Title Science and Technology Indonesia e-ISSN:2580-4391 p-ISSN:2580-4405 Vol. 8, No. 2, April 2023 Research Paper The Relationship of Multiset, Stirling Number, Bell Number, and Catalan Number Wamiliana1*, Attiya Yuliana1,2, Fitriani1 1Department of Mathematics, FMIPA, Universitas Lampung, Lampung, 35145, Indonesia2Senior High School 1, Gedung Tataan, Bandar Lampung, 35362, Indonesia *Corresponding author: wamiliana.1963@fmipa.unila.ac.id AbstractCatalan numbers is not as famous as Fibonacci numbers, however this number has own its beauty and arts. Catalan numbers wasdiscovered by Ming Antu in 1730, however, this numbers is credited to Eugene Catalan when he was studying parentheses in 1838.Catalan numbers mostly occurs in counting or enumeration problems. The Catalan numbers can be defined in more than one forms, and the most famous form is Cn = 1n+1 ( 2n n ) . In this study we will discuss the multiset construction and the relationship of the results of Multiset with Stirling, Bell, and Catalan numbers. KeywordsCounting, Enumeration, Multiset, Catalan Numbers, Stirling Numbers Received: 19 January 2023, Accepted: 13 April 2023 https://doi.org/10.26554/sti.2023.8.2.330-337 1. INTRODUCTION In mathematics, there are many types of sequence of numbers. Each sequence has its own definition, uniqueness and benefits. For example, a very famous number sequence, namely the Fibonacci number sequence, which has a uniqueness, namely the Golden Ratio and its benefits have also been widely applied in life. Catalan numbers are not as popular as the Fibonacci numbers, but this sequence of numbers also has many ben- efits. Catalan numbers are named after the Belgian scientist Eugene C. Catalan based on his work while studying paren- theses sequences in 1838. The parentheses in question are sequences that are well formed parentheses (Pak, 2014) . In addition to brackets, other forms related to Catalan numbers are the triangular forms of the convex polygon (Cayley, 1890) . Actually in 1730 Ming Antu had discovered the Catalan Numbers based on the geometrical models constructed (Koshy, 2008) . However, because the results were in China and not known in the Western world, Catalan was bettern known, and the numbers are called as Catalan Numbers. This number is unique in that it can be defined in several forms and the most famous form is Cn = 1 n+1 ( 2n n ) . The following is the first nine terms of Catalan number: 1, 2, 5, 14, 42, 132, 429, 1430, 4862,· · · . Not as Fibonacci sequence where we can easily pre- dict the nth term if given the (n-1) th and (n-2) th terms using its recursive formula Fn = F(n−1)+F(n−2), in Catalan sequence to predict the nth term by using recursive formula C(n+1) = C0Cn+C1C(n−1)+C2C(n−2)+· · ·+CnC0 is a little bit harder, be- cause we have to know almost all values of the terms. For Catalan sequence it is easier to count the nth term by using the explisit formula Cn = 1 n+1 ( 2n n ) . The specialty of Catalan numbers is that they often appear in different problems with different solutions, but have the same final solution, namely Catalan numbers. Shapiro (1976) investigated lattice paths in the first quad- rant and derive a similar triangle to Pascal’s triangle and called it Catalan’s triangle because it involves the Catalan numbers. Other geometric shapes that turn out to form Catalan numbers include the combination of the Laticce Path based on Catalan numbers (Saračević et al., 2018) , binary tress and triangulation of a convex polygons (Koshy, 2008; Stojadinovic, 2015). Lee and Oh (2018) showed that any binomial coefficient can be written as weighted sums along rows of the Catalan triangle. Ceballos and D’León (2018) investigated a generalization of the Catalan objects indexed by a composition of the form s = (s(1), s(2),...,s(a)), which we call a signature, and the object is combinatoric of planar rooted tree. Catalan numbers are also used to solve the pill problem (Bayer and Brandt, 2014) . Boyadzhiev (2021) explored a for- mula for generating various series, including Catalan, Bernoulli, Harmonic, and Stirling numbers. A new conservative matrix derived by Catalan numbers was investigated by İlkhan (2020) . So, actually there are a lot of benefits from the Catalan num- https://crossmark.crossref.org/dialog/?doi=10.26554/sti.2023.8.2.330-337&domain=pdf https://doi.org/10.26554/sti.2023.8.2.330-337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 bers themselves. In addition, Catalan numbers have also been widely applied in various fields, such as engineering, in com- putational geometry, geographic information systems, crypto- graphic geodesy, and medicine (Selim and Saračević, 2019) . For data security, Saračević et al. (2018) investigated the appli- cation of Catalan numbers and lattice path for cryptography; and Saračević et al. (2017) used Catalan keys based on dy- namic programming for steganography. Moreover, in 2019, Saračević et al. (2019) continued exploring Catalan numbers and Dyck word for steganography. (Ndagijimana, 2016) inves- tigated the properties and application of Catalan number in the RNA (Ribonucleic Acid) secondary structure. For more com- prhehensive discussion about Catalan number can be found in Koshy (2008) and Stanley (2015) . One of the problems whose solution is Catalan numbers is the problem of Catalan numbers in Algebra, namely the Parker permutation problem (Guy, 1993) . In the book Catalan numbers and their applications written by T. Koshy it was informed that this problem had been answered by Ira M. Gessel of Brandies University, Waltham, Massachusetts in 1993, by providing a solution in determining the number of multisets with n-elements a1, a2, a3,· · · , an are members of ai∈Zn such that a1 + a2 + a3+ · · · + an = is the identity of the sum Zn (Koshy, 2008) . In this study we will discuss the multisets of Group Additive Zn with 1 ≤ n ≤ 10, and shows the ralations of those multisets with Bell numbers, Stirling numbers, and Catalan numbers. 2. THE METHOD 2.1 Multiset Let S be a non-empty set. A multiset M from a set S is an ordered pair: M = {(si,ni)|si∈S, ni∈ℤ+,𝕫i≠sj for i≠j}, where ℤ+ = {1,2,. . . }, ni is called as the multiplicity of element si in M. If the underlying set is finite, the multiset is called finite. The size of a finite multiset M is defined as the sum of the multiplicities of all of its elements (Roman, 2005) . For example, M = {(p,3), (q,1), (r,2)} is a multiset of underlying set S = {p, q, r} in which element p has multiplicity 3, q has multiplicity 1 and r has multiplicity 2. Another way to write this multiset is by writing out all elements according the multiplicities, for above example write M as M = {p, p, p, q, r, r}. If X is a set of elements, a multiset A from the set X is represented by a function CA is CA: M→N, where N is the set of non-negative integers. For every x∈X, CA(x) is the charac- teristic value of x in A and shows how many times the element x appears in A. A multiset A is a set if CA(x) = 0 or 1 for all x (Tripathy et al., 2018) . There are a lot of possible permutation of multiset, for example, Albert et al. (2001) investigated a permutation of a multiset which do not contain certain ordered patterns of length 3. The following table shows the example of possible multiset with n elements of additive group ℤn+1, 1 ≤ n ≤ 4 (Koshy, 2008) . Table 1 is the table that shows the number of multisets of group ℤn+1, 1 ≤ n ≤4 where the number of multisets constitute Table 1. The Number of Multisets With n Elements of Group ℤn+1, 1≤ n ≤ 4 n ℤn+1 Multiset with n-elements The number of multisets 1 {0̄, 1̄} 0̄ 1 2 {0̄, 1̄, 2̄} {0̄, 0̄} {1̄, 2̄} 2 3 {0̄, 1̄, 2̄, 3̄} {0̄, 0̄, 0̄} {0̄, 1̄, 3̄} {0̄, 2̄, 2̄} {1̄, 1̄, 2̄} {2̄, 3̄, 3̄} 5 three Catalan numbers which are 1, 2 and 5. 2.2 Catalan Numbers There are some definitions regarding Catalan numbers. How- ever, the most famous way of defining Catalan numbers is Cn = 1 n + 1 ( 2n n ) where n ≥ 0, n ∈ ℤ+ (1) Singmaster (1978) gave a mathematical proof that Equa- tion (1) can also be written as Cn = ( 2n + 1 n ) 2n + 1 (2) Table 2. Second-kind Stirling Number, s(n, k) and Bell number Bn = ∑n k=0S(n,k) n\k 1 2 3 4 5 6 7 8 9 Bell’s number 1 1 1 2 1 1 2 3 1 3 1 5 4 1 7 6 1 15 5 1 15 25 10 1 52 6 1 31 90 65 15 1 203 7 1 63 301 350 140 21 1 877 8 1 127 966 1701 1050 266 28 1 4140 9 1 255 3025 7770 6951 2646 462 36 1 21147 2.3 Bell Numbers A partition of a set S is a collection of non-empty subsets Ai ⊆ S, 1 ≤ i ≤ k, such that ∪ki=1Ai = S and for every i ≠ j Ai∩Aj = ∅. Bell number Bn is the number of partitions of an n-element set and is defined as: Bn = ∑n k=0S(n,k). Since Bn = 1 , then Bn+1 =∑n k=0 ( n k ) Bk. For example, B2 = 2 because the 2-element set {a, b} can be partitioned in 2 distinct ways: {a, b}, and {{a}, {b}}, and b3 = 5, because there are 5 different partitions of 3-element set {a, b, c} which are: {{a}, {b}, {c}}, {{a,b}, {c}}, {{a,c},{b} {{b, c}, {c}, and {a, b, c}. 2.4 Striling Numbers Stirling numbers appear in a variety of combinatorial situations. Stirling numbers are classified into two kinds: Stirling num- bers of the first kind and Stirling numbers of the second kind. © 2023 The Authors. Page 331 of 337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 Table 3. The Multiset Group ℤn+1 with 1 ≤ n ≤7 n ℤn+1 Multiset with n-elements The number of Multiset 1 {0̄,0̄} {0̄} 1 2 {0̄,1̄,2̄ } {0̄,0̄} {1̄,2̄} 2 3 {0̄,1̄,2̄,3̄} {0̄,0̄,0̄} {0̄,1̄,3̄} {0̄,2̄,2̄} {1̄,1̄,2̄} {2̄,3̄,3̄} 5 4 {0̄,1̄,2̄,3̄,4̄} {0̄,0̄,1̄,4̄} {0̄,0̄,2̄,3̄} {0̄,1̄,1̄,3̄} {0̄,1̄,2̄,2̄} {0̄,2̄,4̄,4̄} {0̄,3̄,3̄,4̄} {1̄,1̄,1̄,2̄} {1̄,1̄,4̄,4̄} {1̄,2̄,3̄,4̄} {1̄,3̄,3̄,3̄} {2̄,2̄,2̄,4̄} {2̄,2̄,3̄,3̄} {3̄,4̄,4̄,4̄} 14 5 {0̄,1̄,2̄,3̄,4̄,5̄} {0̄,0̄,0̄,0̄,0̄} {0̄,0̄,0̄,0̄,5̄} {0̄,0̄,0̄,2̄,4̄} {0̄,0̄,0̄,3̄,3̄} {0̄,0̄,1̄,1̄,4̄} {0̄,0̄,1̄,2̄,3̄} {0̄,0̄,2̄,2̄,2̄} {0̄,1̄,1̄,1̄,3̄} {0̄,0̄,2̄,5̄,5̄} {0̄,0̄,3̄,4̄,5̄} {0̄,0̄,4̄,4̄,4̄} {0̄,1̄,1̄,2̄,2̄} {0̄,1̄,1̄,5̄,5̄} {0̄,1̄,3̄,3̄,5̄} {0̄,1̄,3̄,4̄,4̄} {0̄,2̄,2̄,4̄,4̄} {0̄,2̄,2̄,3̄,5̄} {0̄,2̄,3̄,3̄,4̄} {0̄,3̄,3̄,3̄,3̄} {1̄,1̄,1̄,1̄,2̄} {1̄,1̄,2̄,4̄,5̄} {1̄,1̄,2̄,3̄,5̄} {1̄,1̄,2̄,4̄,4̄} {1̄,1̄,3̄,3̄,4̄} {1̄,2̄,2̄,2̄,5̄} {1̄,2̄,2̄,3̄,4̄} {1̄,2̄,3̄,3̄,3̄} {1̄,2̄,5̄,5̄,5̄} {1̄,3̄,4̄,5̄,5̄} {1̄,4̄,4̄,4̄,5̄} {0̄,2̄,2̄,2̄,4̄} {2̄,2̄,2̄,3̄,3̄} {2̄,2̄,4̄,5̄,5̄} {2̄,3̄,3̄,5̄,5̄} {2̄,3̄,4̄,4̄,5̄} {2̄,4̄,4̄,4̄,4̄} {3̄,3̄,3̄,4̄,5̄} {3̄,3̄,4̄,4̄,4̄} {4̄,5̄,5̄,5̄,5̄} 42 6 {0̄,1̄,2̄,3̄,4̄,5̄,6̄} {0̄,0̄,0̄,0̄,0̄,0̄} {0̄,0̄,0̄,0̄,1̄,6̄} {0̄,0̄,0̄,0̄,2̄,5̄} {0̄,0̄,0̄,0̄,3̄,4̄} {0̄,0̄,0̄,1̄,1̄,5̄} {0̄,0̄,0̄,1̄,3̄,3̄} {0̄,0̄,0̄,2̄,2̄,3̄} {0̄,0̄,0̄,2̄,6̄,6̄} {0̄,0̄,0̄,3̄,5̄,6̄} {0̄,0̄,0̄,4̄,4̄,6̄} {0̄,0̄,0̄,4̄,5̄,5̄} {0̄,0̄,1̄,1̄,1̄,4̄} {0̄,0̄,1̄,1̄,2̄,3̄} {0̄,0̄,1̄,2̄,2̄,2̄} {0̄,0̄,1̄,1̄,6̄,6̄} {0̄,0̄,1̄,2̄,5̄,6̄} {0̄,0̄,1̄,3̄,4̄,6̄} {0̄,0̄,1̄,3̄,5̄,5̄} {0̄,0̄,1̄,4̄,4̄,5̄} {0̄,0̄,2̄,2̄,4̄,6̄} {0̄,0̄,2̄,2̄,5̄,5̄} {0̄,0̄,2̄,3̄,3̄,6̄} {0̄,0̄,4̄,5̄,6̄,6̄} {0̄,0̄,5̄,5̄,5̄,6̄} {0̄,1̄,1̄,1̄,1̄,3̄} {0̄,1̄,1̄,1̄,2̄,2̄} {0̄,1̄,1̄,1̄,5̄,6̄} {0̄,1̄,1̄,2̄,4̄,6̄} {0̄,1̄,1̄,2̄,5̄,5̄} {0̄,1̄,1̄,3̄,3̄,6̄} {0̄,1̄,1̄,3̄,4̄,5̄} {0̄,1̄,1̄,4̄,4̄,4̄} {0̄,1̄,2̄,2̄,3̄,6̄} {0̄,1̄,2̄,4̄,4̄,5̄} {0̄,1̄,2̄,3̄,3̄,5̄} {0̄,1̄,3̄,2̄,4̄,4̄} {0̄,1̄,2̄,6̄,6̄,6̄} {0̄,1̄,3̄,3̄,3̄,4̄} {0̄,1̄,4̄,4̄,6̄,6̄} {0̄,1̄,4̄,5̄,5̄,6̄} {0̄,1̄,5̄,5̄,5̄,5̄} {0̄,2̄,2̄,2̄,2̄,6̄} {0̄,2̄,2̄,2̄,3̄,5̄} {0̄,2̄,2̄,2̄,4̄,4̄} {0̄,2̄,2̄,3̄,3̄,4̄} {0̄,2̄,3̄,3̄,3̄,3̄} {0̄,2̄,2̄,5̄,6̄,6̄} {0̄,2̄,3̄,4̄,6̄,6̄} {0̄,2̄,3̄,5̄,6̄,6̄} {0̄,2̄,3̄,4̄,6̄,6̄} {0̄,2̄,3̄,5̄,5̄,6̄} {0̄,2̄,4̄,4̄,5̄,6̄} {0̄,3̄,3̄,4̄,5̄,6̄} {0̄,3̄,3̄,5̄,5̄,5̄} {0̄,3̄,4̄,4̄,4̄,6̄} {0̄,3̄,4̄,4̄,5̄,5̄} {0̄,4̄,4̄,4̄,4̄,5̄} {0̄,4̄,6̄,6̄,6̄,6̄} {0̄,5̄,5̄,6̄,6̄,6̄} {1̄,1̄,1̄,1̄,1̄,2̄} {1̄,1̄,1̄,1̄,4̄,6̄} {1̄,1̄,1̄,1̄,5̄,5̄} {1̄,1̄,1̄,2̄,3̄,6̄} {1̄,1̄,1̄,2̄,4̄,5̄} {1̄,1̄,1̄,3̄,4̄,4̄} {1̄,1̄,1̄,3̄,3̄,5̄} {1̄,1̄,1̄,6̄,6̄,6̄} {1̄,1̄,2̄,2̄,2̄,6̄} {1̄,1̄,2̄,2̄,3̄,5̄} {1̄,1̄,2̄,2̄,4̄,4̄} {1̄,1̄,2̄,3̄,3̄,4̄} {1̄,1̄,3̄,3̄,3̄,3̄} {1̄,1̄,2̄,5̄,6̄,6̄} {1̄,1̄,2̄,5̄,5̄,7̄} {1̄,1̄,3̄,4̄,6̄,6̄} {1̄,1̄,3̄,5̄,5̄,6̄} {1̄,1̄,4̄,4̄,5̄,6̄} {1̄,1̄,4̄,5̄,5̄,5̄} {1̄,2̄,2̄,2̄,2̄,5̄} {1̄,2̄,2̄,2̄,3̄,4̄} {1̄,2̄,2̄,3̄,3̄,3̄} {1̄,2̄,2̄,4̄,6̄,6̄} {1̄,2̄,2̄,5̄,5̄,6̄} {1̄,2̄,3̄,3̄,6̄,6̄} {1̄,2̄,3̄,4̄,5̄,6̄} {1̄,2̄,3̄,5̄,5̄,5̄} {1̄,2̄,4̄,4̄,4̄,6̄} {1̄,2̄,4̄,4̄,5̄,5̄} {1̄,2̄,4̄,4̄,4̄,6̄} {1̄,2̄,4̄,4̄,5̄,5̄} {1̄,3̄,3̄,3̄,5̄,6̄} {1̄,2̄,4̄,4̄,5̄,5̄} {1̄,3̄,3̄,4̄,4̄,6̄} {1̄,3̄,3̄,4̄,5̄,5̄} {1̄,3̄,4̄,4̄,4̄,5̄} {1̄,3̄,6̄,6̄,6̄,6̄} {1̄,4̄,5̄,6̄,6̄,6̄} {1̄,5̄,5̄,5̄,6̄,6̄} {2̄,2̄,2̄,2̄,2̄,4̄} {2̄,2̄,2̄,2̄,3̄,3̄} {2̄,2̄,2̄,3̄,6̄,6̄} {2̄,2̄,2̄,4̄,5̄,6̄} {2̄,2̄,2̄,5̄,5̄,5̄} {2̄,2̄,3̄,3̄,5̄,6̄} {2̄,2̄,3̄,4̄,4̄,6̄} {2̄,2̄,3̄,4̄,5̄,5̄} {2̄,2̄,4̄,4̄,4̄,5̄} {2̄,2̄,6̄,6̄,6̄,6̄} {2̄,3̄,3̄,3̄,4̄,6̄} {2̄,3̄,3̄,3̄,5̄,5̄} {2̄,3̄,3̄,4̄,4̄,5̄} {2̄,3̄,4̄,4̄,4̄,4̄} {2̄,3̄,5̄,6̄,6̄,6̄} {2̄,4̄,4̄,6̄,6̄,6̄} {2̄,4̄,5̄,5̄,6̄,6̄} {2̄,5̄,5̄,5̄,5̄,6̄} {3̄,3̄,3̄,3̄,3̄,6̄} {3̄,3̄,3̄,3̄,4̄,5̄} {3̄,3̄,3̄,4̄,4̄,4̄} {3̄,3̄,4̄,6̄,6̄,6̄} {3̄,3̄,5̄,5̄,6̄,6̄} {3̄,4̄,4̄,5̄,6̄,6̄} {3̄,4̄,5̄,5̄,5̄,6̄} {3̄,5̄,5̄,5̄,5̄,5̄} {4̄,4̄,4̄,4̄,6̄,6̄} {4̄,4̄,4̄,5̄,5̄,6̄} {4̄,4̄,5̄,5̄,5̄,5̄} {5̄,6̄,6̄,6̄,6̄,6̄} 132 © 2023 The Authors. Page 332 of 337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 7 {0̄,1̄,2̄,3̄,4̄,5̄,6̄,7̄} {0̄,0̄,0̄,0̄,0̄,0̄,0̄} {0̄,0̄,0̄,0̄,0̄,1̄,7̄} {0̄,0̄,0̄,0̄,0̄,2̄,6̄} {0̄,0̄,0̄,0̄,0̄,3̄,5̄} {0̄,0̄,0̄,0̄,1̄,2̄,5̄} {0̄,0̄,0̄,0̄,1̄,3̄,4̄} {0̄,0̄,0̄,0̄,2̄,2̄,4̄} {0̄,0̄,0̄,0̄,2̄,3̄,3̄} {0̄,0̄,0̄,0̄,2̄,7̄,7̄} {0̄,0̄,0̄,0̄,3̄,6̄,7̄} {0̄,0̄,0̄,0̄,4̄,5̄,7̄} {0̄,0̄,0̄,0̄,4̄,6̄,6̄} {0̄,0̄,0̄,0̄,5̄,5̄,6̄} {0̄,0̄,0̄,1̄,1̄,1̄,5̄} {0̄,0̄,0̄,1̄,1̄,2̄,4̄} {0̄,0̄,0̄,1̄,1̄,3̄,3̄} {0̄,0̄,0̄,1̄,2̄,2̄,3̄} {0̄,0̄,0̄,2̄,2̄,2̄,2̄} {0̄,0̄,0̄,1̄,1̄,7̄,7̄} {0̄,0̄,0̄,1̄,2̄,6̄,7̄} {0̄,0̄,0̄,1̄,3̄,5̄,7̄} {0̄,0̄,0̄,1̄,3̄,6̄,6̄} {0̄,0̄,0̄,1̄,4̄,4̄,7̄} {0̄,0̄,0̄,1̄,4̄,5̄,6̄} {0̄,0̄,0̄,1̄,5̄,5̄,5̄} {0̄,0̄,0̄,1̄,4̄,4̄,7̄} {0̄,0̄,0̄,1̄,4̄,5̄,6̄} {0̄,0̄,0̄,2̄,3̄,4̄,7̄} {0̄,0̄,0̄,2̄,3̄,5̄,6̄} {0̄,0̄,0̄,2̄,4̄,4̄,6̄} {0̄,0̄,0̄,2̄,4̄,5̄,5̄} {0̄,0̄,0̄,3̄,3̄,3̄,7̄} {0̄,0̄,0̄,3̄,3̄,4̄,6̄} {0̄,0̄,0̄,3̄,3̄,5̄,5̄} {0̄,0̄,0̄,4̄,4̄,4̄,4̄} {0̄,0̄,0̄,4̄,6̄,7̄,7̄} {0̄,0̄,0̄,5̄,5̄,7̄,7̄} {0̄,0̄,0̄,5̄,6̄,6̄,7̄} {0̄,0̄,0̄,6̄,6̄,6̄,6̄} {0̄,0̄,1̄,1̄,1̄,1̄,4̄} {0̄,0̄,1̄,1̄,1̄,2̄,3̄} {0̄,0̄,1̄,1̄,2̄,2̄,2̄} {0̄,0̄,1̄,1̄,1̄,6̄,7̄} {0̄,0̄,1̄,1̄,2̄,5̄,7̄} {0̄,0̄,1̄,1̄,2̄,6̄,6̄} {0̄,0̄,1̄,1̄,3̄,4̄,7̄} {0̄,0̄,1̄,1̄,3̄,5̄,6̄} {0̄,0̄,1̄,1̄,4̄,4̄,6̄} {0̄,0̄,1̄,1̄,4̄,5̄,5̄} {0̄,0̄,1̄,2̄,2̄,4̄,7̄} {0̄,0̄,1̄,2̄,2̄,5̄,6̄} {0̄,0̄,1̄,2̄,3̄,3̄,7̄} {0̄,0̄,1̄,2̄,3̄,4̄,6̄} {0̄,0̄,1̄,2̄,3̄,5̄,5̄} {0̄,0̄,1̄,3̄,3̄,3̄,6̄} {0̄,0̄,1̄,3̄,3̄,5̄,5̄} {0̄,0̄,1̄,3̄,3̄,4̄,5̄} {0̄,0̄,1̄,3̄,4̄,4̄,4̄} {0̄,0̄,1̄,3̄,6̄,7̄,7̄} {0̄,0̄,1̄,4̄,5̄,7̄,7̄} {0̄,0̄,1̄,4̄,6̄,6̄,7̄} {0̄,0̄,1̄,5̄,5̄,6̄,7̄} {0̄,0̄,1̄,5̄,6̄,6̄,6̄} {0̄,0̄,1̄,2̄,7̄,7̄,7̄} {0̄,0̄,1̄,3̄,6̄,7̄,7̄} {0̄,0̄,1̄,4̄,5̄,7̄,7̄} {0̄,0̄,1̄,4̄,6̄,6̄,7̄} {0̄,0̄,1̄,5̄,5̄,6̄,7̄} {0̄,0̄,1̄,5̄,6̄,6̄,6̄} {0̄,0̄,2̄,2̄,2̄,3̄,7̄} {0̄,0̄,2̄,2̄,2̄,4̄,6̄} {0̄,0̄,2̄,2̄,2̄,5̄,5̄} {0̄,0̄,2̄,2̄,3̄,3̄,6̄} {0̄,0̄,2̄,2̄,3̄,4̄,5̄} {0̄,0̄,2̄,2̄,4̄,4̄,4̄} {0̄,0̄,2̄,2̄,6̄,7̄,7̄} {0̄,0̄,2̄,3̄,3̄,3̄,5̄} {0̄,0̄,2̄,3̄,3̄,4̄,4̄} {0̄,0̄,2̄,3̄,5̄,7̄,7̄} {0̄,0̄,2̄,3̄,6̄,6̄,7̄} {0̄,0̄,2̄,4̄,4̄,7̄,7̄} {0̄,0̄,2̄,4̄,5̄,6̄,7̄} {0̄,0̄,2̄,4̄,6̄,6̄,6̄} 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{1̄,1̄,1̄,2̄,2̄,2̄,7̄} {1̄,1̄,1̄,2̄,2̄,3̄,6̄} {1̄,1̄,1̄,2̄,2̄,4̄,5̄} {1̄,1̄,1̄,2̄,3̄,3̄,5̄} {1̄,1̄,1̄,2̄,3̄,4̄,4̄} {1̄,1̄,1̄,3̄,3̄,3̄,4̄} {1̄,1̄,1̄,2̄,5̄,7̄,7̄} {1̄,1̄,1̄,2̄,6̄,6̄,7̄} 429 © 2023 The Authors. Page 333 of 337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 {1̄,1̄,1̄,3̄,4̄,7̄,7̄} {1̄,1̄,1̄,3̄,5̄,6̄,7̄} {1̄,1̄,1̄,3̄,6̄,6̄,6̄} {1̄,1̄,1̄,4̄,4̄,6̄,7̄} {1̄,1̄,1̄,4̄,5̄,5̄,7̄} {1̄,1̄,1̄,4̄,5̄,6̄,6̄} {1̄,1̄,1̄,5̄,5̄,5̄,6̄} {1̄,1̄,2̄,2̄,2̄,2̄,6̄} {1̄,1̄,2̄,2̄,2̄,3̄,5̄} {1̄,1̄,2̄,2̄,2̄,4̄,4̄} {1̄,1̄,2̄,2̄,3̄,3̄,4̄} {1̄,1̄,2̄,3̄,3̄,3̄,3̄} {1̄,1̄,2̄,2̄,4̄,7̄,7̄} {1̄,1̄,2̄,2̄,5̄,6̄,7̄} {1̄,1̄,2̄,2̄,6̄,6̄,6̄} {1̄,1̄,2̄,3̄,3̄,7̄,7̄} {1̄,1̄,2̄,3̄,4̄,6̄,7̄} {1̄,1̄,2̄,3̄,5̄,5̄,7̄} {1̄,1̄,2̄,3̄,5̄,6̄,6̄} {1̄,1̄,2̄,4̄,4̄,5̄,7̄} {1̄,1̄,2̄,4̄,4̄,6̄,6̄} {1̄,1̄,2̄,4̄,5̄,5̄,6̄} {1̄,1̄,2̄,5̄,5̄,5̄,5̄} {1̄,1̄,2̄,7̄,7̄,7̄,7̄} {1̄,1̄,3̄,3̄,3̄,6̄,7̄} {1̄,1̄,3̄,3̄,4̄,5̄,7̄} {1̄,1̄,3̄,3̄,4̄,6̄,6̄} {1̄,1̄,3̄,3̄,5̄,5̄,6̄} {1̄,1̄,3̄,4̄,4̄,4̄,7̄} {1̄,1̄,3̄,4̄,4̄,5̄,6̄} {1̄,1̄,3̄,4̄,5̄,5̄,5̄} {1̄,1̄,3̄,6̄,7̄,7̄,7̄} {1̄,1̄,4̄,4̄,4̄,4̄,6̄} {1̄,1̄,4̄,4̄,4̄,5̄,5̄} {1̄,1̄,4̄,5̄,7̄,7̄,7̄} {1̄,1̄,4̄,6̄,6̄,7̄,7̄} {1̄,1̄,5̄,5̄,6̄,7̄,7̄} {1̄,1̄,5̄,6̄,6̄,6̄,7̄} {1̄,1̄,6̄,6̄,6̄,6̄,7̄} {1̄,2̄,2̄,2̄,2̄,3̄,4̄} {1̄,2̄,2̄,2̄,3̄,3̄,3̄} {1̄,3̄,3̄,3̄,3̄,4̄,7̄} {1̄,3̄,3̄,3̄,3̄,5̄,6̄} {1̄,3̄,3̄,3̄,4̄,4̄,6̄} {1̄,3̄,3̄,3̄,4̄,5̄,5̄} {1̄,3̄,3̄,4̄,4̄,4̄,5̄} {1̄,3̄,3̄,4̄,7̄,7̄,7̄} {1̄,3̄,3̄,5̄,6̄,7̄,7̄} {1̄,3̄,3̄,6̄,6̄,6̄,7̄} {1̄,3̄,4̄,4̄,4̄,4̄,4̄} {1̄,3̄,4̄,4̄,6̄,7̄,7̄} {1̄,3̄,4̄,5̄,5̄,7̄,7̄} {1̄,3̄,4̄,5̄,6̄,6̄,7̄} {1̄,3̄,4̄,6̄,6̄,6̄,6̄} {1̄,3̄,5̄,5̄,5̄,6̄,7̄} {1̄,3̄,5̄,5̄,6̄,6̄,6̄} {1̄,4̄,4̄,4̄,5̄,7̄,7̄} {1̄,4̄,4̄,4̄,6̄,6̄,7̄} {1̄,4̄,4̄,5̄,5̄,6̄,7̄} {1̄,4̄,4̄,5̄,6̄,6̄,6̄} {1̄,4̄,5̄,5̄,5̄,5̄,7̄} {1̄,4̄,5̄,5̄,5̄,6̄,6̄} {1̄,5̄,5̄,5̄,5̄,5̄,6̄} {1̄,5̄,6̄,7̄,7̄,7̄,7̄} {1̄,6̄,6̄,6̄,7̄,7̄,7̄} {2̄,2̄,2̄,2̄,2̄,2̄,4̄} {2̄,2̄,2̄,2̄,2̄,3̄,3̄} {2̄,2̄,2̄,2̄,2̄,7̄,7̄} {2̄,2̄,2̄,2̄,3̄,6̄,7̄} {2̄,2̄,2̄,2̄,4̄,5̄,7̄} {2̄,2̄,2̄,2̄,4̄,6̄,6̄} {2̄,2̄,2̄,2̄,5̄,5̄,6̄} {2̄,2̄,2̄,3̄,3̄,6̄,6̄} {2̄,2̄,2̄,3̄,4̄,4̄,7̄} {2̄,2̄,2̄,3̄,4̄,5̄,6̄} {2̄,2̄,2̄,3̄,5̄,5̄,5̄} {2̄,2̄,2̄,4̄,4̄,4̄,6̄} {2̄,2̄,2̄,4̄,4̄,5̄,5̄} {2̄,2̄,2̄,5̄,7̄,7̄,7̄} {2̄,2̄,2̄,6̄,6̄,7̄,7̄} {2̄,2̄,3̄,3̄,3̄,4̄,7̄} {2̄,2̄,3̄,3̄,3̄,5̄,6̄} {2̄,2̄,3̄,3̄,4̄,4̄,6̄} {2̄,2̄,3̄,3̄,4̄,5̄,5̄} {2̄,2̄,3̄,4̄,4̄,4̄,5̄} {2̄,2̄,4̄,4̄,4̄,4̄,4̄} {2̄,3̄,3̄,3̄,3̄,3̄,7̄} {2̄,3̄,3̄,3̄,3̄,4̄,6̄} {2̄,3̄,3̄,3̄,3̄,5̄,5̄} {2̄,3̄,3̄,3̄,4̄,4̄,5̄} {2̄,3̄,3̄,4̄,4̄,4̄,4̄} {2̄,3̄,3̄,3̄,7̄,7̄,7̄} {2̄,3̄,3̄,4̄,6̄,7̄,7̄} {2̄,3̄,3̄,5̄,5̄,7̄,7̄} {2̄,3̄,3̄,5̄,6̄,6̄,7̄} {2̄,3̄,3̄,6̄,6̄,6̄,6̄} {2̄,3̄,4̄,4̄,5̄,7̄,7̄} {2̄,3̄,4̄,4̄,6̄,6̄,7̄} {2̄,3̄,4̄,5̄,5̄,6̄,7̄} {2̄,3̄,4̄,5̄,6̄,6̄,6̄} {2̄,3̄,5̄,5̄,5̄,5̄,7̄} {2̄,3̄,7̄,7̄,7̄,7̄,7̄} {2̄,4̄,4̄,4̄,4̄,7̄,7̄} {2̄,4̄,4̄,4̄,5̄,6̄,7̄} {2̄,4̄,4̄,4̄,6̄,6̄,6̄} {2̄,4̄,4̄,5̄,5̄,5̄,7̄} {2̄,4̄,4̄,5̄,5̄,6̄,6̄} {2̄,4̄,5̄,5̄,5̄,5̄,6̄} {2̄,4̄,6̄,7̄,7̄,7̄,7̄} {2̄,5̄,5̄,5̄,5̄,5̄,5̄} {2̄,5̄,5̄,7̄,7̄,7̄,7̄} {2̄,5̄,6̄,6̄,7̄,7̄,7̄} {2̄,6̄,6̄,6̄,6̄,6̄,7̄} {3̄,3̄,3̄,3̄,3̄,3̄,6̄} {3̄,3̄,3̄,3̄,3̄,4̄,5̄} {3̄,3̄,3̄,3̄,4̄,4̄,4̄} {3̄,3̄,3̄,3̄,6̄,7̄,7̄} {3̄,3̄,3̄,4̄,5̄,7̄,7̄} {3̄,3̄,3̄,4̄,6̄,6̄,7̄} {3̄,3̄,3̄,5̄,5̄,6̄,7̄} {3̄,3̄,3̄,5̄,6̄,6̄,6̄} {3̄,3̄,4̄,4̄,4̄,7̄,7̄} {3̄,3̄,4̄,4̄,5̄,6̄,7̄} {3̄,3̄,4̄,4̄,6̄,6̄,6̄} {3̄,3̄,4̄,5̄,5̄,5̄,7̄} {3̄,3̄,4̄,5̄,5̄,6̄,6̄} {3̄,3̄,5̄,5̄,5̄,5̄,6̄} {3̄,3̄,6̄,7̄,7̄,7̄,7̄} {3̄,4̄,4̄,4̄,4̄,6̄,7̄} {3̄,4̄,4̄,4̄,5̄,5̄,7̄} {3̄,4̄,4̄,4̄,5̄,6̄,6̄} {3̄,4̄,4̄,5̄,5̄,5̄,6̄} {3̄,4̄,5̄,5̄,5̄,5̄,5̄} {3̄,4̄,5̄,7̄,7̄,7̄,7̄} {3̄,4̄,6̄,6̄,7̄,7̄,7̄} {3̄,5̄,5̄,6̄,7̄,7̄,7̄} {3̄,5̄,6̄,6̄,6̄,7̄,7̄} {3̄,6̄,6̄,6̄,6̄,6̄,7̄} {4̄,4̄,4̄,4̄,5̄,5̄,6̄} {4̄,4̄,4̄,5̄,5̄,5̄,5̄} {4̄,4̄,4̄,7̄,7̄,7̄,7̄} {4̄,4̄,5̄,6̄,7̄,7̄,7̄} {4̄,4̄,6̄,6̄,6̄,7̄,7̄} {4̄,5̄,5̄,5̄,7̄,7̄,7̄} {4̄,5̄,5̄,6̄,6̄,7̄,7̄} {4̄,5̄,6̄,6̄,6̄,6̄,7̄} {4̄,6̄,6̄,6̄,6̄,6̄,6̄} {5̄,5̄,5̄,5̄,6̄,7̄,7̄} {5̄,5̄,5̄,6̄,6̄,6̄,7̄} {5̄,5̄,6̄,6̄,6̄,6̄,6̄} © 2023 The Authors. Page 334 of 337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 Table 4. Multiset of Additive Group ℤn+1 with 1 ≤ n ≤ 7 which is the Partition of ∑n−1 k=0 p(k(n + 1); n, n) n ℤn+1 k Multiset The number of Multisets Total 1 {0̄,1̄} 0 {0̄} 1 1 2 {0̄,1̄,2̄} 0 {0̄,0̄} 1 2 1 {1̄,2̄} 1 3 {0̄,1̄,2̄,3̄} 0 {0̄,0̄,0̄} 1 1 {0̄,1̄,3̄} {0̄,2̄,2̄} {1̄,1̄,2̄} 3 5 2 {2̄,3̄,3̄} 1 4 {0̄,1̄,2̄,3̄,4̄} 0 {0̄,0̄,0̄,0̄} 1 14 1 {0̄,0̄,1̄,4̄} {0̄,0̄,2̄,3̄} {0̄,1̄,1̄,3̄} {0̄,1̄,2̄,3̄} {1̄,1̄,1̄,2̄} 5 2 {0̄,2̄,4̄,4̄} {0̄,3̄,3̄,4̄} {1̄,1̄,4̄,4̄} {1̄,2̄,3̄,4̄} {1̄,3̄,3̄,3̄} {2̄,2̄,2̄,4̄} {2̄,2̄,3̄,3̄} 7 3 {3̄,4̄,4̄,4̄} 1 5 {0̄,1̄,2̄,3̄,4̄,5̄} 0 {0̄,0̄,0̄,0̄,0̄} 1 181 1 {0̄,0̄,0̄,1̄,5̄} {0̄,0̄,0̄,2̄,4̄} {0̄,0̄,0̄,3̄,3̄} {0̄,0̄,1̄,1̄,4̄} {0̄,0̄,1̄,2̄,3̄} {0̄,0̄,2̄,2̄,2̄} {0̄,1̄,1̄,1̄,3̄} {0̄,1̄,1̄,2̄,2̄} {1̄,1̄,1̄,1̄,2̄} 9 2 {0̄,0̄,2̄,5̄,5̄} {0̄,0̄,3̄,4̄,5̄} {0̄,0̄,4̄,4̄,4̄} {0̄,1̄,1̄,5̄,5̄} {0̄,1̄,2̄,4̄,5̄} {0̄,1̄,3̄,3̄,5̄} {0̄,1̄,3̄,4̄,4̄} {0̄,2̄,2̄,4̄,4̄} {0̄,2̄,2̄,3̄,5̄} {0̄,2̄,3̄,3̄,4̄} {0̄,3̄,3̄,3̄,3̄} {1̄,1̄,1̄,4̄,5̄} {1̄,1̄,2̄,3̄,5̄} {1̄,1̄,2̄,4̄,4̄} {1̄,1̄,3̄,3̄,4̄} {1̄,2̄,2̄,2̄,5̄} {1̄,2̄,2̄,3̄,4̄} {1̄,2̄,3̄,3̄,3̄} {2̄,2̄,2̄,2̄,4̄} {2̄,2̄,2̄,3̄,3̄} 20 3 {0̄,3̄,5̄,5̄,5̄} {0̄,4̄,4̄,5̄,5̄} {1̄,2̄,5̄,5̄,5̄} {1̄,3̄,4̄,5̄,5̄} {1̄,4̄,4̄,4̄,5̄} {2̄,2̄,4̄,5̄,5̄} {2̄,3̄,3̄,5̄,5̄} {2̄,3̄,4̄,4̄,5̄} {2̄,4̄,4̄,4̄,4̄} {3̄,3̄,3̄,4̄,5̄} {3̄,3̄,4̄,4̄,4̄} 11 4 {4̄,5̄,5̄,5̄,5̄} 1 4 {0̄,0̄,4̄,7̄,7̄,7̄,7̄} {0̄,0̄,5̄,6̄,7̄,7̄,7̄} {0̄,0̄,6̄,6̄,6̄,7̄,7̄} {0̄,1̄,3̄,7̄,7̄,7̄,7̄} {0̄,1̄,4̄,6̄,7̄,7̄,7̄} {0̄,1̄,5̄,5̄,7̄,7̄,7̄} {0̄,1̄,5̄,6̄,6̄,7̄,7̄} {0̄,1̄,6̄,6̄,6̄,6̄,7̄} {0̄,2̄,2̄,7̄,7̄,7̄,7̄} {0̄,2̄,3̄,6̄,7̄,7̄,7̄} {0̄,2̄,4̄,5̄,7̄,7̄,7̄} {0̄,2̄,4̄,6̄,6̄,7̄,7̄} {0̄,2̄,5̄,5̄,6̄,7̄,7̄} {0̄,2̄,5̄,6̄,6̄,6̄,7̄} {0̄,2̄,6̄,6̄,6̄,6̄,6̄} {0̄,3̄,3̄,5̄,7̄,7̄,7̄} {0̄,3̄,3̄,6̄,6̄,7̄,7̄} {0̄,3̄,4̄,4̄,7̄,7̄,7̄} {0̄,3̄,4̄,5̄,6̄,7̄,7̄} {0̄,3̄,4̄,6̄,6̄,6̄,7̄} {0̄,3̄,5̄,5̄,5̄,7̄,7̄} {0̄,3̄,5̄,5̄,6̄,6̄,7̄} {0̄,3̄,5̄,6̄,6̄,6̄,6̄} {0̄,4̄,4̄,4̄,6̄,7̄,7̄} {0̄,4̄,4̄,5̄,5̄,7̄,7̄} {0̄,4̄,4̄,5̄,6̄,6̄,7̄} {0̄,4̄,4̄,6̄,6̄,6̄,6̄} {0̄,4̄,5̄,5̄,5̄,6̄,7̄} {0̄,4̄,5̄,5̄,6̄,6̄,6̄} {0̄,5̄,5̄,5̄,5̄,5̄,7̄} {0̄,5̄,5̄,5̄,5̄,6̄,7̄} {0̄,5̄,5̄,5̄,6̄,6̄,6̄} {1̄,1̄,2̄,7̄,7̄,7̄,7̄} {1̄,1̄,3̄,6̄,7̄,7̄,7̄} {1̄,1̄,4̄,5̄,7̄,7̄,7̄} {1̄,1̄,4̄,6̄,6̄,7̄,7̄} {1̄,1̄,5̄,5̄,6̄,7̄,7̄} {1̄,1̄,6̄,6̄,6̄,6̄,7̄} {1̄,2̄,2̄,6̄,7̄,7̄,7̄} {1̄,2̄,3̄,5̄,7̄,7̄,7̄} {1̄,2̄,3̄,6̄,6̄,7̄,7̄} {1̄,2̄,4̄,4̄,7̄,7̄,7̄} {1̄,2̄,4̄,5̄,6̄,7̄,7̄} {1̄,2̄,4̄,6̄,6̄,6̄,7̄} {1̄,2̄,5̄,5̄,5̄,7̄,7̄} {1̄,2̄,5̄,5̄,6̄,6̄,7̄} {1̄,2̄,5̄,6̄,6̄,6̄,6̄} {1̄,3̄,3̄,4̄,7̄,7̄,7̄} {1̄,3̄,3̄,5̄,6̄,7̄,7̄} {1̄,3̄,3̄,6̄,6̄,6̄,7̄} {1̄,3̄,4̄,4̄,6̄,7̄,7̄} {1̄,3̄,4̄,5̄,5̄,7̄,7̄} {1̄,3̄,4̄,5̄,6̄,6̄,7̄} {1̄,3̄,4̄,6̄,6̄,6̄,6̄} {1̄,3̄,5̄,5̄,5̄,6̄,7̄} {1̄,3̄,5̄,5̄,6̄,6̄,6̄} {1̄,4̄,4̄,4̄,5̄,7̄,7̄} {1̄,4̄,4̄,4̄,6̄,6̄,7̄} {1̄,4̄,4̄,5̄,5̄,6̄,7̄} {1̄,4̄,4̄,5̄,6̄,6̄,6̄} {1̄,4̄,5̄,5̄,5̄,5̄,7̄} {1̄,4̄,5̄,5̄,5̄,6̄,6̄} {1̄,5̄,5̄,5̄,5̄,5̄,6̄} {2̄,2̄,2̄,5̄,7̄,7̄,7̄} {2̄,2̄,2̄,6̄,6̄,7̄,7̄} {2̄,2̄,3̄,4̄,7̄,7̄,7̄} {2̄,2̄,3̄,5̄,6̄,7̄,7̄} {2̄,2̄,3̄,6̄,6̄,6̄,7̄} {2̄,2̄,4̄,4̄,6̄,7̄,7̄} {2̄,2̄,4̄,5̄,5̄,7̄,7̄} {2̄,2̄,4̄,5̄,6̄,6̄,7̄} {2̄,2̄,4̄,6̄,6̄,6̄,6̄} {2̄,2̄,5̄,5̄,5̄,6̄,7̄} {2̄,2̄,5̄,5̄,6̄,6̄,6̄} {2̄,3̄,3̄,3̄,7̄,7̄,7̄} {2̄,3̄,3̄,4̄,6̄,7̄,7̄} {2̄,3̄,3̄,5̄,5̄,7̄,7̄} {2̄,3̄,3̄,5̄,6̄,6̄,7̄} {2̄,3̄,3̄,6̄,6̄,6̄,6̄} {2̄,3̄,4̄,4̄,5̄,7̄,7̄} {2̄,3̄,4̄,4̄,6̄,6̄,7̄} {2̄,3̄,4̄,5̄,5̄,6̄,7̄} {2̄,3̄,4̄,5̄,6̄,6̄,6̄} {2̄,3̄,5̄,5̄,5̄,5̄,7̄} {2̄,4̄,4̄,4̄,4̄,7̄,7̄} {2̄,4̄,4̄,4̄,5̄,6̄,7̄} {2̄,4̄,4̄,4̄,6̄,6̄,6̄} {2̄,4̄,4̄,5̄,5̄,5̄,7̄} {2̄,4̄,4̄,5̄,5̄,6̄,6̄} {2̄,4̄,5̄,5̄,5̄,5̄,6̄} {2̄,5̄,5̄,5̄,5̄,5̄,5̄} {3̄,3̄,3̄,3̄,6̄,7̄,7̄} {3̄,3̄,3̄,4̄,5̄,7̄,7̄} {3̄,3̄,3̄,4̄,6̄,6̄,7̄} {3̄,3̄,3̄,5̄,5̄,6̄,7̄} {3̄,3̄,3̄,5̄,6̄,6̄,6̄} 112 © 2023 The Authors. Page 335 of 337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 {3̄,3̄,4̄,4̄,4̄,7̄,7̄} {3̄,3̄,4̄,4̄,5̄,6̄,7̄} {3̄,3̄,4̄,4̄,6̄,6̄,6̄} {3̄,3̄,4̄,5̄,5̄,5̄,7̄} {3̄,3̄,4̄,5̄,5̄,6̄,6̄} {3̄,3̄,5̄,5̄,5̄,5̄,6̄} {3̄,4̄,4̄,4̄,4̄,6̄,7̄} {3̄,4̄,4̄,4̄,5̄,5̄,7̄} {3̄,4̄,4̄,4̄,5̄,6̄,6̄} {3̄,4̄,4̄,5̄,5̄,5̄,6̄} {3̄,4̄,5̄,5̄,5̄,5̄,5̄} {4̄,4̄,4̄,4̄,4̄,5̄,7̄} {4̄,4̄,4̄,4̄,4̄,6̄,6̄} {4̄,4̄,4̄,4̄,5̄,5̄,6̄} {4̄,4̄,4̄,5̄,5̄,5̄,5̄} 5 {0̄,5̄,7̄,7̄,7̄,7̄,7̄} {0̄,6̄,6̄,7̄,7̄,7̄,7̄} {1̄,4̄,7̄,7̄,7̄,7̄,7̄} {1̄,5̄,6̄,7̄,7̄,7̄,7̄} {1̄,6̄,6̄,6̄,7̄,7̄,7̄} {2̄,3̄,7̄,7̄,7̄,7̄,7̄} {2̄,4̄,6̄,7̄,7̄,7̄,7̄} {2̄,5̄,5̄,7̄,7̄,7̄,7̄} {2̄,5̄,6̄,6̄,7̄,7̄,7̄} {2̄,6̄,6̄,6̄,6̄,6̄,7̄} {3̄,3̄,6̄,7̄,7̄,7̄,7̄} {3̄,4̄,5̄,7̄,7̄,7̄,7̄} {3̄,4̄,6̄,6̄,7̄,7̄,7̄} {3̄,5̄,5̄,6̄,7̄,7̄,7̄} {3̄,5̄,6̄,6̄,6̄,7̄,7̄} {3̄,6̄,6̄,6̄,6̄,6̄,7̄} {4̄,4̄,4̄,7̄,7̄,7̄,7̄} {4̄,4̄,5̄,6̄,7̄,7̄,7̄} {4̄,4̄,6̄,6̄,6̄,7̄,7̄} {4̄,5̄,5̄,5̄,7̄,7̄,7̄} {4̄,5̄,5̄,6̄,6̄,7̄,7̄} {4̄,5̄,6̄,6̄,6̄,6̄,7̄} {4̄,6̄,6̄,6̄,6̄,6̄,6̄} {5̄,5̄,5̄,5̄,6̄,7̄,7̄} {5̄,5̄,5̄,6̄,6̄,6̄,7̄} {5̄,5̄,6̄,6̄,6̄,6̄,6̄} 26 6 {6̄,7̄,7̄,7̄,7̄,7̄,7̄} 1 The Stirling number of the first kind c(n, k) is the number of permutations of an n-element set with exactly cycles. Second- kind Stirling number S(n, k) counts the number of ways that n distinct objects can be partitioned into k indistinguishable subsets, with each subset containing at least one object. Accord- ing to Riordan (2012) , Stirling numbers of the first kind is s(n, k), where s(n, k) satisfies the recursion relation s(n, k) = (n-1) s(n−1, k)+ s(n−1, k−1), where k and n are integers, 1 ≤ n ≤ k−1 with initial conditions s(n, 0) = 0, for n ≥ 1 and s(n, n) = 1, for n ≥ 0. Table 2 show the relationship of the Second-kind Stirling number, s(n, k) and Bell number Bn. The sum of the Second- kind Stirling number (the sum in every row) is Bell number (the number in the last column). 3. RESULTS AND DISCUSSION To determine the number of multisets we can enumerate or use source code. For this study we developed a source code to determine a multiset. The following is an algorithm and the source code for determining all multisets of the additive group ℤn+1 using Python. #Multiset Grup Aditif ℤn+1 inisiasi Input: n, k, n, k integer Output : Multisets, number of multiset, multiset for every k and number of multisets for every k. Implementation Begin Read (n, k) Set ℤn+1 ← Set {0, 1, 2, · · · , n} Max k ← (0, 1, 2,· · · , n-1) Target (n, k) repeat (n+1)*k until Max k candidates←set(Combination with raplecement (ℤn+1, Max k)); targets ← [Target (n, k) for k in max k]; multiset (candidates, targets): for candidate in candidates if (sum(candidate) in targets) then add multiset in candidate repeat multisets until Target (n, k) multiset← multiset (candidates, targets); Stirling(multisets, targets) for multiset in multisets if (sum(multiset) in targets) then add stirling in multiset repeat stirling until Target k2← target (n, k) for k = k input; stirling←Stirling (multiset, k2); print("ℤn+1 : ", Set ℤn+1) print("k : ", Max k) print("multisets : ", multisets) print("Bilangan Stirling k = ",k,":", stirling) print("Jumlah Bilangan Stirling k = ",k," : ", len(stirling)) print("Total Bilangan Catalan : ", len(multisets)) End The multisets generated by that source code are grouped and put in Table 3. From the result of Table 3, we do partition p(m;n,n) on the number of multisets, where p(m;n,n) is the number of partitions of m to at most n parts and every element in each part cannot exceed n, with m = k(n+1). Thus, the number of multisets can be written as ∑n−1 k=0 p(k(n+1); n, n). The result are shown on Table 4. Based on the computational results where 0 ≤ k ≤ n-1, a sequence of numbers is obtained, which is contained in Table 5 below: Based on Table 5, many possible multisets are formed from the additive group ℤ10 forming a Catalan number for each n. In addition, if the multiset additive group ℤ10 is divided into several parts where each part depends on the value of k(n+1) with 0 ≤ k ≤ n-1, then the number of multisets formed will form a new second type of the Stirling number. The Stirling number in the left-hand side (in Table 5, the yellow row) will © 2023 The Authors. Page 336 of 337 Wamiliana et. al. Science and Technology Indonesia, 8 (2023) 330-337 Table 5. Number Sequences of the Possible Multiset Sums of the Additive Group ℤ10 k\n 0 1 2 3 4 5 6 7 8 Catalan number 1 1 1 2 1 1 2 3 1 3 1 5 4 1 5 7 1 14 5 1 9 20 11 1 42 6 1 13 48 51 18 1 132 7 1 20 100 169 112 26 1 429 8 1 28 194 461 486 221 38 1 1430 9 1 40 352 1128 1667 1210 411 52 1 4862 be smaller, and the rightmost portion (in red ) will be larger than the Stirling number of the second type that is known. Moreover, the last column which sum of the Stirling’s number and originally is Bell’s, now was replaced by a Catalan number. 4. CONCLUSION We can conclude that the multiset formed from the additive group ℤ10 is a Catalan number based on the results. In addition, if the multiset additive group ℤ10 is divided into several parts where each part depends on the value of k(n+1) with 0 ≤ k ≤ n-1, then the number of multisets will form a new Stirling number of the second kind. However, the second kind of Stirling number formed based on the multiset additive group ℤ10 has a different initial value, namely S(n,0) = 1 and the value S(n,n) = 0. For S(n,1) to S(n,n-2) the value is smaller, while for S(n,n-1) it has a larger value. Moreover, the sum of the Stirling number and originally is Bell number, now is changed to be Catalan number. 5. ACKNOWLEDGMENT The authors gratefully acknowledge the help and support from Postgraduate Study Program in Mathematics and Department of Mathematics Universitas Lampung. REFERENCES Albert, M. H., R. E. Aldred, M. D. Atkinson, C. Handley, and D. Holton (2001). Permutations of a Multiset Avoiding Permutations of Length 3. European Journal of Combinatorics, 22(8); 1021–1031 Bayer, M. and K. Brandt (2014). The Pill Problem, Lattice Paths, and Catalan Numbers. Mathematics Magazine, 87(5); 388–394 Boyadzhiev, K. N. (2021). 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Discrete Mathematics, 14(1); 83–90 Singmaster, D. (1978). An Elementary Evaluation of the Cata- lan Numbers. The American Mathematical Monthly, 85(5); 366–368 Stanley, R. P. (2015). Catalan Numbers. Cambridge University Press Stojadinovic, T. (2015). On Catalan Numbers. The Teaching of Mathematics, 18(1); 16–24 Tripathy, B. C., S. Debnath, and D. Rakshit (2018). On Mul- tiset Group. Proyecciones (Antofagasta), 37(3); 479–489 © 2023 The Authors. Page 337 of 337 INTRODUCTION THE METHOD Multiset Catalan Numbers Bell Numbers Striling Numbers RESULTS AND DISCUSSION CONCLUSION ACKNOWLEDGMENT