Orientador: Marcelo Firer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-15T01:48:53Z (GMT). No. of bitstreams: 1 Moura_AllandeOliveira1_D.pdf: 766044 bytes, checksum: e752134a3aa77aa9bf3559d18a7f0a12 (MD5) Previous issue date: 2010 / Resumo: Considerando uma generalização da métrica de Hamming, a métrica ponderada por uma ordem parcial, fazemos uma descrição sistemática para os espaços com a métrica ponderada, dando ênfase aos códigos poset e à hierarquia de pesos contextualizada nesse novo ambiente. Técnicas de multiconjunto, para códigos ponderados, são utilizadas para estender o Teorema da Dualidade de Wei, uma relação entre as hierarquias do código e do seu dual. Como consequência desta Dualidade estendemos certos resultados sobre a discrepância, códigos MDS e uma relação entre a condição cadeia do código e do seu dual. / Abstract: Considering a generalization of the Hamming metric, the metric weighted by a partial order, we make a systematic description of the spaces with those metrics, emphasizing poset codes and the weight hierarchy of weights of those codes. Techniques of multiset, for weighted codes, are used to extend the Duality Theorem of Wei, a relationship between the hierarchy of a code and its dual. As a consequence of Duality we extend some results about the discrepancy, MDS codes and a relationship between a chain code and its dual. / Doutorado / Matematica / Doutor em Matemática

Orientador: Marcelo Firer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T19:55:05Z (GMT). No. of bitstreams: 1 Felix_LucianoVianna_D.pdf: 818003 bytes, checksum: fdff7cac576e6c465521860f01c9fc96 (MD5) Previous issue date: 2014 / Resumo: Neste trabalho são estudados diversos aspectos de códigos em espaços munidos de métricas poset. Considerando posets hierárquicos é determinada uma forma canônica-sistemática de um código linear. Esta forma permite calcular os principais invariantes métricos da teoria de códigos nestes espaços, incluindo distância mínima, pesos generalizados, hierarquia de pesos e o raio de empacotamento. Esta forma canônica-sistemática também permite, considerando métricas poset hierárquicas, classificar códigos MDS e códigos perfeitos e reduzir significativamente a complexidade do algoritmo de decodificação por síndrome. Considerando posets genéricos, são estabelecidas condições necessárias e suficientes para que órbitas de grupos de isometrias lineares sejam determinadas pelas classes de isomorfismos de ideais (propriedade de extensão de ideais). São apresentadas algumas famílias de posets que satisfazem essa condição, incluindo posets cujo diagrama de Hasse é uma árvore uni-raiz, regular por nível. Neste caso específico de árvore, é determinada um invariante que caracteriza estas órbitas. Considerando as operações clássicas entre posets, é demonstrado que apenas a soma ordinal preserva a propriedade de extensão de ideais / Abstract: In this work we consider vector spaces over a finite field equipped with a metric induced by a partial order (poset) and study several aspects of codes embedded in those. Considering hierarchical posets, a canonical-systematic form for linear codes is determined. With this form it is possible to calculate the main metric invariants of coding theory, such as minimal distance, generalized weights, weight hierarchy and the packing radius. This canonical-systematic form also allows to classify MDS and perfect codes and to significantly decrease the complexity of syndrome decoding algorithm. Considering generic posets, necessary and sufficient conditions are established to ensure the orbits of the group of linear isometries to be determined by the ideals isomorphisms classes (ideal extension property). Some families of posets that satisfy those conditions are presented, including posets that have as a Hasse diagram a level-wise regular rooted tree. In this particular case of trees, it is established an invariant that classifies those orbits. Considering classic operations over posets, it is proofed that only ordinal sum preserves the ideal extension property / Doutorado / Matematica / Doutor em Matemática

Made available in DSpace on 2015-03-26T13:45:36Z (GMT). No. of bitstreams: 1 texto completo.pdf: 1755688 bytes, checksum: 33e268f82618cf29e2d1fa6df5c6fa6c (MD5) Previous issue date: 2013-07-30 / Fundação de Amparo a Pesquisa do Estado de Minas Gerais / A poset metric is the generalization of the Hamming metric. In this work we make a detailed study of poset spaces, hierarchy of I -weights and I -distribution of P P weights, emphasizing the non-degenerate poset codes. We verify the duality relation between the hierarchy weights of poset code and its dual. In the sequel two new parameters are defined to a class of poset codes non-degenerate with dual code is too non-degenerate in the environment. As a result enunciated in the Minimality Theorem, the Variance Theorem and the Minimality Identity in the poset spaces. / Uma generalização da métrica de Hamming é a métrica poset. Faremos um estudo detalhado dos espaços poset, hierarquia de I-pesos e a I-distribuição de pesos, dando ênfase aos códigos poset não degenerados. Verificamos a relação de dualidade poset entre as hierarquias de um código e seu dual. Definimos dois novos parâmetros para a classe de códigos dualmente não degenerados no ambiente poset. Como consequência, enunciamos e mostramos o Teorema da Minimalidade, o Teorema da e Variância e a Identidade de Minimalidades no espaço poset.

<p>Ako je&nbsp; Lalgebarska mreža i&nbsp; a kodistributivan elemenat u L, onda sve klase kongruencije&nbsp; (pa indukovane homomorfizmom&nbsp; ma : xi&mdash; &gt;&nbsp; a&nbsp;&nbsp; A x imaju najveće elemente. Najveći elemenat klase kojoj&nbsp; x E Lpripada je označen sa&nbsp; x.Ako je&nbsp; *a binarna op&shy;eracija definisana sa x *ay = (x A y)&nbsp; V&nbsp; (x A y),onda je istraživana struktura&nbsp; (L, *a), i odgovarajući poset ( L, &lt;&raquo;). Kao primer takve strukture posmatrana je algebra slabih kongruencija&nbsp; (CwA, *a), gde je *a specijalna grafička kompozicija. Dobijeni rezultati daju prirodne posledice u strukturi slabih kongruencija. Data je primena ovih rezultata u univerzalnoj algebri. Njihovom primenom karakterizuje se CEP i Hamiltonovo svojstvo. Dat je potreban i dovoljan uslov da poset (L, &lt; -) bude mreža i ovi rezultati su primenjeni na mrežu slabih kongruencija.</p> / <p>If&nbsp; Lis an algebraic lattice and&nbsp; a codistributive element in&nbsp; L,then all the classes of the congruences&nbsp; 4&gt;a determined by the homomorphism&nbsp; ma :&nbsp; x&nbsp; \&mdash; &gt;&nbsp; a Ax&nbsp;&nbsp; have top elements. The top element of the class which to belongs an&nbsp; x&nbsp;&nbsp; &euro;&nbsp; Lis denoted by&nbsp; x.&nbsp;&nbsp; If *a is a binary operation defined by&nbsp; x&nbsp; *ay=&nbsp; (xA y)&nbsp; V&nbsp; (xA y),then we investigate the<br />structure&nbsp; (L,*a), and the corresponding poset&nbsp; (L, &lt; t ). Asan example of such a structure we observe an algebra of weak congruences ( C w A , * a),where *a is a special graphical composition. We obtain natural conse&shy; quences of the mentioned results to the structure weak congruences. An application in universal algebra is presented, for example, we characterized CEP and Hamiltonian property. Necessary and sufficient conditions for a poset&nbsp; (L,&lt;*) to be a lattice are given, and the results are applied in the case of weak congruence lattices.</p>

Orientador: Marcelo Muniz Silva Alves / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-12T16:23:24Z (GMT). No. of bitstreams: 1 Ritter_Donizete_M.pdf: 621556 bytes, checksum: 2bf0368b784f3a2be59ca3c2552f4908 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho abordamos a teoria dos Códigos Corretores de Erros clássica e também os códigos sobre ordens parciais, com algumas comparações entre os dois casos. Enfocamos, particularmente, a definição de Alfabeto, a distância de Hamming, os códigos lineares e a definição de matriz geradora de um código; o estudo dos limitantes de Singleton e de Hamming, além de tratar dos Códigos de Hamming. Em relação aos Códigos em Conjuntos Parcialmente Ordenados, apresentamos a definição de ordens parciais, métricas sobre conjuntos ordenados, contagem dos elementos da "bola", resultados sobre Ideais e o Código de Hamming Estendido; estudamos o caso da ordem cadeia ("chain poset"), analisando os códigos de uma cadeia e os códigos de duas cadeias de mesmo comprimento e, por fim, nos dedicamos ao estudo das "Métricas POSET", que admitem códigos binários perfeitos de codi-mensão m, caracterizando assim os Códigos Posets m-corretores de erros. Nosso objetivo é apresentar um texto, acessível a alunos de graduação, que contemple a teoria básica dos Códigos Corretores de Erros, no entanto, forneça uma noção sobre os códigos sobre ordens parciais. / Abstract: In this work, we address the classical theory of error-correcting codes and the theory of codes over poset spaces, also known as poset codes, establishing comparisons between these two cases. In particular, we present the definition of alphabet, the Hamming distance, linear codes and the definition of a generating matrix for a linear code; we also present the Singleton and Hamming bounds, alongside with the Hamming codes. With respect to poset codes, we present the definitions of partial orders and of the poset metric, the counting of the number of elements in a ball in a poset space, some results on ideals in posets and the extended Hamming code; we study the chain poset case, analysing the cases of codes over a chain poset and codes over a union of two chains of the same length and, finally, we study the poset metrics that allow m-perfect binary codes of codimension m, thus characterizing these codes. Our aim is to present a text, accessible for undergraduates, that encompasses the basic theory of error-correcting codes and, nonetheless, also provides some notions on poset codes. / Mestrado / Teoria dos Erros / Mestre em Matemática

Orientador: Marcelo Firer / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T06:58:11Z (GMT). No. of bitstreams: 1 Spreafico_MarcosViniciusPereira_M.pdf: 558857 bytes, checksum: a9d033bd1132fd1bc42fcb1aa2296ef5 (MD5) Previous issue date: 2009 / Resumo: Neste trabalho fazemos uma apresentação dos espaços poset, introduzidos por Brualdi (1995), apresentamos os conceitos necessarios da teoria de conjuntos parcialmente ordenados e da teoria de codigos. Trabalhamos com uma questão de caráter amplo e estrutural deste contexto, o problema da determinação da ordem atraves da distribuição de pesos. A distribuição de pesos é essencialmente o conjunto das cardinalidades das esferas métricas e a pergunta que se coloca é em que medida este invariante determina a métrica em questão. Demonstramos que para as classes de codigos, cadeia, anticadeia, coroa e hierárquico, classes importantes no contexto da teoria de codigos, o problema possui uma resposta positiva e justificamos algumas conjecturas que relacionam este problema ao da reconstrução de grafos. / Abstract: In this work, we introduce the concept of poset codes (Brualdi - 1995) and in this context we study the weight distribution problem, presenting the necessary concepts of the partially ordered set and error correcting codes theory. The weight distribution is the cardinality of metric-spheres in finite dimensional vector space over a finite field endowed with a poset metric. The weight distribution problem asks for conditions to ensure that the weight distribution determines the metric. In this work we show that the weight distribution of some families of posets, namely the classes of anti-chain, chain, crown and hierarchical posets, determines the metric. We also show that the weight distribution determines some known invariants of posets. Finally, we present some conjectures relating the weight distribution problem and the reconstruction problem of graphs. / Mestrado / Mestre em Matemática

Orientador: Marcelo Firer / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-21T02:49:59Z (GMT). No. of bitstreams: 1 LucasD'Oliveira_RafaelGregorio_M.pdf: 16647897 bytes, checksum: a2258aca5a39f0a7d0bd2243b905a772 (MD5) Previous issue date: 2012 / Resumo: Até o trabalho presente, só era conhecido o raio de empacotamento de um código poset nos casos do poset ser uma cadeia, hierárquico, a união disjunta de cadeias do mesmo tamanho, e para algumas famílias de códigos. Nosso objetivo é abordar o caso geral de um poset qualquer. Para isso, iremos dividir o problema em dois. A primeira parte consiste em encontrar o raio de empacotamento de um único vetor. Veremos que este problema é equivalente à uma generalização de um problema NP-difícil famoso conhecido como \o problema da partição". Veremos então os principais resultados conhecidos sobre este problema dando atenção especial aos algoritmos para resolvê-lo. A receita principal destes algoritmos é o método da diferenciação, e sendo assim, iremos estendê-la para o caso geral. A segunda parte consiste em encontrar o vetor que determina o raio de empacotamento do código. Para isso, mostraremos como é as vezes possível comparar o raio de empacotamento de dois vetores sem calculá-los explicitamente / Abstract: Until the present work, the packing radius of a poset code was only known in the cases where the poset was a chain, hierarchy, a union of disjoint chains of the same size, and for some families of codes. Our objective is to approach the general case of any poset. To do this, we will divide the problem into two parts. The first part consists in finding the packing radius of a single vector. We will show that this is equivalent to a generalization of a famous NP-hard problem known as \the partition problem". Then, we will review the main results known about this problem giving special attention to the algorithms to solve it. The main ingredient to these algorithms is what is known as the differentiating method, and therefore, we will extend it to the general case. The second part consists in finding the vector that determines the packing radius of the code. For this, we will show how it is sometimes possible to compare the packing radius of two vectors without calculating them explicitly / Mestrado / Matematica / Mestre em Matemática

In this thesis we study the following dimensional parameters : boxicity, cubicity, threshold dimension and poset dimension. While the first three parameters are defined on graphs, poset dimension is defined on partially ordered sets (or posets). We only consider finite graphs and posets. In addition, we assume that the graphs are simple and undirected. Boxicity and Cubicity: A k-box (k-cube) is a Cartesian product of closed intervals(unit-intervals) [a1,b1]x…x [ak,bk]. The boxicity (cubicity) of a graph G,box (G) (cub(G)) is the minimum integer k such that every vertex in G is mapped to a k-box(k-cube) in the k-dimensional Euclidean space and two boxes(cubes) intersect if and only if their corresponding vertices are adjacent in G. Boxicity and cubicity can be considered as extensions of the concept of interval graphs and unit-interval graphs respectively. Threshold Dimension: A graph G is a threshold graph if there is a real number p and a weight function w: V→ R such that for any two vertices u,,v ε V(G),{ u, v }is an edge if and only if w(u)+w(v) ≥ p. The threshold dimension of a graph G is the minimum integer k such that there exist k threshold graphs Gi, i =1,2,...,k which satisfy E(G)= E(G1)U E(G2)U….UE(Gk). Poset Dimension: Let P = (S, P)be a poset where S is a finite non-empty set and P is a reflexive, anti-symmetric and transitive binary relation on S. P is a total order if every pair of elements in S is comparable in P. The dimension of P , denoted by dim(P )is the minimum integer k such that there exist k total orders on S, L1,...,Lk and for two distinct elements x,y ε S: x < y in P if and only if x < y in each Li,i ε ,{1. 2,...,k } All the four dimensional parameters that we have considered are very hard to compute. It is NP-complete to even determine if the boxicity of a graph is at most 2, if its cubicity is at most 3, if its threshold dimension is at most 3 and if the dimension of a poset is at most 3. Also it is hard to design an approximation algorithm within √n factor for computing the dimension of a poset. OurResults We state some of our main results: 1. Lower bounds for boxicity: We have developed two general methods based on certain vertex isoperimetric properties of graphs for deriving lower bounds. Application of these methods has led to some significant results. We mention a few of them here: ( a) Almost all graphs have boxicity Ω(n). (b) For a fixed k, boxicity of random k-regular graphs is Ω(k/log k). 2. Consider a poset P = (S,P) and let GP be its underlying comparability graph. We show that for any poset P, box(GP)/(χ(GP) - 1) ≤ dim(P) ≤ 2box (GP), where χ(GP) is the chromatic number of GP and χ(GP) = 1. Some important consequences of this result are: (a) It allows us to derive hitherto unknown upper bounds for poset dimension such as dim(P) ≤ 2tree-width (GP) + 4. (b) The boxicity of any graph with maximum degree Δ is O (Δlog2 Δ) which is an improvement over the best known upper bound of Δ2 +2. (c) There exist graphs with boxicity Ω(ΔlogΔ). This disproves a conjecture that the boxicity of a graph is O(Δ). (d)There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices within a factor of O(n0.5−ε)for any ε > 0, unless NP = ZPP. 3.We show that every poset can be associated with a split graph such that the threshold dimension of the complement of the split graph is equal to the dimension of the poset. As a consequence we show that there exists no polynomial-time algorithm to approximate the threshold dimension of a split graph on n vertices with a factor of O(n0.5−ε)for any ε > 0, unless NP= ZPP. 4.We have given an upper bound for the cubicity of interval graphs. Claw number of a graph G, ψ(G) is the largest positive integer m such that K1,m is an induced subgraph of G. If G is an interval graph, we show that [log2 ψ(G)] ≤ cub(G) ≤ min([log2 α ], [log2 ψ(G)] +2), where α is the independence number of G. 5.We have improved upper bounds for the dimension of incidence posets and interval orders which are among the well-studied classes of posets.

Dans cette thèse, nous abordons deux problèmes de théorie des graphes liés à deux problèmes biologiques de symbiose (deux organismes vivent en symbiose s'ils ont une interaction étroite et à long terme). Le premier problème est lié au phénomène de l'Incompatibilité cytoplasmique (IC) induit par certaines bactéries parasites chez leurs hôtes. L'IC se traduit par l'impossibilité de donner naissance à une progéniture saine lorsqu'un mâle infecté s'accouple avec une femelle non infectée. En termes de graphe ce problème peut s'interpréter comme la recherche d'une couverture minimum par des "sous-graphes des chaînes" d'un graphe biparti. Un graphe des chaînes est un graphe biparti dont les noeuds peuvent être ordonnés selon leur voisinage.En terme biologique, la taille minimale représente le nombre de facteurs génétiques impliqués dans le phénomène de l'IC. Dans la première moitié de la thèse, nous abordons trois problèmes connexes à ce modèle de la théorie des graphes. Le premier est l'énumération de tous les graphes des chaînes maximaux arêtes induits d'un graphe biparti G, pour lequel nous fournissons un algorithme en delai polynomial avec un retard de O(n^2m) où n est le nombre de noeuds et m le nombre d'arêtes de G. Dans la même section, nous montrons que (n/2)! et 2^(\sqrt{m}\log m) bornent le nombre de sous-graphes de chaînes maximales de G et nous les utilisons pour établir la complexité "input-sensitive" de notre algorithme. Le deuxième problème que nous traitons est de trouver le nombre minimum de graphes des chaînes nécessaires pour couvrir tous les bords d'un graphe biparti.Pour résoudre ce problème NP-hard, en combinant notre algorithme avec la technique d'inclusion-exclusion, nous fournissons un algorithme exponentiel exact en O^*((2+c)^m), pour chaque c > 0 (par O^* on entend la notation O standard mais en omettant les facteurs polynomiaux). Le troisième problème est l'énumération de toutes les couvertures minimales par des sous-graphes des chaînes. Nous montrons qu'il est possible d'énumérer toutes les couvertures minimales de G en temps O([(M + 1) |S|] ^ [\ log ((M + 1) |S|)]) où S est le nombre de couvertures minimales de G et M le nombre maximum des sous-graphes des chaînes dans une couverture minimale. Nous présentons ensuite la relation entre le second problème et le calcul de la dimension intervallaire d'un poset biparti. Nous donnons une interprétation de nos résultats dans le contexte de la dimension d'ordre / In this thesis, we address two graph theoretical problems connected to two different biological problems both related to symbiosis (two organisms live in symbiosis if they have a close and long term interaction). The first problem is related to the size of a minimum cover by "chain subgraphs" of a bipartite graph. A chain graph is a bipartite graph whose nodes can be ordered by neighbourhood inclusion. In biological terms, the size of a minimum cover by chain subgraphs represents the number of genetic factors involved in the phenomenon of Cytoplasmic Incompatibility (CI) induced by some parasitic bacteria in their insect hosts. CI results in the impossibility to give birth to an healthy offspring when an infected male mates with an uninfected female. In the first half of the thesis we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph G, for which we provide a polynomial delay algorithm with a delay of O(n^2m) where n is the number of nodes and m the number of edges of G. Furthermore, we show that (n/2)! and 2^(\sqrt{m} \log m) bound the number of maximal chain subgraphs of G and use them to establish the input-sensitive complexity of the algorithm. The second problem we treat is finding the minimum number of chain subgraphs needed to cover all the edges of a bipartite graph. To solve this NP-hard problem, we provide an exact exponential algorithm which runs in time O^*((2+c)^m), for every c>0, by a procedure which uses our algorithm and an inclusion-exclusion technique (by O^* we denote standard big O notation but omitting polynomial factors). Notice that, since a cover by chain subgraphs is a family of subsets of edges, the existence of an algorithm whose complexity is close to 2^m is not obvious. Indeed, the basic search space would have size 2^(2^m), which corresponds to all families of subsets of edges of a graph on $m$ edges. The third problem is the enumeration of all minimal covers by chain sugbgraphs. We show that it is possible to enumerate all such minimal covers of G in time O([(M+1)|S|]^[\log((M+1)|S|)]) where S is the number of minimal covers of G and M the maximum number of chain graphs in a minimal cover. We then present the relation between the second problem and the computation of the interval order dimension of a bipartite poset. We give an interpretation of our results in the context of poset and interval poset dimension... [etc]

Let <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D%20=%20(%5Cmathbb%7BP%7D,%20%5Cpreceq)" data-classname="equation_inline" data-title="" /> be a finite poset (partially ordered set) with cardinality <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?n" data-classname="equation_inline" data-title="" />. A linear extension of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" /> is an order-preserving bijection <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csigma" data-classname="equation_inline" data-title="" />: <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D%20%5Crightarrow%20%5Bn%5D" data-classname="equation_inline" data-title="" />, that is, if <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?x%20%5Cpreceq%20y" data-classname="equation_inline" data-title="" /> in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" /> then <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csigma(x)%20%5Cle%20%5Csigma(y)" data-classname="equation_inline" data-title="" />. We define the poset probability <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)" data-classname="equation_inline" data-title="" /> as the proportion of linear extensions where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Csigma(%5Calpha)%20%5Cle%20%5Csigma(%5Cbeta)" data-classname="equation_inline" data-title="" />. We are primarily interested in <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)" data-classname="equation_inline" data-title="" /> for incomparable elements <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha%20%5Cparallel%20%5Cbeta" data-classname="equation" data-title="" />. The probability has significance in areas such as information theory. Let <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?e(%5Cmathbb%7BP%7D)" data-classname="equation_inline" data-title="" /> denote the total number of linear extensions of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" />. We prove that the poset probability can be evaluated as <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)%20=%20%5Cfrac%7B%20%5Csum_%7BT%20%5Cin%20B(%5Calpha,%5Cbeta)%7D%20e(T)%20e(%5Cmathbb%7BP%7D%20%5Csetminus%20(T%20%5Ccup%20%5C%7B%5Calpha%5C%7D))%7D%7Be(%5Cmathbb%7BP%7D)%7D" data-classname="equation" data-title="" /> where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?B(%5Calpha,%5Cbeta)" data-classname="equation_inline" data-title="" /> is the set of order ideals of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" /> without <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" data-classname="equation" data-title="" /> or <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cbeta" data-classname="equation" data-title="" />, where we can add <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Calpha" data-classname="equation_inline" data-title="" /> to get a new order ideal of <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Cmathbb%7BP%7D" data-classname="equation_inline" data-title="" />. The practicality of the preceding formula is explored and we show that <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?T%20%5Cin%20B(%5Calpha,%5Cbeta)%20%5CLeftrightarrow%20%5Cleft%5C%7B%20x%20%7C%20x%20%5Cprec%20%5Calpha%20%5Cright%5C%7D%20%5Csubseteq%20T%20%5Ctext%7B%20and%20%7D%20T%20%5Ctext%7B%20order%20ideal%20of%20%7D%0A%5Cleft%5C%7B%20x%20%7C%20%5Calpha%20%5Cnot%20%5Cpreceq%20x,%5C%20%5Cbeta%20%5Cnot%20%5Cpreceq%20x%7D" data-classname="equation" /> The formula is particularly useful for certain classes of posets such as partition posets which are examined in further detail. We apply the formula to prove that, for all partition posets of shape <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Bn,n%5D" data-classname="equation_inline" data-title="" />, the probability obeys <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P((2,a)%20%5Cpreceq%20(1,a+1))%20=%20%5Cfrac%7B%20C_a%20C_%7Bn-a%7D%7D%20%7BC_n%7D" data-classname="equation" data-title="" /> where <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?C_n" data-classname="equation_inline" data-title="" /> is the nth Catalan number and <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?a%20%3C%20n" data-classname="equation_inline" data-title="" />. We also explore how Monte Carlo methods can be used to estimate <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?P(%5Calpha%20%5Cpreceq%20%5Cbeta)" data-classname="equation_inline" data-title="" />.