### 321-polygon-avoiding permutations and Chebyshev polynomials.

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In this paper, a direct combinatorial proof is given of a result on permutation pairs originally due to Carlitz, Scoville, and Vaughan and later extended. It concerns showing that the series expansion of the reciprocal of a certain multiply exponential generating function has positive integer coefficients. The arguments may then be applied to related problems, one of which concerns the reciprocal of the exponential series for Fibonacci numbers.

We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer $k\ge 2$ and every set $A$ of words over $k$ satisfying $\mathrm{lim}\phantom{\rule{4pt}{0ex}}{\mathrm{sup}}_{n\to \infty}|A\cap {\left[k\right]}^{n}|/{k}^{n}>0$ there exist a word $c$ over $k$ and a sequence $\left({w}_{n}\right)$ of left variable words over $k$ such that the set $c\cup \{{c}^{}{w}_{0}{\left({a}_{0}\right)}^{}..{.}^{}{w}_{n}\left({a}_{n}\right):n\in \mathbb{N}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4.0pt}{0ex}}{a}_{0},...,{a}_{n}\in \left[k\right]\}$ is contained in $A$. While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.