2i × (AIK(k-1)/2a)n/2-i)
if n,k odd: bn = sum{i>0 and i<n/2} (a2i-1 × (AIK(k-1)/2a)(n+1)/2-i)

BIKk
bn = ((AIKka)n + (LPALka)n) / 2

BHKk
k=1: bn = an
k>1: bn = ((AIKka)n - (LPALka)n) / 2

CHKk
bn = (MÖBIUS · AIK)kan / n

CIK
CIK = MÖBIUS-1 · CHK

CPALk (Circular palindrome)
CPAL1 = IDENTITY
CPAL2 = CIK2
k>2:
If n, k even: bn = (I+J)/2+K+L+M where:
(No boxes joined)
I=(AIKk/2a)n/2
(2 boxes joined are identical)
J=sum{i=1 to n/2}(AIKk/2-1a)(n-2i)/2
(2 boxes joined are even and different sizes)
K=sum{i,j even, j>i, i+j<n} (ai × aj × (AIKk/2-1a)(n-i-j)/2)
(2 boxes joined are odd and different sizes)
L=sum{i,j odd, j>i, i+j<n} (ai × aj × (AIKk/2-1a)(n-i-j)/2)
(2 boxes joined are the same size and different colors)
M=sum{i>0 and i<n/2} ((ai2-ai)/2 × (AIKk/2-1a)(n-2i)/2)
If n odd, k even:
bn = sum{i odd, j even, i+j<n} (ai × aj × ((AIKk/2-1a)(n-i-j)/2)
If n even, k odd:
bn = sum{i>0 and i<n/2} (a2i × (AIK(k-1)/2a)n/2-i)
if n,k odd:
bn = sum{i>0 and i<n/2} (a2i-1 × (AIK(k-1)/2a)(n+1)/2-i)

DIKk
bn = ((CIKka)n + (CPALka)n) / 2

DHKk
DHK1 = IDENTITY
DHK2 = CHK2
For k>2:
DHKk = (MÖBIUS · (CIK - CPAL)/2)k

If EXX is one of: {EFJ, EFK, EGJ, EGK, EIJ} then:
AXXk = k! × EXXk
BXXk = max(1,k!/2) × EXXk
CXXk = (k-1!) × EXXk
DXXk = max(1,(k-1)!/2) × EXXk

To calculate (EFXka)n, enumerate the distinct partitions of n into k parts as terms of the following form:
p1+p2+...+pk
Sum the terms calculated as follows:
EFJk: prod{i=1 to k}api × n! / prod{i=1 to k}pi!
EFKk: prod{i=1 to k}api

EFK can also be calculated as:
B(x)=prod{k=1 to infinity}(1+akxk).

To calculate (AIJka)n, (BHJka)n, (CHJka)n or (EGXka)n, enumerate the partitions of of n into k parts as terms of the following form:
p1q1+p2q2+...+pjqj where all the pi's are distinct.
Sum the terms calculated as follows:
AIJk: prod{i=1 to j}apiqi × n! × k! / ((prod{i=1 to j}pi!qi) × (prod{i=1 to j}qi!))
BHJk:
term1 = prod{i=1 to j}apiqi × k! / (prod{i=1 to j}qi!)
term2 = prod{i=1 to j}api[qi/2] × [k/2]! / (prod{i=1 to j}[qi/2]!)
If more than 1 qi is odd: term3 = term1
otherwise: term3 = term1 - term2
term = term3 × n! / prod{i=1 to j}pi!qi / 2
CHJk:
term2 = sum{d|qm for all m} (µ(d) × prod{i=1 to j}api[qi/d] × [k/d]! / (prod{i=1 to j}[qi/d]!))
term = term2 × n! / prod{i=1 to j}pi!qi / k
EGJk: prod{i=1 to j}C(api,qi) × n! / prod{i=1 to j}pi!qi
EGKk: prod{i=1 to j}C(api,qi)

DHJ:
Work is in progress.


Part 3: Catalogue of sequences

This table identifies a formula for each sequence, usually based on one of the transforms. This should provide a convenient way to browse the sequences and see how the transforms apply to a broad class of mathematics.

The base sequences:

These transforms have been applied to one of the base sequences defined in the following table or to sequences in the On-line Encyclopedia of Integer Sequences, identified by number.

s1, s2, s3... sk1 = k, skn=0 for n>1
all1, all2, all3,... allkn = k for all n
codd (characteristic of odd) coddn = 1 if n is odd, 0 otherwise
noone noone1=0, noonen=1 for n>1
twoone twoone1=2, twoonen=1 for n>1
iden idenn=n
odd oddn=2n-1
even evenn=2n

If T is a transform:

Left(n;k1, k2,..., kn)T is the eigensequence that shifts left n places under T and has ai=ki for 1<=i<=n.
M2(n)T is the eigensequence that doubles the terms whose indices are greater than 1 under T.

AFJ sequences

A032000

AFJ all2

A032001

AFJ twoone

A032002

AFJ iden

A032003

AFJ odd

A032004

Left(1;1)AFJ

AFK sequences

A032005

AFK all2

A032006

AFK twoone

A032007

AFK iden

A032008

AFK odd

A032009

Left(1;1)AFK

A032010

(CFK A032009 )n-1

AGJ sequences

A032011

AGJ all1

A032012

AGJ codd

A032013

AGJ noone

A032014

AGJ all2

A032015

AGJ twoone

A032016

AGJ iden

A032017

AGJ odd

A032018

Left(1;1)AGJ

A032019

M2(2)AGJ

AGK sequences

A032020

AGK all1

A032021

AGK codd

A032022

AGK noone

A032023

AGK all2

A032024

AGK twoone

A032025

AGK iden

A032026

AGK odd

A032027

Left(1;1)AGK

A032028

(CGK A032027 )n-1

A032029

Left(2;1,1)AGK

A032030

M2(2)AGK

AIJ sequences

A000142

AIJ s1

A000165

AIJ s2

A032031

AIJ s3

A000670

AIJ all1

A000918

AIJ2 all1

A001117

AIJ3 all1

A000919

AIJ4 all1

A001118

AIJ5 all1

A000920

AIJ6 all1

A006154

AIJ codd

A032032

AIJ noone

A004123

AIJ all2

A006155

AIJ twoone

A032033

AIJ all3

A006153

AIJ iden

A000354

AIJ odd

A001147

Left(1;1)AIJ

A032034

Left(1;2)AIJ

A032035

Left(2;1,1)AIJ

A032036

Left(3;1,1,1)AIJ

A032037

M2(1)AIJ

BFJ sequences

A032038

BFJ all2

A032039

BFJ twoone

A032040

BFJ iden

A032041

BFJ odd

A032042

Left(1;1)BFJ

BFK sequences

A032043

BFK all2

A032044

BFK twoone

A032045

BFK iden

A032046

BFK odd

A032047

Left(1;1)BFK

A032048

(CFK A032047 )n-1

BGJ sequences

A032049

BGJ all1

A032050

BGJ codd

A032051

BGJ noone

A032052

BGJ all2

A032053

BGJ twoone

A032054

BGJ iden

A032055

BGJ odd

A032056

Left(1;1)BGJ

A032057

M2(2)BGJ

BGK sequences

A032058

BGK all1

A032059

BGK codd

A032060

BGK noone

A032061

BGK all2

A032062

BGK twoone

A032063

BGK iden

A032064

BGK odd

A032065

Left(1;1)BGK

A032066

(CGK A032065 )n-1

A032067

Left(2;1,1)BGK

A032068

M2(2)BGK

BHJ sequences

A032069

BHJ s2

A032070

BHJ s3

A032071

BHJ s4

A032072

BHJ s5

A032073

BHJ all1

A032074

BHJ codd

A032075

BHJ noone

A032076

BHJ all2

A032077

BHJ twoone

A032078

BHJ all3

A032079

BHJ iden

A032080

BHJ odd

A032081

Left(1;1)BHJ

A032082

Left(1;2)BHJ

A032083

Left(2;1,1)BHJ

A032084

M2(2)BHJ

BHK sequences

A032085

BHK s2

A032086

BHK s3

A032087

BHK s4

A032088

BHK s5

A032089

BHK codd

A032090

BHK noone

A002620

(BHK3 all1)n+2

A006584

(BHK4 all1)n+2

A032091

BHK5 all1

A032092

BHK6 all1

A032093

BHK7 all1

A032094

BHK8 all1

A032095

(BHKn all1)2n-1

A032096

BHK all2

A032097

BHK twoone

A032098

BHK all3

A032099

BHK iden

A032100

BHK odd

A032101

Left(1;1)BHK

A032102

(DHK A032101 )n-1

A032103

Left(1;2)BHK

A032104

Left(1;1,1)BHK

A032105

M2(2)BHK

A032106

(BHKn all1)2n

BIJ sequences

A001710

BIJ s1

A032107

BIJ s2

A032108

BIJ s3

A032109

BIJ all1

A009568

(-1)n+1 × BIJ codd

A032110

BIJ noone

A032111

BIJ all2

A032112

BIJ twoone

A032113

BIJ all3

A032114

BIJ iden

A032115

BIJ odd

A032116

Left(1;1)BIJ

A032117

Left(1;2)BIJ

A032118

Left(2;1,1)BIJ

A032119

M2(1)BIJ

BIK sequences

A005418

(BIK s2)n-1

A005418

BIK all1

A032120

BIK s3

A032121

BIK s4

A032122

BIK s5

A001224

(BIK codd)n+1

A001224

(BIK noone)n+2

A002620

(BIK3 all1)n+1

A005993

(BIK4 all1)n+4

A005994

(BIK5 all1)n+5

A005995

(BIK6 all1)n+6

A018210

(BIK7 all1)n+7

A018211

(BIK8 all1)n+8

A018212

(BIK9 all1)n+9

A018213

(BIK10 all1)n+10

A018214

(BIK11 all1)n+11

A032123

(BIKn all1)2n-1

A005654

(BIKn all1)2n

A005656

(BIKn-3 all1)2n-3

A032124

BIK all2

A032125

BIK all3

A005207

BIK twoone

A032126

BIK iden

A032127

BIK odd

A032128

Left(1;1)BIK

A032129

(DIK A032128 )n-1

A032130

Left(1;2)BIK

A032131

Left(2;1,1)BIK

A032132

M2(1)BIK

A032133

M2(2)BIK

CFJ sequences

A032134

CFJ all2

A032135

CFJ twoone

A032136

CFJ iden

A032137

CFJ odd

A032138

Left(1;1)CFJ

CFK sequences

A032139

CFK all2

A032140

CFK twoone

A032141

CFK iden

A032142

CFK odd

A032143

Left(1;1)CFK

CGJ sequences

A032144

CGJ all1

A032145

CGJ codd

A032146

CGJ noone

A032147

CGJ all2

A032148

CGJ twoone

A032149

CGJ iden

A032150

CGJ odd

A032151

Left(1;1)CGJ

A032152

M2(2)CGJ

CGK sequences

A032153

CGK all1

A032154

CGK codd

A032155

CGK noone

A032156

CGK all2

A032157

CGK twoone

A032158

CGK iden

A032159

CGK odd

A032160

Left(1;1)CGK

A032161

Left(1;2)CGK

A032162

Left(2;1,1)CGK

A032163

M2(2)CGK

CHJ sequences

A032321

CHJ s2

A032322

CHJ s3

A032323

CHJ s4

A032264

CHJ s5

A032325

CHJ all1

A032326

CHJ codd

A032327

CHJ noone

A032328

CHJ all2

A032329

CHJ twoone

A032330

CHJ all3

A032331

CHJ iden

A032332

CHJ odd

A032333

Left(1;1)CHJ

A032334

Left(1;2)CHJ

A032335

Left(2;1,1)CHJ

A032336

M2(2)CHJ

CHK sequences

A001037

CHK s2

A001037

(CHK all1) + s1

A027376

CHK s3

A027376

(CHK all2) + s1

A027376

(CHK odd) + s2

A027377

CHK s4

A027377

(CHK all3) + s1

A001692

CHK s5

A027378

CHK s5

A032164

CHK s6

A001693

CHK s7

A027379

CHK s7

A027380

CHK s8

A027381

CHK s9

A032165

CHK s10

A032166

CHK s11

A032167

CHK s12

A006206

(CHK codd) + CHAR({2})

A006206

(CHK noone) + s1

A001840

(CHK3 all1)n+4

A006918

(CHK4 all1)n+4

A011795

(CHK5 all1)n+1

A011796

(CHK6 all1)n+6

A011797

(CHK7 all1)n+1

A031164

(CHK8 all1)n+9

A011845

CHK9 all1

A032168

CHK10 all1

A032169

CHK11 all1

A000108

(CHKn+1 all1)2n+1

A022553

(CHKn+1 all1)2n+2

A022553

(CHK A000108 )n-1

A032170

CHK iden

A032170

CHK twoone + s1

A032171

Left(1;1)CHK

A032172

Left(1;2)CHK

A032173

Left(2;1,1)CHK

A032174

M2(2)CHK

A032175

CHK A004111

A032176

WEIGH A032175

A032177

A032176 - A004111

A032178

WEIGH A032177

CIJ sequences

A000142

(CIJ s1)n+1

A000165

(CIJ s2)n+1 × 2

A032179

CIJ s3

A000629

CIJ all1

A000225

(CIJ2 all1)n+1

A028243

CIJ3 all1

A028244

CIJ4 all1

A028245

CIJ5 all1

A032180

CIJ6 all1

A003704

(-1)n+1 × (CIJ codd)

A032181

CIJ noone

A027882

CIJ all2

A032182

CIJ twoone

A032183

CIJ all3

A009444

(-1)n+1 × (CIJ iden)

A032184

CIJ odd

A029768

Left(1;1)CIJ

A032185

Left(1;2)CIJ

A032186

Left(2;1,1)CIJ

A032187

Left(3;1,1,1)CIJ

A032188

M2(1)CIJ

CIK sequences

A000031

CIK s2

A000031

(CIK all1) + all1

A008965

CIK all1

A008965

(CIK s2) - all1

A001867

CIK s3

A001867

(CIK all2) + all1

A001868

CIK s4

A001868

(CIK all3) + all1

A001869

CIK s5

A001869

(CIK all4) + all1

A032189

CIK codd

A032190

CIK noone

A000358

(CIK noone) + all1

A007997

(CIK3 all1)n+3

A008610

(CIK4 all1)n+4

A008646

(CIK5 all1)n+5

A032191

CIK6 all1

A032192

CIK7 all1

A032193

CIK8 all1

A032194

CIK9 all1

A032195

CIK10 all1

A032196

CIK11 all1

A032197

CIK12 all1

A000108

(CHKn+1 all1)2n+1

A003239

(CIKn-1 all1)2n-2

A003239

(CIK A000108 n-1)n-1

A005594

CIK twoone

A032198

CIK iden

A032199

CIK odd

A032200

Left(1;1)CIK

A032201

Left(1;2)CIK

A032202

Left(2;1,1)CIK

A032203

M2(1)CIK

A032204

M2(2)CIK

A002861

CIK A000081

A027852

CIK2 A000081

A029852

CIK3 A000081

A029853

CIK4 A000081

A029868

CIK5 A000081

A029869

CIK6 A000081

A029870

CIK7 A000081

A029871

CIK8 A000081

A032205

CIK9 A000081

A032206

CIK10 A000081

A032207

CIK11 A000081

A032208

CIK12 A000081

DFJ sequences

A032209

DFJ all2

A032210

DFJ twoone

A032211

DFJ iden

A032212

DFJ odd

A032213

Left(1;1)DFJ

DFK sequences

A032214

DFK all2

A032215

DFK twoone

A032216

DFK iden

A032217

DFK odd

A032218

Left(1;1)DFK

DGJ sequences

A032219

DGJ all1

A032220

DGJ codd

A032221

DGJ noone

A032222

DGJ all2

A032223

DGJ twoone

A032224

DGJ iden

A032225

DGJ odd

A032226

Left(1;1)DGJ

A032227

M2(2)DGJ

DGK sequences

A032228

DGK all1

A032229

DGK codd

A032230

DGK noone

A032231

DGK all2

A032232

DGK twoone

A032233

DGK iden

A032234

DGK odd

A032235

Left(1;1)DGK

A032236

Left(1;2)DGK

A032237

Left(2;1,1)DGK

A032238

M2(2)DGK

DHJ sequences

A032337

DHJ s2

A032338

DHJ s3

A032339

DHJ s4

A032340

DHJ s5

DHK sequences

A032239

DHK s2

A032240

DHK s3

A032241

DHK s4

A032242

DHK s5

A032243

DHK codd

A032244

DHK noone

A032245

DHK all1

A001399

(DHK3 all1)n+6

A018845

(DHK3 all1)n+6

A026809

(DHK3 all1)n+3

A008804

(DHK4 all1)n+7

A032246

DHK5 all1

A032247

DHK6 all1

A032248

DHK7 all1

A032249

DHK8 all1

A032250

(DHKn all1)2n

A032251

DHK all2

A032252

DHK twoone

A032253

DHK all3

A032254

DHK iden

A032255

DHK odd

A032256

Left(1;1)DHK

A032257

Left(1;2)DHK

A032258

Left(2;1,1)DHK

A032259

M2(2)DHK

A032260

(DHKn all1)2n-1

DIJ sequences

A001710

(DIJ s1)n+1

A000165

(DIJ s2)n+1 - s2

A032261

DIJ s3

A032262

DIJ all1

A000225

(DIJ2 all1)n+1

A000392

DIJ3 all1

A032263

DIJ4 all1

A032264

DIJ codd

A032265

DIJ noone

A032266

DIJ all2

A032267

DIJ twoone

A032268

DIJ all3

A032269

DIJ iden

A032270

DIJ odd

A032271

Left(1;1)DIJ

A032272

Left(1;2)DIJ

A032273

Left(2;1,1)DIJ

A032274

M2(1)DIJ

DIK sequences

A000029

DIK s2

A000029

(DIK all1) + all1

A027671

DIK s3

A032275

DIK s4

A032276

DIK s5

A032277

DIK codd

A032278

DIK noone

A001399

(DIK3 all1)n+3

A018845

(DIK3 all1)n+3

A026809

DIK3 all1

A005232

DIK4 all1

A032279

DIK5 all1

A005513

DIK6 all1

A032280

DIK7 all1

A005514

DIK8 all1

A032281

DIK9 all1

A005515

DIK10 all1

A032282

DIK11 all1

A005516

DIK12 all1

A005648

(DIKn all1)2n

A007123

(DIKn all1)2n-1

A032283

DIK all2

A032284

DIK all3

A032285

DIK all4

A032286

DIK all5

A005595

DIK twoone

A032287

DIK iden

A032288

DIK odd

A032289

Left(1;1)DIK

A032290

Left(1;2)DIK

A032291

Left(2;1,1)DIK

A032292

M2(1)DIK

A032293

M2(2)DIK

A001371

MÖBIUS A000029

A032294

MÖBIUS A027671

A032295

MÖBIUS A032275

A032296

MÖBIUS A032276

EFJ sequences

A032297

EFJ all2

A032298

EFJ twoone

A032299

EFJ iden

A032300

EFJ odd

A032301

Left(1;1)EFJ

EFK sequences

A032302

EFK all2

A032303

EFK twoone

A022629

EFK iden

A032304

EFK odd

A032305

Left(1;1)EFK

A032306

Left(1;2)EFK

A032307

Left(2;1,1)EFK

A032308

EFK all3

A032309

EFK even

EGJ sequences

A007837

EGJ all1

A032310

EGJ codd

A032311

EGJ noone

A032312

EGJ all2

A032313

EGJ twone

A032314

EGJ all3

A032315

EGJ iden

A032316

EGJ odd

A032317

Left(1;1)EGJ

A032318

Left(1;2)EGJ

A032319

Left(2;1,1)EGJ

A032320

M2(2)EGJ