aarjaneiro/planetmath huggingface dataset:
consequence operator determined by a class of subsets

consequence operator determined by a class of subsets

Theorem 1

.

Let

L

be a set and let

K

be a subset of

P

⁢

(

L

)

. The the

mapping

C

:

P

⁢

(

L

)

→

P

⁢

(

L

)

defined as

C

⁢

(

X

)

=

∩

{

Y

∈

K

∣

X

⊆

Y

}

is a

consequence operator

.

Proof.

We need to check that

C

satisfies

the

defining properties

.

Property

1:

Since every

element of

the set

{

Y

∈

K

∣

X

⊆

Y

}

contains

X

, we have

X

⊆

C

⁢

(

X

)

.

Property 2:

For every

element

Y

of

K

such that

X

⊆

Y

, it also is the case that

C

⁢

(

X

)

⊆

Y

because an

intersection

of a family of sets is a subset of
any member of the family. In other words (or rather, symbols),

{

Y

∈

K

∣

X

⊆

Y

}

⊆

{

Y

∈

K

∣

C

⁢

(

X

)

⊆

Y

}

,

hence

C

⁢

(

C

⁢

(

X

)

)

⊆

C

⁢

(

X

)

. By the first property proven above,

C

⁢

(

X

)

⊆

C

⁢

(

C

⁢

(

X

)

)

so

C

⁢

(

C

⁢

(

X

)

)

=

C

⁢

(

X

)

. Thus,

C

∘

C

=

C

.

Property 3:

Let

X

and

Y

be two subsets of

L

such that

X

⊆

Y

. Then if,
for some other subset

Z

of

L

, we have

Y

⊂

Z

, it follows that

X

⊂

Z

. Hence,

{

Z

∈

K

∣

Y

⊆

Z

}

⊆

{

Z

∈

K

∣

X

⊆

Z

}

,

so

C

⁢

(

X

)

⊆

C

⁢

(

Y

)

.

∎

Title

consequence operator determined by a class of subsets

Canonical

name

ConsequenceOperatorDeterminedByAClassOfSubsets

Date of creation

2013-03-22 16:29:45

Last modified on

2013-03-22 16:29:45

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

10

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03G25

Classification

msc 03G10

Classification

msc 03B22

Generated on Thu Feb 8 20:29:14 2018 by

LaTeXML
superfluity of the third defining property for finite consequence operator

superfluity of the third defining property for finite consequence operator

In this entry, we demonstrate the claim made in

section

1 of the

http://planetmath.org/node/8646

parent entry that the defining conditions for

finitary consequence operator

given there are redundant because
one of them may be derived from the other two.

Theorem

.

Let

L

be a set. Suppose that a

mapping

C

:

P

⁢

(

L

)

→

P

⁢

(

L

)

satisfies

the following three

properties

:

1.

For all

X

⊆

L

, it happens that

X

⊆

C

⁢

(

X

)

.

2.

C

∘

C

=

C

3.

For all

X

∈

L

, it happens that

C

⁢

(

X

)

=

⋃

Y

∈

ℱ

⁢

(

X

)

C

⁢

(

Y

)

.

Then

C

also satisfies the following property: For all

X

,

Y

⊆

L

, if

X

⊆

Y

, then

C

⁢

(

X

)

⊆

C

⁢

(

Y

)

.

Title

superfluity of the third

defining property

for finite consequence operator

Canonical

name

SuperfluityOfTheThirdDefiningPropertyForFiniteConsequenceOperator

Date of creation

2013-03-22 16:30:13

Last modified on

2013-03-22 16:30:13

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 03G25

Classification

msc 03G10

Classification

msc 03B22

Generated on Thu Feb 8 20:29:21 2018 by

LaTeXML
G.W. Leibniz’ s quote

G.W. Leibniz’ s quote

0.1

“Does Mathematics need any Philosophy ?”

0.1.1

Famous quote from Gottfried Wilhelm (von) Leibniz – both mathematician and philosopher:

A liberal

translation

from the original in French quoted below is:

1.

Without Mathematics one cannot understand the fundamentals of Philosophy.

2.

Without Philosophy we cannot reach the

Foundation of Mathematics

.

3.

Without both (Mathematics and Philosophy) one cannot reach anything that is fundamental.

1.

“Sans les mathématiques on ne pénétre point au fond de la philosophie.

2.

Sans la philosophie on ne pénétre point au fond des mathématiques.

3.

Sans les deux on ne pénétre au fond de rien.”

-by G. W. Leibniz-

References

http://www.dpmms.cam.ac.uk/ wtg10/

Timothy Gowers,

“Does mathematics need a philosophy?”

http://www.dpmms.cam.ac.uk/ wtg10/philosophy.html

Plato vs.

Formalism

,

Logicism

,… and so on

1

Title

G.W. Leibniz’ s quote

Canonical name

GWLeibnizSQuote

Date of creation

2013-03-22 18:26:10

Last modified on

2013-03-22 18:26:10

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

9

Author

bci1 (20947)

Entry type

Topic

Classification

msc 03G30

Classification

msc 03B50

Classification

msc 03B15

Related topic

PlatosMathematics

Related topic

FormalLogicsAndMetaMathematics

Generated on Thu Feb 8 20:29:26 2018 by

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biextensional collapse

biextensional collapse

If

𝒞

=

(

𝒜

,

r

,

𝒳

)

is a

Chu space

, we can define the

biextensional collapse

of

𝒞

to be

(

r

^

⁢

[

A

]

,

r

′

,

r

ˇ

⁢

[

X

]

)

where

r

′

⁢

(

r

^

⁢

(

a

)

,

r

ˇ

⁢

(

x

)

)

=

r

⁢

(

a

,

x

)

.

That is, to name the rows of the biextensional collapse, we just use

functions

representing the actual rows of the original Chu space (and similarly for the columns). The effect is to merge

indistinguishable

rows and columns.

We say that two Chu spaces are

equivalent

if their biextensional collapses are

isomorphic

.

Title

biextensional collapse

Canonical name

BiextensionalCollapse

Date of creation

2013-03-22 13:04:54

Last modified on

2013-03-22 13:04:54

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03G99

Defines

equivalent Chu space

Generated on Thu Feb 8 20:29:27 2018 by

LaTeXML
Chu space

Chu space

A

Chu space

over a set

Σ

is a triple

(

𝒜

,

r

,

𝒳

)

with

r

:

𝒜

×

𝒳

→

Σ

.

𝒜

is called the

carrier

and

𝒳

the

cocarrier

.

Although the definition is symmetrical, in practice asymmetric uses are common. In particular, often

𝒳

is just taken to be a set of

function

from

𝒜

to

Σ

, with

r

⁢

(

a

,

x

)

=

x

⁢

(

a

)

(such a Chu space is called

normal

and is abbreviated

(

𝒜

,

𝒳

)

).

We define the

perp

of a Chu space

𝒞

=

(

𝒜

,

r

,

𝒳

)

to be

𝒞

⟂

=

(

𝒳

,

r

⌣

,

𝒜

)

where

r

⌣

⁢

(

x

,

a

)

=

r

⁢

(

a

,

x

)

.

Define

r

^

and

r

ˇ

to be functions defining the

rows

and

columns

of

𝒞

respectively, so that

r

^

⁢

(

a

)

:

𝒳

→

Σ

and

r

ˇ

⁢

(

x

)

:

𝒜

→

Σ

are given by

r

^

⁢

(

a

)

⁢

(

x

)

=

r

ˇ

⁢

(

x

)

⁢

(

a

)

=

r

⁢

(

a

,

x

)

. Clearly the rows of

𝒞

are the columns of

𝒞

⟂

.

Using these definitions, a Chu space can be represented using a matrix.

If

r

^

is

injective

then we call

𝒞

separable

and if

r

ˇ

is injective we call

𝒞

extensional

. A Chu space which is both separable and extensional is

biextensional

.

Title

Chu space

Canonical name

ChuSpace

Date of creation

2013-03-22 13:04:51

Last modified on

2013-03-22 13:04:51

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

6

Author

Henry (455)

Entry type

Definition

Classification

msc 03G99

Defines

perp

Defines

carrier

Defines

cocarrier

Defines

normal

Defines

normal Chu space

Defines

separable

Defines

extensional

Defines

biextensional

Defines

row

Defines

column

Generated on Thu Feb 8 20:29:30 2018 by

LaTeXML
Chu transform

Chu transform

If

𝒞

=

(

𝒜

,

r

,

𝒳

)

and

𝒟

=

(

ℬ

,

s

,

𝒴

)

are

Chu spaces

then we say a pair of

functions

f

:

𝒜

→

ℬ

and

g

:

𝒴

→

𝒳

form a

Chu transform

from

𝒞

to

𝒟

if for any

(

a

,

y

)

∈

𝒜

×

𝒴

we have

r

⁢

(

a

,

g

⁢

(

y

)

)

=

s

⁢

(

f

⁢

(

a

)

,

y

)

.

Title

Chu transform

Canonical name

ChuTransform

Date of creation

2013-03-22 13:05:03

Last modified on

2013-03-22 13:05:03

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

4

Author

Henry (455)

Entry type

Definition

Classification

msc 03G99

Generated on Thu Feb 8 20:29:31 2018 by

LaTeXML
example of Chu space

example of Chu space

Any set

A

can be represented as a

Chu space

over

{

0

,

1

}

by

(

A

,

r

,

𝒫

⁢

(

A

)

)

with

r

⁢

(

a

,

X

)

=

1

iff

a

∈

X

. This Chu space

satisfies

only the

trivial

property

2

A

, signifying the fact that sets have no internal

structure

. If

A

=

{

a

,

b

,

c

}

then the

matrix representation

is:

{}

{a}

{b}

{c}

{a,b}

{a,c}

{b,c}

{a,b,c}

a

0

1

0

0

1

1

0

1

b

0

0

1

0

1

0

1

1

c

0

0

0

1

0

1

1

1

Increasing the structure of a Chu space, that is, adding properties, is

equivalent

to deleting columns. For

instance

we can delete the columns named

{

c

}

and

{

b

,

c

}

to turn this into the

partial order

satisfying

c

≤

a

. By deleting more columns, we can further increase the structure. For example, if we require that the set of rows be

closed under

the bitwise or

operation

(and delete those columns which would prevent this) then we can it will define a

semilattice

, and if it is closed under both bitwise or and bitwise and then it will define a

lattice

. If the rows are also closed under

complementation

then we have a

boolean algebra

.

Note that these are not arbitrary connections: the

Chu transforms

on each of these classes of Chu spaces correspond to the appropriate notion of

homomorphism

for those classes.

For instance, to see that Chu transforms are order preserving on Chu spaces viewed as partial orders, let

𝒞

=

(

𝒜

,

r

,

𝒳

)

be a Chu space satisfying

b

≤

a

. That is, for any

x

∈

X

we have

r

⁢

(

b

,

x

)

=

1

→

r

⁢

(

a

,

x

)

=

1

. Then let

(

f

,

g

)

be a Chu transform to

𝒟

=

(

ℬ

,

s

,

𝒳

)

, and suppose

s

⁢

(

f

⁢

(

b

)

,

y

)

=

1

. Then

r

⁢

(

b

,

g

⁢

(

y

)

)

=

1

by the definition of a Chu transform, and then we have

r

⁢

(

a

,

g

⁢

(

y

)

)

=

1

and so

s

⁢

(

f

⁢

(

a

)

,

y

)

=

1

, demonstrating that

f

⁢

(

b

)

≤

f

⁢

(

a

)

.

Title

example of Chu space

Canonical name

ExampleOfChuSpace

Date of creation

2013-03-22 13:05:00

Last modified on

2013-03-22 13:05:00

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Example

Classification

msc 03G99

Generated on Thu Feb 8 20:29:34 2018 by

LaTeXML
property of a Chu space

property of a Chu space

A

property

of a Chu space over

Σ

with carrier

𝒜

is some

Y

⊆

Σ

𝒜

. We say that a Chu space

𝒞

=

(

𝒜

,

r

,

𝒳

)

satisfies

Y

if

X

⊆

Y

.

For example, every Chu space satisfies the property

Σ

𝒜

.

Title

property of a Chu space

Canonical

name

PropertyOfAChuSpace

Date of creation

2013-03-22 13:04:57

Last modified on

2013-03-22 13:04:57

Owner

Henry (455)

Last modified by

Henry (455)

Numerical id

5

Author

Henry (455)

Entry type

Definition

Classification

msc 03G99

Defines

satisfies as a Chu space

Generated on Thu Feb 8 20:29:35 2018 by

LaTeXML
infinitesimal

infinitesimal

Let

R

be a real closed field, for example the reals thought of as a

structure

in

L

, the

language

of

ordered rings

. Let

B

be some set
of parameters from

R

. Consider the following set of

formulas

in

L

(

B

):

{

x

<

b

:

b

∈

B

∧

b

>

0

}

Then this set of formulas is finitely satisfied, so by

compactness

is

consistent

. In fact this set of formulas extends to a unique type

p

over

B

, as it defines a

Dedekind cut

. Thus there is some model

M

containing

B

and some

a

∈

M

so that the type of

a

over

B

is

p

.

Any such

element

will be called

B-infinitesimal

. In
particular, suppose

B

=

∅

. Then the

definable

closure

of

B

is the

intersection

of the reals with the

algebraic numbers

.
Then a

∅

-

infinitesimal

(or simply

infinitesimal

) is
any element of any real closed field that is

positive

but smaller than
every real

algebraic

(positive) number.

As noted above such models exist, by compactness. One can construct
them using

ultraproducts

; see the entry “Hyperreal (

http://planetmath.org/Hyperreal

)” for more
details. This is due to
Abraham Robinson, who used such fields to formulate

nonstandard
analysis

.

Let

K

be any ordered ring. Then

K

contains

𝐍

.
We say

K

is

archimedean

if and only if for every

a

∈

K

there is some

n

∈

𝐍

so that

a

<

n

. Otherwise

K

is

non-archimedean

.

Real closed fields with infinitesimal elements are non-archimedean:
for any infinitesimal

a

we have

a

<

1

/

n

and thus

1

/

a

>

n

for each

n

∈

𝐍

.

References

1

Robinson, A.,

Selected papers of Abraham
Robinson. Vol. II. Nonstandard analysis and philosophy

, New Haven,
Conn., 1979.

Title

infinitesimal

Canonical name

Infinitesimal

Date of creation

2013-03-22 13:22:59

Last modified on

2013-03-22 13:22:59

Owner

mps (409)

Last modified by

mps (409)

Numerical id

12

Author

mps (409)

Entry type

Definition

Classification

msc 03H05

Classification

msc 06F25

Classification

msc 03C64

Related topic

Hyperreal

Defines

infinitesimal

Defines

Archimedean

Generated on Thu Feb 8 20:29:37 2018 by

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counting compositions of an integer

counting compositions of an integer

A

composition

of a

nonnegative integer

n

is a

sequence

(

a

1

,

…

,

a

k

)

of

positive integers

with

∑

a

i

=

n

. Denote by

C

n

the number of compositions of

n

, and denote by

S

n

the set of those compositions. (Note that this is a very different - and simpler -

concept

than the number of

partitions

of an

integer

; here the order (

http://planetmath.org/PartialOrder

) matters).

For some small values of

n

, we have

C

0

=

1

C

1

=

1

C

2

=

2

(

2

)

,

(

1

,

1

)

C

3

=

4

(

3

)

,

(

1

,

2

)

,

(

2

,

1

)

,

(

1

,

1

,

1

)

In fact, it is easy to see that

C

n

=

2

⁢

C

n

-

1

for

n

>

1

: each composition

(

a

1

,

…

,

a

k

)

of

n

-

1

can be associated with two different compositions of

n

(

a

1

,

a

2

,

…

,

a

k

,

1

)

(

a

1

,

a

2

,

…

,

a

k

+

1

)

We thus get a map

φ

:

S

n

-

1

×

{

0

,

1

}

→

S

n

given by

φ

⁢

(

(

a

1

,

…

,

a

k

)

,

0

)

=

(

a

1

,

…

,

a

k

,

1

)

φ

⁢

(

(

a

1

,

…

,

a

k

)

,

1

)

=

(

a

1

,

…

,

a

k

+

1

)

and this map is clearly

injective

. But it is also clearly

surjective

, for given

(

a

1

,

…

,

a

k

)

∈

S

n

, if

a

k

=

1

then the composition is the image of

(

(

a

1

,

…

,

a

k

-

1

)

,

0

)

while if

a

k

>

1

, then it is the image of

(

(

a

1

,

…

,

a

k

-

1

)

,

1

)

. This proves that (for

n

>

1

)

C

n

=

2

⁢

C

n

-

1

.

We can also figure out how many compositions there are of

n

with

k

parts. Think of a box with

n

sections

in it, with dividers between each pair of sections and a chip in each section; there are thus

n

chips and

n

-

1

dividers. If we leave

k

-

1

of the dividers in place, the result is a composition of

n

with

k

parts; there are obviously

(

n

-

1

k

-

1

)

ways to do this, so the number of compositions of

n

into

k

parts is simply

(

n

-

1

k

-

1

)

. Note that this gives even a simpler proof of the first result, since

∑

k

=

1

n

(

n

-

1

k

-

1

)

=

∑

k

=

0

n

-

1

(

n

-

1

k

)

=

2

n

-

1

Title

counting compositions of an integer

Canonical name

CountingCompositionsOfAnInteger

Date of creation

2013-03-22 17:37:55

Last modified on

2013-03-22 17:37:55

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

5

Author

rm50 (10146)

Entry type

Result

Classification

msc 05-00

Defines

composition

Generated on Thu Feb 8 20:34:44 2018 by

LaTeXML
example of pigeonhole principle

example of pigeonhole principle

A example.

Theorem

.

For any set of

8

integers

, there exist at least two of them
whose

difference

is

divisible

by

7

.

Proof.

The

residue classes

modulo

7

are

0

,

1

,

2

,

3

,

4

,

5

,

6

.
We have seven and eight integers. So it must be the case that 2 integers fall on the same
residue class, and therefore their difference will be divisible by

7

.
∎

Title

example of

pigeonhole principle

Canonical

name

ExampleOfPigeonholePrinciple

Date of creation

2013-03-22 12:41:32

Last modified on

2013-03-22 12:41:32

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

8

Author

Mathprof (13753)

Entry type

Example

Classification

msc 05-00

Generated on Thu Feb 8 20:34:51 2018 by

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multi-index derivative of a power

multi-index derivative of a power

Theorem

If

i

,

k

are

multi-indices

in

ℕ

n

, and

x

=

(

x

1

,

…

,

x

n

)

,
then

∂

i

⁡

x

k

=

{

k

!

(

k

-

i

)

!

⁢

x

k

-

i

if

⁢

i

≤

k

,

0

otherwise

.

Proof.

The proof follows from the corresponding rule for
the ordinary

derivative

; if

i

,

k

are in

0

,

1

,

2

,

…

, then

d

i

d

⁢

x

i

⁢

x

k

=

{

k

!

(

k

-

i

)

!

⁢

x

k

-

i

if

⁢

i

≤

k

,

0

otherwise.

(1)

Suppose

i

=

(

i

1

,

…

,

i

n

)

,

k

=

(

k

1

,

…

,

k

n

)

, and

x

=

(

x

1

,

…

,

x

n

)

.
Then we have that

∂

i

⁡

x

k

=

∂

|

i

|

∂

⁡

x

1

i

1

⁢

⋯

⁢

∂

⁡

x

n

i

n

⁢

x

1

k

1

⁢

⋯

⁢

x

n

k

n

=

∂

i

1

∂

⁡

x

1

i

1

⁢

x

1

k

1

⋅

⋯

⋅

∂

i

n

∂

⁡

x

n

i

n

⁢

x

n

k

n

.

For each

r

=

1

,

…

,

n

, the

function

x

r

k

r

only depends on

x

r

.
In the above, each
partial differentiation

∂

/

∂

⁡

x

r

therefore
reduces to the corresponding
ordinary

differentiation

d

/

d

⁢

x

r

.
Hence, from equation

1

, it follows that

∂

i

⁡

x

k

vanishes

if

i

r

>

k

r

for any

r

=

1

,

…

,

n

. If this is not the case, i.e.,
if

i

≤

k

as multi-indices, then for each

r

,

d

i

r

d

⁢

x

r

i

r

⁢

x

r

k

r

=

k

r

!

(

k

r

-

i

r

)

!

⁢

x

r

k

r

-

i

r

,

and the theorem follows.

□

Title

multi-index derivative of a power

Canonical name

MultiindexDerivativeOfAPower

Date of creation

2013-03-22 13:42:01

Last modified on

2013-03-22 13:42:01

Owner

matte (1858)

Last modified by

matte (1858)

Numerical id

9

Author

matte (1858)

Entry type

Theorem

Classification

msc 05-00

Generated on Thu Feb 8 20:34:54 2018 by

LaTeXML
topic entry on discrete mathematics

topic entry on discrete mathematics

This is an entry-in-progress for a new

topic entry on discrete mathematics

.

1.

Combinatorics

2.

Graph theory

3.

Cryptography

4.

Game theory

5.

Mathematical programming

Title

topic entry on discrete mathematics

Canonical name

TopicEntryOnDiscreteMathematics

Date of creation

2013-03-22 18:00:23

Last modified on

2013-03-22 18:00:23

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

4

Author

CWoo (3771)

Entry type

Topic

Classification

msc 05-00

Classification

msc 90C99

Classification

msc 91A99

Generated on Thu Feb 8 20:35:07 2018 by

LaTeXML
one-line notation for permutations

one-line notation for permutations

One-line notation

is a system for representing

permutations

on a

collection

of symbols by words over the

alphabet

consisting of those symbols. First we show how the notation works in an example, and then we show that the notation can be made to work for any

symmetric group

.

First consider the permutation

π

=

(

134

)

⁢

(

25

)

in the symmetric group

𝔖

5

. Here

π

is written in

cycle notation

, so

π

⁢

(

1

)

=

3

,

π

⁢

(

2

)

=

5

,

π

⁢

(

3

)

=

4

,

π

⁢

(

4

)

=

1

, and

π

⁢

(

5

)

=

2

. We can record this information in the following table:

i

1

2

3

4

5

π

⁢

(

i

)

3

5

4

1

2

Finally, we read off the one-line notation as the second row of the table. Thus we write

π

=

35412

.

Now we define one-line notation for arbitrary finite symmetric groups. Let

X

be a set of finite

cardinality

n

and let

𝔖

X

be the group of permutations on

X

. Fix once and for all a

total order

<

on

X

. Using this order, we may say that

X

=

{

x

1

<

x

2

<

…

<

x

n

}

.

For an arbitrary

π

∈

𝔖

X

, the one-line notation for

π

is then

π

=

π

⁢

(

x

1

)

⁢

π

⁢

(

x

2

)

⁢

⋯

⁢

π

⁢

(

x

n

)

.

Observe that if

π

and

σ

are distinct permutations in

𝔖

X

, then there is some

i

for which

π

⁢

(

x

i

)

≠

σ

⁢

(

x

i

)

. Hence the one-line notations for

π

and

σ

will differ in the

i

th position. This shows that one-line notation is an injective map.
Furthermore, we can immediately recover

π

from its one-line notation. If

π

has one-line notation

a

1

⁢

a

2

⁢

…

⁢

a

n

, then we know that

π

⁢

(

x

i

)

=

a

i

for all

i

. For example, consider the permutation in

𝔖

7

written in one-line notation as

π

=

1732654

. We immediately obtain the following:

i

1

2

3

4

5

6

7

π

⁢

(

i

)

1

7

3

2

6

5

4

So now we can translate the permutation into cycle notation:

π

=

(

1

)

⁢

(

274

)

⁢

(

56

)

.

If we are willing to allow words of

infinite

length, we can even extend one-line notation to symmetric groups of arbitrary cardinality. Let

X

be a set and

𝔖

X

its symmetric group. Apply the

axiom of choice

to select a

well-ordering

on

X

. So we may write the elements of

X

in order as the tuple

(

x

1

,

x

2

,

…

,

x

α

,

x

α

+

1

,

…

)

. Then for each

π

∈

𝔖

X

the one-line notation for

π

is the tuple

(

π

⁢

(

x

1

)

,

π

⁢

(

x

2

)

,

…

,

π

⁢

(

x

α

)

,

π

⁢

(

x

α

+

1

)

,

…

)

.

The same

analysis

as in the finite case shows that a permutation is uniquely recoverable from its

representation

in one-line notation.

Title

one-line notation for permutations

Canonical name

OnelineNotationForPermutations

Date of creation

2013-03-22 16:24:38

Last modified on

2013-03-22 16:24:38

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 05A05

Related topic

Permutation

Related topic

CycleNotation

Defines

one-line notation

Generated on Thu Feb 8 20:35:10 2018 by

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permutation pattern

permutation pattern

A

permutation pattern

is simply a

permutation

viewed as its

representation

in

one-line notation

. Let

π

=

π

1

⁢

π

2

⁢

…

⁢

π

k

be a permutation pattern on

k

symbols. Then for any permutation

σ

=

σ

1

⁢

σ

2

⁢

…

⁢

σ

n

∈

𝔖

n

, we say that

σ

contains

π

if there is a (not necessarily contiguous)

subword

of

σ

of length

k

that is order-isomorphic to

π

. More formally,
for any subset

J

=

{

j

1

,

…

,

j

k

}

⊂

{

1

,

…

,

n

}

of

cardinality

k

, write

σ

J

=

σ

j

1

⁢

σ

j

2

⁢

…

⁢

σ

j

k

.

There is a

isomorphism

(

http://planetmath.org/GroupHomomorphism

)

𝔖

J

→

𝔖

k

. We say that

σ

contains

π

if there is some

J

⊂

{

1

,

…

,

n

}

of cardinality

k

such that

σ

J

↦

π

under the isomorphism.

If a permutation

σ

does not contain

π

, then we say that

σ

avoids

the pattern

π

.

For example, let

π

=

132

. Then the permutation

σ

=

1234

avoids

π

, since

σ

is strictly increasing but

π

has a descent. On the other hand,

τ

=

1423

contains

π

twice, once as the subword

142

and once as the subword

143

.

Knuth showed that a permutation is stack-sortable if and only if it avoids the permutation pattern

231

.

Title

permutation pattern

Canonical name

PermutationPattern

Date of creation

2013-03-22 16:24:41

Last modified on

2013-03-22 16:24:41

Owner

mps (409)

Last modified by

mps (409)

Numerical id

4

Author

mps (409)

Entry type

Definition

Classification

msc 05A05

Classification

msc 05A15

Defines

pattern avoidance

Generated on Thu Feb 8 20:35:16 2018 by

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