1,816 research outputs found

    A Note about Iterated Arithmetic Functions

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    Let f ⁣:NN0f\colon\mathbb{N}\rightarrow\mathbb{N}_0 be a multiplicative arithmetic function such that for all primes pp and positive integers α\alpha, f(pα)<pαf(p^{\alpha})<p^{\alpha} and f(p)f(pα)f(p)\vert f(p^{\alpha}). Suppose also that any prime that divides f(pα)f(p^{\alpha}) also divides pf(p)pf(p). Define f(0)=0f(0)=0, and let H(n)=limmfm(n)H(n)=\displaystyle{\lim_{m\rightarrow\infty}f^m(n)}, where fmf^m denotes the mthm^{th} iterate of ff. We prove that the function HH is completely multiplicative.Comment: 5 pages, 0 figure

    Motzkin Intervals and Valid Hook Configurations

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    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 312312-avoiding permutations and intervals in these new posets. We also show that valid hook configurations of permutations avoiding 132132 (or equivalently, 231231) 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 312312-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