The rank enumeration of certain parabolic non-crossing partitions
Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 437-468.

We consider m-divisible non-crossing partitions of {1,2,...,mn} with the property that for some tn no block contains more than one of the integers 1,2,...,t. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapoton’s M-triangle in this setting and conjecture a combinatorial interpretation for the H-triangle. This conjecture is proved for m=1.

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DOI: 10.5802/alco.219
Classification: 05A15,  05A18,  06A07
Keywords: Non-crossing partition, generating function, Lagrange inversion, zeta polynomial, Dyck path, ballot path.
Krattenthaler, Christian 1; Mühle, Henri 2

1 Universität Wien Fakultät für Mathematik Oskar-Morgenstern-Platz 1 A-1090 Vienna Austria
2 Technische Universität Dresden Institut für Algebra Zellescher Weg 12–14 01069 Dresden Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Krattenthaler, Christian; Mühle, Henri. The rank enumeration of certain parabolic non-crossing partitions. Algebraic Combinatorics, Volume 5 (2022) no. 3, pp. 437-468. doi : 10.5802/alco.219. https://alco.centre-mersenne.org/articles/10.5802/alco.219/

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