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Register a new userAnnular noncrossing permutations and minimal transitive factorizations
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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 $B$.
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A notion of $t$-designs in the symmetric group on $n$ letters was introduced by Godsil in 1988. In particular $t$-transitive sets of permutations form a $t$-design. We derive special lower bounds for $t=1$ and $t=2$ by a power moment method. For general $n,t$ we give a %linear programming lower bound . For $nge 4$ and $t=2,$ this bound is strong enough to show a lower bound on the size of such $t$-designs of $n(n-1)dots (n-t+1),$ which is best possible when sharply $t$-transitive sets of permutations exist. This shows, in particular, that tight $2$-designs do not exist.
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.
A detailed description of the structure of two-ended arc-transitive digraphs is given. It is also shown that several sets of conditions, involving such concepts as Property Z, local quasi-primitivity and prime out-valency, imply that an arc-transitive digraph must be highly-arc-transitive. These are then applied to give a complete classification of two-ended highly-arc-transitive digraphs with prime in- and out-valencies.
For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haimans notion of a parking function symmetric function.
In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contains any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA pseudoknot structures cite{Stadler:99,Reidys:07pseu,Reidys:07lego} and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied cite{Waterman:78b, Waterman:79,Waterman:93, Schuster:98}. Let ${sf Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${sf Q}_k(n)sim c_k n^{-(k-1)^2-frac{k-1}{2}}gamma_{k}^{-n}$ for $k=3,..., 9$ and derive a new proof of the formula ${sf Q}_2(n)sim 1.4848, n^{-3/2},1.8489^{-n}$ cite{Schuster:98}.
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