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Register a new userRanks of Strictly Minimal Reaction Systems Induced by Permutations and Cartesian Product
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Reaction system is a computing model inspired by the biochemical interaction taking place within the living cells. Various extended or modified frameworks motivated by biological, physical, or purely mathematically considerations have been proposed and received significant amount of attention, notably in the recent years. This study, however, takes after particular early works that concentrated on the mathematical nature of minimal reaction systems in the context-free basic framework and motivated by a recent result on the sufficiency of strictly minimal reaction systems to simulate every reaction system. This paper focuses on the largest reaction system rank attainable by strictly minimal reaction systems, where the rank pertains to the minimum size of a functionally equivalent reaction system. Precisely, we provide a very detailed study for specific strictly minimal reaction system induced by permutations, up to the quaternary alphabet. Along the way, we obtain a general result about reaction system rank for Cartesian product of functions specified by reaction systems.
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We give two combinatorial proofs of Goulden and Jacksons formula for the number of minimal transitive factorizations of a permutation when the permutation has two cycles. We use the recent result of Goulden, Nica, and Oancea on the number of maximal chains of annular noncrossing partitions of type $B$.
Let $G=(V,E)$ be a graph and $Gamma $ an Abelian group both of order $n$. A $Gamma$-distance magic labeling of $G$ is a bijection $ell colon Vrightarrow Gamma $ for which there exists $mu in Gamma $ such that $% sum_{xin N(v)}ell (x)=mu $ for all $vin V$, where $N(v)$ is the neighborhood of $v$. Froncek %(cite{ref_CicAus}) showed that the Cartesian product $C_m square C_n$, $m, ngeq3$ is a $mathbb{Z}_{mn}$-distance magic graph if and only if $mn$ is even. It is also known that if $mn$ is even then $C_m square C_n$ has $mathbb{Z}_{alpha}times mathcal{A}$-magic labeling for any $alpha equiv 0 pmod {{rm lcm}(m,n)}$ and any Abelian group $mathcal{A}$ of order $mn/alpha$. %cite{ref_CicAus} However, the full characterization of group distance magic Cartesian product of two cycles is still unknown. In the paper we make progress towards the complete solution this problem by proving some necessary conditions. We further prove that for $n$ even the graph $C_{n}square C_{n}$ has a $Gamma$-distance magic labeling for any Abelian group $Gamma$ of order $n^{2}$. Moreover we show that if $m eq n$, then there does not exist a $(mathbb{Z}_2)^{m+n}$-distance magic labeling of the Cartesian product $C_{2^m} square C_{2^{n}}$. We also give necessary and sufficient condition for $C_{m} square C_{n}$ with $gcd(m,n)=1$ to be $Gamma$-distance magic.
The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_nsquare H$ is Hamiltonian; (ii) $P_nsquare H$ is $1$-tough; and (iii) $H$ has a path factor.
We find a formula for the number of permutations of $[n]$ that have exactly $s$ runs up and down. The formula is at once terminating, asymptotic, and exact.
We use moment method to understand the cycle structure of the composition of independent invariant permutations. We prove that under a good control on fixed points and cycles of length 2, the limiting joint distribution of the number of small cycles is the same as in the uniform case i.e. for any positive integer k, the number of cycles of length k converges to the Poisson distribution with parameter 1/k and is asymptotically independent of the number of cycles of length k different from k.
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