Abstract
A partition of \(\{1,\ldots,n\}\) has an m-nesting if it contains at least m disjoint blocks, and a subset of 2m points \(i_{1} < i_{2} <\ldots < i_{m} < j_{m} < j_{m-1} <\ldots < j_{1}\), such that i l and j l are in the same block for all 1 ≤ l ≤ m, but no other pairs are in the same block. In this note, we use generating trees to construct the class of partitions with no m-nesting, determine functional equations satisfied by the associated generating functions, and generate enumerative data for m ≥ 4.
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Acknowledgements
We are grateful to an anonymous referee for many constructive suggestions, to Mireille Bousquet-Mélou for her suggestions, and to Mogens Lemvig Hansen for his tireless generation of numbers with Maple. The first author is partially supported by an Natural Sciences and Engineering Research Council of Canada Discovery Grant.
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Mishna, M., Yen, L. (2013). Set Partitions with No m-Nesting. In: Kotsireas, I., Zima, E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30979-3_13
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DOI: https://doi.org/10.1007/978-3-642-30979-3_13
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