2005/02/17 by Dmitri I. Panyushev, Panyushev, Dmitri I.
Computer Science · Mathematics · #06A07 #17B20 #20F55 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT) #math.CO #math.RT #msc:06A07 #msc:17B20 #msc:20F55
paper · pdf · doi:10.48550/arxiv.math/0502386
23 pp, v3: considerable revision; v4: final version, to appear in" Journal of Combinatorics"
openalex publication_date 2005/02/17 · arxiv created 2012/05/21 · arxiv updated 2012/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.