The Rank Enumeration of Certain Parabolic Non-Crossing Partitions

TL;DR

This paper studies a special class of non-crossing partitions with divisibility and block constraints, providing formulas for multi-chains, computing related combinatorial invariants, and proposing a conjecture with a proof for the base case.

Contribution

It introduces a closed-form enumeration of multi-chains of certain parabolic non-crossing partitions and explores their combinatorial invariants, extending understanding in this area.

Findings

Derived a closed formula for the number of multi-chains with prescribed blocks.

Computed Chapoton's M-triangle in this setting.

Conjectured and proved a combinatorial interpretation for the H-triangle when m=1.

Abstract

We consider $m$-divisible non-crossing partitions of $\{1,2,\ldots,mn\}$ with the property that for some $t\leq n$ 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 $m=1$.