Related papers: Permutation classes of every growth rate above 2.4…

Polignac [1] conjectured that for every even natural number $2k (k\geq1)$, there exist infinitely many consecutive primes $p_n$ and $p_{n+1}$ such that $p_{n+1}-p_n=2k$. A weakened form of this conjecture states that for every $k\geq1$,…

We analyse the recursive formula found for various Ext groups for $\SL_2(k)$, $k$ a field of characteristic $p$, and derive various generating functions for these groups. We use this to show that the growth rate for the cohomology of…

We show that for any $\varepsilon > 0$, prime $q$ sufficiently large with respect to $1 / \varepsilon$ and residue class $(a,q) = 1$, there exist two integers $m, n \leq q^{5/2 + \varepsilon}$ with $m \equiv n \equiv a \pmod{q}$ such that…

We revisit the problem of influencing the sex ratio of a population by subjecting reproduction of each family to some stopping rule. As an easy consequence of the strong law of large numbers, no such modification is possible in the sense…

Let $E_n^r=\{[\tau]_a=(\tau_1^{(a_1)},...,\tau_n^{(a_n)})| \tau\in S_n,\ 1\leq a_i\leq r\}$ be the set of all signed permutations on the symbols 1,2,...,n with signs 1,2,...,r. We prove, for every 2-letter signed pattern $[\tau]_a$, that…

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm PSL}_n(q)$ is prime. We…

We consider permutation classes having two basis elements of size three and one further basis element. We completely classify the possible enumeration sequences of such classes and demonstrate that there are far fewer of them than might be…

We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the…

We consider a population evolving under mutation and selection. The genotype of an individual is a word of length $\ell$ over a finite alphabet. Mutations occur during reproduction, independently on each locus; the fitness depends on the…

We prove the existence of a limit shape and give its explicit description for certain probability distribution on signatures (or highest weights for unitary groups). The distributions have representation theoretic origin-they encode…

This paper is a sequel to arXiv:2501.14444, in which we shall give proofs of several results stated in arXiv:2501.14444 (Theorems D--L) which, for brevity and clarity, we postponed to this sequel paper. These results were the following: for…

As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side…

We develop a new, powerful method for counting elements in a multiset. As a first application, we use this algorithm to study the number of occurrences of patterns in a permutation. For patterns of length 3 there are two Wilf classes, and…

A pattern class is a set of permutations closed under the formation of subpermutations. Such classes can be characterised as those permutations not involving a particular set of forbidden permutations. A simple collection of necessary and…

We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. $$ where $\la\geq 0$, $A(x)$ is a…

We obtain an upper and lower bound for the number of reduced words for a permutation in terms of the number of braid classes and the number of commutation classes of the permutation. We classify the permutations that achieve each of these…

Weprovide an upper bound for generalized Littlewood-Richardson coefficients $c^w_{uv}$, where $u$ is a two-row Young diagram corresponding to a Grassmannian permutation. We end with a conjecture on the upper bounds for all such structure…

We introduce the notion of a weighted inversion statistic on the symmetric group, and examine its distribution on each conjugacy class. Our work generalizes the study of several common permutation statistics, including the number of…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

As a consequence of the classification of finite simple groups, the classification of permutation groups of prime degree is complete, apart from the question of when the natural degree $(q^n-1)/(q-1)$ of ${\rm L}_n(q)$ is prime. We present…