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nt.quantity concept – What is the cardinality of the clique of Dyck unaffected numbers of semilength $ok$? Answer

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nt.quantity concept – What is the cardinality of the clique of Dyck unaffected numbers of semilength $ok$?

In arXiv:2102.02777 (“Recursive Prime Factorizations: Dyck Words as Numbers”), I display that there’s a 1:1 correspondence between $mathbb{N} = {0,1,2,3,4,ldots}$ and $mathcal{D}_{r_{textual content{min}}} = {epsilon,(),(()),()(()),((())), ldots}$. So what precisely is $mathcal{D}_{r_{textual content{min}}}$? It’s the clique of Dyck unaffected numbers, i.e., the clique of recursive prime factorizations ${gamma’_{mathbb{N}_{r}}(n) mid n in mathbb{N}}$, the place $gamma’_{mathbb{N}_{r}}$ is given by Definition 3 beneath (Definitions 1 and a couple of insert notation for employ in Definition 3):

Definition 1. The attribute $glower$ is the string concatenation operator; $a glower b$ denotes the concatenation of $a$ and $b$. The concatenation of $a$ and $b$ might too breathe written in accustomed style as $ab$, offered that doing so doesn’t incur ambiguity.

Definition 2. Let $j,ok in mathbb{N}_{+}$. Then we will grasp $bigoplus_{i=j}^{ok}s_{i}$ to indicate the string concatenation $s_{j} ldots s_{ok}$ if $j le ok$, in any other case the vacant string $epsilon$.

Definition 3. Let $Sigma^*$ breathe the Kleene closure of the clique ${(,)}$. Then the benchmark nonsurjective recursive prime factorization unaffected transcription duty, denoted by $gamma’_{mathbb{N}_{r}}$, is given by $gamma’_{mathbb{N}_{r}}: mathbb{N} rightarrow Sigma^*$, the place

  • For $n = 0$, $gamma’_{mathbb{N}_{r}}(n)$ is the vacant string $epsilon$.
  • For $n = 1$, $gamma’_{mathbb{N}_{r}}(n)$ is the string $()$.
  • For $n > 1$, let $p_{i}$ breathe the $i$th prime quantity, let $p_{m}$ breathe the best prime issue of $n$, and let $a = (a_{1}, ldots, a_{m})$ breathe the integer sequence satisfying
    commence{equation}
    n = prod_{i=1}^{m}p_{i}^{a_{i}}.
    aim{equation}

    Then
    commence{equation}
    gamma’_{mathbb{N}_{r}}(n) = bigoplus_{i=1}^{m}(;'(‘glower gamma’_{mathbb{N}_{r}}(a_{i}) glower ‘)’;).
    aim{equation}

Note: It is $gamma’_{mathbb{N}_{r}}$ somewhat than $gamma_{mathbb{N}_{r}}$ as a result of I limit its codomain to relent the bijection $gamma_{mathbb{N}_{r}}$ (which I convene the benchmark RPF unaffected spelling duty; I do that in order that my spelling duty may have an inverse, permitting me to map from $mathcal{D}_{r_{textual content{min}}}$ to $N$). The subscript $mathbb{N}$ tells us that the duty maps unaffected numbers to Dyck phrases, in organize to differentiate the duty from one other one which maps rationals to Dyck phrases. Finally, the subscript $r$ signifies that it entails the right-ascending sequence of prime numbers $(2,3,5,7, ldots)$, somewhat than another prime permutation equivalent to $(ldots,7,5,3,2)$ or $(3,2,7,5,ldots)$.

Now let $S_{ok}$ breathe the subset of ${0,1,2,3,ldots}$ such that each member of $S_{ok}$ is represented by a Dyck unaffected variety of semilength $ok$. Then

$S_{0} = {0}$

$S_{1} = {1}$

$S_{2} = {2}$

$S_{3} = {3,4}$

$S_{4} = {5,6,8,9,16}$

$S_{5} = {7,10,12,15,18,25,27,32,64,81,256,512,65536}$

Notice that the biggest members in units $S_{2}$ by $S_{5}$ can breathe expressed as $2$, $2^{2}$, $2^{2^2}$ and $2^{2^{2^{2^2}}}$, respectively. If the sample holds, then the best quantity in $S_{6}$ is the same as $2^{65536}$; thus I cannot attempt to listing the members in additional $S_{ok}$. But we’ve sufficient info to narrate the sequence as $(1,1,1,2,5,13, ldots)$, and there are a number of candidates within the Online Encyclopedia of Integer Sequences. So what is the sequence?

IMPORTANT NOTE: $mathcal{D}_{r_{textual content{min}}}$ has another nonnumerical definition (behold Theorem 2.7 in Section 2.8 of arXiv:2102.02777):

Definition 4. Let $mathcal{D}$ denote the Dyck language. The benchmark minimal RPF language, denoted by $mathcal{D}_{r_{textual content{min}}}$, is the clique
commence{equation}
{d in mathcal{D} mid (;{‘})()){‘} textual content{ isn’t a substring of } d;);wedge;(;{‘})(){‘} textual content{ isn’t a suffix of } d;) }.
aim{equation}

Thus my query can breathe answered by offering a components for the variety of Dyck phrases of semilength $ok$ not containing the substring ${)())}$ or the suffix ${)()}$, the place $ok ge 0$.

Thank you…

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