Journal of Combinatorics
Volume 3 (2012)
Two covering polynomials of a finite poset, with applications to root systems and ad-nilpotent ideals
Pages: 63 – 89
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.
root system, ad-nilpotent ideal, graded poset, Hasse diagram
2010 Mathematics Subject Classification
06A07, 17B20, 20F55