Barred Preferential Arrangements

A preferential arrangement of a set is a total ordering of the elements of that set with ties allowed. A barred preferential arrangement is one in which the tied blocks of elements are ordered not only amongst themselves but also with respect to one or more bars. We present various combinatorial identities for r_m‚_ℓ, the number of barred preferential arrangements of ℓ elements with m bars, using both algebraic and combinatorial arguments. Our main result is an expression for r_m,_ℓ as a linear combination of the r_k (= r_0,_k,, the number of unbarred preferential arrangements of k elements) for ℓ ≤ k ≤ ℓ + m. We also enumerate those arrangements in which the sections, into which the blocks are segregated by the bars, must be nonempty. We conclude with an expression of r_ℓ as an infinite series that is both convergent and asymptotic

A study of barred preferential arrangements with applications to numerical approximation in electric circuits

In 1854 Cayley proposed an interesting sequence 1,1,3,13,75,541,... in connection with analytical forms called trees. Since then there has been various combinatorial interpretations of the sequence. The sequence has been interpreted as the number of preferential arrangements of members of a set with n elements. Alternatively the sequence has been interpreted as the number of ordered partitions; the outcomes in races in which ties are allowed or geometrically the number of vertices, edges and faces of simplicial objects. An interesting application of the sequence is found in combination locks. The idea of a preferential arrangement has been extended to a wider combinatorial object called barred preferential arrangement with multiple bars. In this thesis we study barred preferential arrangements combinatorially with application to resistance of certain electrical circuits. In the process we derive some results on cyclic properties of the last digit of the number of barred preferential arrangements. An algorithm in python has been developed to find the number of barred preferential arrangements

A combinatorial analysis of barred preferential arrangements

For a non-negative integer n an ordered partition of a set Xn with n distinct elements is called a preferential arrangement (PA). A barred preferential arrangement (BPA) is a preferential arrangement with bars in between the blocks of the partition. An integer sequence an associated with the counting PA's of Xn has been intensely studied over a century and a half in many different contexts. In this thesis we develop a unified combinatorial framework to study the enumeration of BPAs and a special subclass of BPAs. The results of the study lead to a positive settlement of an open problem and a conjecture by Nelsen. We derive few important identities pertaining to the number of BPAs and restricted BPAs of an n element set using generating- functionology. Later we show that the number of restricted BPAs of Xn are intricately related to well-known numbers such as Eulerian numbers, Bell numbers, Poly-Bernoulli numbers and the number of equivalence classes of fuzzy subsets of Xn under some equivalent relation