1,816 research outputs found

### A Note about Iterated Arithmetic Functions

Let $f\colon\mathbb{N}\rightarrow\mathbb{N}_0$ be a multiplicative arithmetic
function such that for all primes $p$ and positive integers $\alpha$,
$f(p^{\alpha})<p^{\alpha}$ and $f(p)\vert f(p^{\alpha})$. Suppose also that any
prime that divides $f(p^{\alpha})$ also divides $pf(p)$. Define $f(0)=0$, and
let $H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}$, where $f^m$ denotes
the $m^{th}$ iterate of $f$. We prove that the function $H$ is completely
multiplicative.Comment: 5 pages, 0 figure

### Motzkin Intervals and Valid Hook Configurations

We define a new natural partial order on Motzkin paths that serves as an
intermediate step between two previously-studied partial orders. We provide a
bijection between valid hook configurations of $312$-avoiding permutations and
intervals in these new posets. We also show that valid hook configurations of
permutations avoiding $132$ (or equivalently, $231$) are counted by the same
numbers that count intervals in the Motzkin-Tamari posets that Fang recently
introduced, and we give an asymptotic formula for these numbers. We then
proceed to enumerate valid hook configurations of permutations avoiding other
collections of patterns. We also provide enumerative conjectures, one of which
links valid hook configurations of $312$-avoiding permutations, intervals in
the new posets we have defined, and certain closed lattice walks with small
steps that are confined to a quarter plane.Comment: 22 pages, 8 figure

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