# 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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