The Rank Enumeration of Certain Parabolic Non-Crossing Partitions

Abstract

We consider $m$-divisible non-crossing partitions of $\{1,2,\dots ,mn\}$ with the property that for some $t\le n$ no block contains more than one of the first $t$ integers. 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 $m=1$.

arXiv comment34 pages, 7 figures. Comments are welcome

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• Hochschild lattices and shuffle lattices
• Parabolic Tamari Lattices in Linear Type B
• $F$- and $H$-Triangles for $\nu$-Associahedra
• Refined Lattice Path Enumeration and Combinatorial Reciprocity