In this note, we introduce two polynomials associated with a finite poset P , study their properties, and then compute these polynomials in the case, where P is one of the posets considered in [13]. Consider two statistics κ and ι on P . By definition, κ(x), x ∈ P , is the number of elements of P that are covered by x, and ι(x) is the number of elements that covers x. The generating function associated with κ (resp. ι) is the upper (resp. lower) covering polynomial of P . That is, KP(q) = ∑ x∈P q κ(x) and ǨP(q) = ∑ x∈P q . The upper covering polynomial, KP , has briefly been considered, without adjective ‘upper’, in [13, Section 5]. It immediately follows from the definition that KP(1) = ǨP(1) and K ′ P(1) = Ǩ ′ P(1). The last equality follows from the observation that both values equal the number of edges in the Hasse diagram of P . Hence KP(q)− ǨP(q) = (q − 1) DP(q) for some polynomial DP , which is said to be the deviation polynomial of P . We begin with a simple observation that KP ≡ ǨP whenever P is a distributive lattice or admits an order-reversing involution. Let ∆ be an irreducible root system, ∆ a subset of positive roots, and Π ⊂ ∆ is the set of simple roots. If ∆ is reduced, then g denotes the respective simple Lie algebra, with fixed Borel subalgebra b corresponding to ∆. We determine KP , ǨP , and DP in the following three cases: 1) P = ∆; 2)P = J(∆), the poset of dual order ideals in∆, orP = J(∆\Π), the poset of dual order ideals in∆ \Π. In the Lie algebra case, these posets are isomorphic to the poset of all and strictly positive ad-nilpotent ideals in b. These two posets are also denoted by Ad and Ad0, respectively; 3) P is the poset of Abelian ideal in J(∆), denoted Ab. Let us briefly describe our results. For ∆, it is shown that deg K∆+ = deg Ǩ∆+ 6 3 and the coefficients of q in K∆+ and Ǩ∆+ are equal. This readily implies that the deviation polynomial is quite simple; namely, D∆+(q) ≡ rk∆− 1.