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# Journal of Combinatorics

## Volume 3 (2012)

### Number 1

### Two covering polynomials of a finite poset, with applications to root systems and ad-nilpotent ideals

Pages: 63 – 89

DOI: http://dx.doi.org/10.4310/JOC.2012.v3.n1.a3

#### Author

#### Abstract

We introduce two polynomials (in $q$) associated with a finite poset $P$ that encode some information on the covering relation in $P$. If $P$ is a distributive lattice, and hence $P$ is isomorphic to the poset of dual order ideals in a poset $L$, then these polynomials coincide and the coefficient of $q$ equals the number of $k$-element antichains in $L$. In general, these two covering polynomials are different, and we introduce a deviation polynomial of $P$, which measures the difference between these two. We then compute all these polynomials in the case, where $P$ is one of the posets associated with an irreducible root system. These are 1) the posets of positive roots, 2) the poset of ad-nilpotent ideals, and 3) the poset of Abelian ideals.

#### Keywords

root system, ad-nilpotent ideal, graded poset, Hasse diagram

#### 2010 Mathematics Subject Classification

06A07, 17B20, 20F55