arXiv subjects
Publications and source records attributed to Arnold Knopfmacher.
In a Dyck path a peak which is (weakly) higher than all the preceding peaks is called a strict (weak) left to right maximum. We obtain explicit generating functions for both weak and strict left to right maxima in Dyck paths. The proofs of the associated asymptotics make use of analytic techniques such as Mellin transforms, singularity analysis and formal residue calculus.
A sequence of geometric random variables of length $n$ is a sequence of $n$ independent and identically distributed geometric random variables ($Γ_1, Γ_2, \dots, Γ_n$) where $\mathbb{P}(Γ_j=i)=pq^{i-1}$ for $1~\leq~j~\leq~n$ with $p+q=1.$ We study the number of distinct adjacent two letter patterns in such sequences. Initially we directly count the number of distinct pairs in words of short length. Because of the rapid growth of the number of word patterns we change our approach to this problem by obtaining an expression for the expected number of distinct pairs in words of length $n$. We also obtain the asymptotics for the expected number as $n \to \infty$.
A word $σ=σ_1...σ_n$ over the alphabet $[k]=\{1,2,...,k\}$ is said to be {\em smooth} if there are no two adjacent letters with difference greater than 1. A word $σ$ is said to be {\em smooth cyclic} if it is a smooth word and in addition satisfies $|σ_n-σ_1|\le 1$. We find the explicit generating functions for the number of smooth words and cyclic smooth words in $[k]^n$, in terms of {\it Chebyshev polynomials of the second kind}. Additionally, we find explicit formula for the numbers themselves, as trigonometric sums. These lead to immediate asymptotic corollaries. We also enumerate smooth necklaces, which are cyclic smooth words that are not equivalent up to rotation.
We study M(n), the number of distinct values taken by multinomial coefficients with upper entry n, and some closely related sequences. We show that both pP(n)/M(n) and M(n)/p(n) tend to zero as n goes to infinity, where pP(n) is the number of partitions of n into primes and p(n) is the total number of partitions of n. To use methods from commutative algebra, we encode partitions and multinomial coefficients as monomials.
For words of length n, generated by independent geometric random variables, we consider the average value and the average position of the r-th left-to-right maximum, for fixed r and large n.