منابع مشابه
Records in Set Partitions
A partition of [n] = {1, 2, . . . , n} is a decomposition of [n] into nonempty subsets called blocks. We will make use of the canonical representation of a partition as a word over a finite alphabet, known as a restricted growth function. An element ai in such a word π is a strong (weak) record if ai > aj (ai > aj) for all j = 1, 2, . . . , i−1. Furthermore, the position of this record is i. We...
متن کاملSet Partitions in R
This short paper introduces a code snippet in the form of an R function that enumerates all possible partitions of a finite set given the sizes of the equivalence classes. Three combinatorial problems are solved using the software: one from bioinformatics, one from scheduling theory, and one from forensic science.
متن کاملPattern Avoidance in Set Partitions
The study of patterns in permutations is a very active area of current research. Klazar defined and studied an analogous notion of pattern for set partitions. We continue this work, finding exact formulas for the number of set partitions which avoid certain specific patterns. In particular, we enumerate and characterize those partitions avoiding any partition of a 3-element set. This allows us ...
متن کاملSet partitions as geometric words
Using an analytic method, we derive an alternative formula for the probability that a geometrically distributed word of length n possesses the restricted growth property. Equating our result with a previously known formula yields an algebraic identity involving alternating sums of binomial coefficients via a probabilistic argument. In addition, we consider refinements of our formula obtained by...
متن کاملFront Representation of Set Partitions
Let π be a set partition of [n] = {1, 2, . . . , n}. The standard representation of π is the graph on the vertex set [n] whose edges are the pairs (i, j) of integers with i < j in the same block which does not contain any integer between i and j. The front representation of π is the graph on the vertex set [n] whose edges are the pairs (i, j) of integers with i < j in the same block whose small...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2010
ISSN: 1077-8926
DOI: 10.37236/381