Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is known that the set of permutations, under the pattern containment ordering, is not a partial ll-order.  Characterizing the partially ll-ordered closed sets (equivalently: down sets or ideals) in this poset remains a wide-open problem.  Given a $0/\pm1$ matrix $M$,  define a closed set of permutations called the profile class of $M$.  These sets are generalizations of sets considered by Atkinson, Murphy, and Ruskuc.   show that the profile class of $M$ is partially ll-ordered if and only if a related graph is a forest. Related to the antichains  construct to prove one of the directions of this result,  construct exotic fundamental antichains, which lack the periodicity exhibited by all previously known fundamental antichains of permutations.
['Maximillian M. Murphy', 'Vincent Vatter']
I am writing a research paper on a idea Profile Classes and Partial Well-Order for Permutations
[{"from": "human", "value": "I am writing a research paper on a idea Profile Classes and Partial Well-Order for Permutations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It is known that the set of permutations, under the pattern containment ordering, is not a partial ll-order.  Characterizing the partially ll-ordered closed sets (equivalently: down sets or ideals) in this poset remains a wide-open problem.  Given a $0/\\pm1$ matrix $M$,  define a closed set of permutations called the profile class of $M$.  These sets are generalizations of sets considered by Atkinson, Murphy, and Ruskuc.   show that the profile class of $M$ is partially ll-ordered if and only if a related graph is a forest. Related to the antichains  construct to prove one of the directions of this result,  construct exotic fundamental antichains, which lack the periodicity exhibited by all previously known fundamental antichains of permutations."}]