Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  give an explicit formula for the number of permutations that cyclically avoid a consecutive pattern in terms of the spectrum of the associated operator of the consecutive pattern. As an example, the number of cyclically consecutive 123-avoiding permutations in ${\mathfrak S}_{n}$ is given by $n!$ times the convergent series $\sum_{k=-\infty}^{\infty} (\frac{\sqrt{3}}{2\pi(k+1/3)})^{n}$ for $n \geq 2$.
['Richard Ehrenborg']
I am writing a research paper on a idea Cyclically Consecutive Permutation Avoidance
[{"from": "human", "value": "I am writing a research paper on a idea Cyclically Consecutive Permutation Avoidance"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  give an explicit formula for the number of permutations that cyclically avoid a consecutive pattern in terms of the spectrum of the associated operator of the consecutive pattern. As an example, the number of cyclically consecutive 123-avoiding permutations in ${\\mathfrak S}_{n}$ is given by $n!$ times the convergent series $\\sum_{k=-\\infty}^{\\infty} (\\frac{\\sqrt{3}}{2\\pi(k+1/3)})^{n}$ for $n \\geq 2$."}]