6 VOLUME 23, No. 2, 2019 Acta Mechanica Slovaca 23 (2): 6 - 19, June 2019 https://doi.org/10.21496/ams.2019.015 * Corresponding author: Samuel Asefa Fufa, E-mail address: samuel.asefa@aau.edu.et Acta Mechanica Slovaca ISSN 1335-2393 www.actamechanica.sk Received: 2019.02.21 Revised: 2019.03.28 Accepted: 2019.05.14 Available online: 2019.06.15 Shellability of Pointed Integer Partition Samuel Asefa Fufa 1 *, Ayichew Abebe Gebeyehu 1 1 Department of Mathematics, Adiss Ababa University, Ethiopia Bonga Collages of Teachers Education Abstract: In this paper we study, pointed integer partition defined as a pair { } { } 1 2 , , , , , r um uu u m =  where { } 1 2 , , , r u uu u =  is an integer partition of n - m, and m is a non-negative integer ≤ n. Shellability of pointed integer partition with Möbius values -1 and +1 denoted by n R . We determine the cardinality of n I • and n R for 1 ≤ n ≤ 10 and n ≥ 1 respectively and compute the Möbius number of n I • for 1 ≤ n ≤ 6. We have shown hat n R admit an EL-labeling which is EL-shellable. Keywords: Pointed Integer Partition, Hassee Diagram, Möbius Values and EL-Shellability. 1. Preliminary 1.1 Notations Through out this paper we will use the following notations: 1) [] { } 1, 2,3, , . n n =  2) X = Cardinality of X. 3) Partially ordered set (or poset) ≡ . 4)  indicates cover relation. 5)  indicates vertex of the Hasse diagram. 6) On the Hasse diagram marked by red color 0, +1 or -1 are Möbius numbers. 7) ˆ 0 and ˆ 1 stand minimal and maximal elements of poset respectively. 8) m denoted Möbius function. 1.2 Partially Ordered Sets and Möbius Function 1.2.1 Equivalence Relations and Partitions Definition 1.2.1. A binary relation R on a set X is said to be a. reflexive if xRx for all x in X, b. symmetric if xRy implies , , yRx xy X ∀ ∈ c. transitive if xRy and yRz imply , , . xRz xyz X ∀ ∈ A relation R is called an equivalence relation if it is reflexive, symmetric and transi- tive, and in this case, we say that x and y are equivalent, if xRy. Definition 1.2.2. For an equivalence relation R on a set A, the set of the elements of A that are related to an element, say a, of A is called the equivalence class of element a and it is denoted by [a]. Definition 1.2.3. A partition of a positive integer n is a way of writing n as a sum of positive integers. The summands of the partition are known as parts. 1.2.2 Partially Ordered Sets Definition 1.2.4. A partially ordered set P (or poset, for short) is a set (which by abuse of notation we also call P), together with a binary relation denoted ≤ (or p ≤ p when there is a possibility of confusion), satisfying the following three axioms: 1. For all t P ∈ , t ≤ t (reflexivity ) 2. If s ≤ t and t ≤ s, then s = t (antisymmetry ) 3. If s ≤ t and t ≤ u, then s ≤ u (transitivity ) We use the obvious notation t ≥ s to mean s ≤ t, s < t to mean s ≤ t and , s t ≠ and t > s to mean s < t. We say that two elements s and t of P are comparable if s ≤ t or t AMS _2-2019.indd 6 12.12.2019 9:32:22