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Mieli dalyviai! Visa mano kūryba ir kartu visi šie puslapiai yra visuomenės turtas, kuriuo visi kviečiami laisvai naudotis, dalintis, visaip perkurti. - Andrius

## Mintys.Catalan istorija

2017 rugsėjo 20 d., 20:33 atliko AndriusKulikauskas -
Pakeistos 1-67 eilutės iš
Žr. [[Matematika]]

>>bgcolor=#FFFFC0<<

Consider how to relate the Catalan generating function to a complex number and a context-free grammar.

How does binary tree height correspond to the difference between left-right parentheses?

Look at q-catalan numbers. How do they relate to parentheses discrepancy?

What are the q-t-catalan numbers, what do they count, and how do they relate to the Macdonald polynomials?

Consider how to relate the boundedness of the generating function to the idea that there is a finite mismatch between left and right parentheses, whereas if the number of left parentheses grows without bound, then the function diverges to infinity.

Consider what we know about the boundedness of the generating function in the complex plane.

Dievas: walks on binary trees (C,I,U). Ar tai tas pats kaip "ordered binary trees"? Ar juos skaičiuoja Catalan skaičiai?
* There is one correct progression: C->I->U->C... Do the convergent sequences represent finite mistakes which are corrected. And then is there a distinction between finite (bounded) and infinite (divergent) mistakes?

Jeigu pridėti c + d (delta) ką sužinotumėme apie mažą paklaidą?

>><<

The issue of memory required by automata - that this is "explicit" truth needed to supplement "implicit" truth.

[[http://algo.inria.fr/flajolet/Publications/book.pdf | Analytic Combinatorics]] by Philippe Flajolet, Robert Sedgewick, 2009. They briefly mention the connection between Mandelbrot sets and Catalan numbers as noting "the speed of convergence" for "the generating functions of binary trees of bounded height". Look for Catalan numbers:
* Section V.2 - supercritical sequence schema: compositions, surjections, alignments. (Regard in terms of duality of top-down and bottom-up, addition and partition.)
* Section V.3 - regular expressions as by regular automata (Figure V.5 on asymptotics).
* Section V.4 - lattice paths as by context-free grammars, Continued Fraction Theorem. See especially the figure linking V.11 linking objects, weights, counting numbers (Catlan numbers), orthogonal polynomials (Chebyshev).
* see also 279 exponential growth for tree families, including 3, e, 4.
* Transfer matrix models
* VI.18 Tolls and costs for Binary tree recurrence. "This example is also of interest since it furnishes an analytically tractable model of a coalescence-fragmentation process, which is of great interest in several areas of science, for which we refer to Aldous’ survey ."
* VII.3.1 Asymptotic counting
* Proposition VII.2 links between tree types and probability distributions.
* Theorem VII.6 on Irreducible positive polynomial systems.
* VII.27 Catalan and Jacobian determinant
* Figure VII.16 Catalan curve
* VII.21 Trees and Lukasiewicz codes
* VII.27 Height of binary trees, mentions Mandelbrot set. Each polynomial has degree 2^(h-1)-1. "When
|z| ≤ r < 1/4,
simple majorant series considerations show that the convergence
yh(z)→y(z)
is uniformly geometric. When
z ≥ s > 1/4
it can be checked that the yh(z) grow doubly exponentially. What happens in-between, in a delta–domain, needs to be quantified. We do so following Flajolet, Gao, Odlyzko, and Richmond [230, 246]. (They analyze the convergence in the interesting domain.)
* Figure VII.24 A collection of universality laws.
* Page 538 Nice relation of tree heights for different kinds of trees.
* Figure IX.13 Singularities of generating functions of Catalan trees, square-root type.

* Flajolet, P., Gao, Z.,Odlyzko, A.,M., and Richmond, B. The distribution of heights of binary trees and other simple trees. Combinatorics, Probability and Computing 2 (1993), 145–15
*  Flajolet, P.,Odlyzko, A.,M. The average height of binary trees and other simple trees. Journal of Computer and System Sciences 25 (1982), 171–213

It seems that the Catalan numbers serve to count both of these cases:
* Any well ordered string of parentheses.
* Only those well ordered strings of parentheses that have no superluous (repeated) parentheses. (Full binary trees.)
* And also this case: of all ordered trees.
Whereas "all binary trees" (all parentheses) are given by binomial theorem, as can be seen by expanding ( + ) to the N.

Mandelbrot set rule: The simplest "mixing rule": "Add the input (complex vector c), add the output ("multiply by 2" by complex rotating and squaring)."

[[http://www.amazon.com/-Catalan-Numbers-Diagonal-Harmonics-University/dp/0821844113/ref=sr_1_5?ie=UTF8&qid=1463389854&sr=8-5&keywords=%22catalan+numbers%22 | The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics (University Lecture Series)]] by James Haglund. With an appendix on the MacDonald polynomials (linked to root weights).

Square root as a sign of "dual" mid-point (reflection point) in a duality.

* http://mathforum.org/kb/message.jspa?messageID=22222
* [[http://link.springer.com/book/10.1007%2F978-3-319-10094-4 | The Problem of
Catalan]]
į:
Žr.[[Book/Catalan]]
2016 birželio 09 d., 16:54 atliko AndriusKulikauskas -
Pridėtos 19-20 eilutės:

Jeigu pridėti c + d (delta) ką sužinotumėme apie mažą paklaidą?
2016 gegužės 16 d., 15:52 atliko AndriusKulikauskas -
Pridėta 18 eilutė:
* There is one correct progression: C->I->U->C... Do the convergent sequences represent finite mistakes which are corrected. And then is there a distinction between finite (bounded) and infinite (divergent) mistakes?
2016 gegužės 16 d., 15:26 atliko AndriusKulikauskas -
Pridėtos 17-18 eilutės:
Dievas: walks on binary trees (C,I,U). Ar tai tas pats kaip "ordered binary trees"? Ar juos skaičiuoja Catalan skaičiai?
Pakeista 64 eilutė iš:
* [[http://link.springer.com/book/10.1007%2F978-3-319-10094-4 | The Problem of Catalan]]
į:
* [[http://link.springer.com/book/10.1007%2F978-3-319-10094-4 | The Problem of Catalan]]
2016 gegužės 16 d., 14:40 atliko AndriusKulikauskas -
Pridėtos 7-12 eilutės:
How does binary tree height correspond to the difference between left-right parentheses?

Look at q-catalan numbers. How do they relate to parentheses discrepancy?

What are the q-t-catalan numbers, what do they count, and how do they relate to the Macdonald polynomials?
Ištrintos 13-14 eilutės:

Look at q-catalan numbers.
2016 gegužės 16 d., 14:36 atliko AndriusKulikauskas -
Pridėtos 44-50 eilutės:
It seems that the Catalan numbers serve to count both of these cases:
* Any well ordered string of parentheses.
* Only those well ordered strings of parentheses that have no superluous (repeated) parentheses. (Full binary trees.)
* And also this case: of all ordered trees.
Whereas "all binary trees" (all parentheses) are given by binomial theorem, as can be seen by expanding ( + ) to the N.

Mandelbrot set rule: The simplest "mixing rule": "Add the input (complex vector c), add the output ("multiply by 2" by complex rotating and squaring)."
2016 gegužės 16 d., 14:16 atliko AndriusKulikauskas -
Pakeista 36 eilutė iš:
it can be checked that the yh(z) growdoubly exponentially. What happens in-between, in a delta–domain, needs to be quantified. We do so following Flajolet, Gao, Odlyzko, and Richmond [230, 246].
į:
it can be checked that the yh(z) grow doubly exponentially. What happens in-between, in a delta–domain, needs to be quantified. We do so following Flajolet, Gao, Odlyzko, and Richmond [230, 246]. (They analyze the convergence in the interesting domain.)
2016 gegužės 16 d., 14:15 atliko AndriusKulikauskas -
Pakeistos 23-24 eilutės iš
* VI.18 Tolls and costs for Binary tree recurrence. "This example is also of interest since it furnishes an analytically
tractable model of a coalescence-fragmentation process, which is of great interest in several areas of science, for which we refer to Aldous’ survey ."
į:
* VI.18 Tolls and costs for Binary tree recurrence. "This example is also of interest since it furnishes an analytically tractable model of a coalescence-fragmentation process, which is of great interest in several areas of science, for which we refer to Aldous’ survey ."
2016 gegužės 16 d., 14:12 atliko AndriusKulikauskas -
Pakeistos 38-39 eilutės iš
į:
* Figure VII.24 A collection of universality laws.
* Page 538 Nice relation of tree heights for different kinds of trees.
* Figure IX.13 Singularities of generating functions of Catalan trees, square-root type.

* Flajolet, P., Gao, Z.,Odlyzko, A.,M., and Richmond, B. The distribution of heights of binary trees and other simple trees. Combinatorics, Probability and Computing 2 (1993), 145–15
*  Flajolet, P.,Odlyzko, A.,M. The average height of binary trees and other simple trees. Journal of Computer and System Sciences 25 (1982), 171–213
Pridėtos 47-48 eilutės:

Square root as a sign of "dual" mid-point (reflection point) in a duality.
2016 gegužės 16 d., 14:02 atliko AndriusKulikauskas -
Pakeistos 31-37 eilutės iš
* VII.27 Height of binary trees, mentions Mandelbrot set. Each polynomial has degree 2^(h-1)-1.
į:
* VII.27 Height of binary trees, mentions Mandelbrot set. Each polynomial has degree 2^(h-1)-1. "When
|z| ≤ r < 1/4,
simple majorant series considerations show that the convergence
yh(z)→y(z)
is uniformly geometric. When
z ≥ s > 1/4
it can be checked that the yh(z) growdoubly exponentially. What happens in-between, in a delta–domain, needs to be quantified. We do so following Flajolet, Gao, Odlyzko, and Richmond [230, 246]
.
2016 gegužės 16 d., 13:59 atliko AndriusKulikauskas -
Pridėtos 30-31 eilutės:
* VII.21 Trees and Lukasiewicz codes
* VII.27 Height of binary trees, mentions Mandelbrot set. Each polynomial has degree 2^(h-1)-1.
2016 gegužės 16 d., 13:55 atliko AndriusKulikauskas -
Pakeistos 27-29 eilutės iš
į:
* Theorem VII.6 on Irreducible positive polynomial systems.
* VII.27 Catalan and Jacobian determinant
* Figure VII.16 Catalan curve
2016 gegužės 16 d., 13:49 atliko AndriusKulikauskas -
2016 gegužės 16 d., 13:49 atliko AndriusKulikauskas -
Pridėtos 25-26 eilutės:
* VII.3.1 Asymptotic counting
* Proposition VII.2 links between tree types and probability distributions.
2016 gegužės 16 d., 13:47 atliko AndriusKulikauskas -
Pakeistos 18-19 eilutės iš
* Section V.3 - regular expressions as by regular automata
* Section V.4 - lattice paths as by context
-free grammars, see especially the figure linking
į:
* Section V.2 - supercritical sequence schema: compositions, surjections, alignments. (Regard in terms of duality of top-down and bottom-up, addition and partition.)
* Section V.3 - regular expressions as by regular automata (Figure V.5 on asymptotics).
* Section V.4 - lattice paths as by context-free grammars, Continued Fraction Theorem. See especially the figure linking V.11 linking objects, weights, counting numbers (Catlan numbers), orthogonal polynomials (Chebyshev).
Pakeistos 22-23 eilutės iš
[[http://www.amazon.com/-Catalan-Numbers-Diagonal-Harmonics-University/dp/0821844113/ref=sr_1_5?ie=UTF8&qid=1463389854&sr=8-5&keywords=%22catalan+numbers%22 | The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics (University Lecture Series)]] by James Haglund. With an appendix on the MacDonald polynomials.
į:
* Transfer matrix models
* VI
.18 Tolls and costs for Binary tree recurrence. "This example is also of interest since it furnishes an analytically
tractable model of a coalescence
-fragmentation process, which is of great interest in several areas of science, for which we refer to Aldous’ survey ."

[[http://www.amazon.com/-Catalan-Numbers-Diagonal-Harmonics-University/dp/0821844113/ref=sr_1_5?ie=UTF8&qid=1463389854&sr=8-5&keywords=%22catalan+numbers%22 | The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics (University Lecture Series)]] by James Haglund. With an appendix on the MacDonald polynomials (linked to root weights)
.
2016 gegužės 16 d., 13:37 atliko AndriusKulikauskas -
Pridėtos 14-20 eilutės:

The issue of memory required by automata - that this is "explicit" truth needed to supplement "implicit" truth.

[[http://algo.inria.fr/flajolet/Publications/book.pdf | Analytic Combinatorics]] by Philippe Flajolet, Robert Sedgewick, 2009. They briefly mention the connection between Mandelbrot sets and Catalan numbers as noting "the speed of convergence" for "the generating functions of binary trees of bounded height". Look for Catalan numbers:
* Section V.3 - regular expressions as by regular automata
* Section V.4 - lattice paths as by context-free grammars, see especially the figure linking
* see also 279 exponential growth for tree families, including 3, e, 4.
2016 gegužės 16 d., 12:58 atliko AndriusKulikauskas -
Pridėtos 1-2 eilutės:
Žr. [[Matematika]]
2016 gegužės 16 d., 12:58 atliko AndriusKulikauskas -
Pridėtos 12-13 eilutės:

[[http://www.amazon.com/-Catalan-Numbers-Diagonal-Harmonics-University/dp/0821844113/ref=sr_1_5?ie=UTF8&qid=1463389854&sr=8-5&keywords=%22catalan+numbers%22 | The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics (University Lecture Series)]] by James Haglund. With an appendix on the MacDonald polynomials.
2016 gegužės 16 d., 10:58 atliko AndriusKulikauskas -
Pridėtos 1-15 eilutės:
>>bgcolor=#FFFFC0<<

Consider how to relate the Catalan generating function to a complex number and a context-free grammar.

Consider how to relate the boundedness of the generating function to the idea that there is a finite mismatch between left and right parentheses, whereas if the number of left parentheses grows without bound, then the function diverges to infinity.

Look at q-catalan numbers.

Consider what we know about the boundedness of the generating function in the complex plane.

>><<

* http://mathforum.org/kb/message.jspa?messageID=22222
* [[http://link.springer.com/book/10.1007%2F978-3-319-10094-4 | The Problem of Catalan]]

#### Catalan

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