In combinatorial mathematics, the Bell numbers count the possible partitions of a set. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s.
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.
In physics, an anyon is a type of quasiparticle that occurs only in two-dimensional systems, with properties much less restricted than fermions and bosons. In general, the operation of exchanging two identical particles may cause a global phase shift but cannot affect observables. Anyons are generally classified as abelian or non-abelian. Abelian anyons have been detected and play a major role in the fractional quantum Hall effect. Non-abelian anyons have not been definitively detected, although this is an active area of research.
In mathematics, a combinatorial class is a countable set of mathematical objects, together with a size function mapping each object to a non-negative integer, such that there are finitely many objects of each size.
The class of Sudoku puzzles consists of a partially completed row-column grid of cells partitioned into N regions each of size N cells, to be filled in ("solved") using a prescribed set of N distinct symbols, so that each row, column and region contains exactly one of each element of the set. The properties of Sudoku puzzles and their solutions can be investigated using mathematics and algorithms.
In combinatorics, the twelvefold way is a systematic classification of 12 related enumerative problems concerning two finite sets, which include the classical problems of counting permutations, combinations, multisets, and partitions either of a set or of a number. The idea of the classification is credited to Gian-Carlo Rota, and the name was suggested by Joel Spencer.
In the mathematical study of permutations and permutation patterns, a superpattern or universal permutation is a permutation that contains all of the patterns of a given length. More specifically, a k-superpattern contains all possible patterns of length k.
The Stanley–Wilf conjecture, formulated independently by Richard P. Stanley and Herbert Wilf in the late 1980s, states that the growth rate of every proper permutation class is singly exponential. It was proved by Adam Marcus and Gábor Tardos (2004) and is no longer a conjecture. Marcus and Tardos actually proved a different conjecture, due to Zoltán Füredi and Péter Hajnal (1992), which had been shown to imply the Stanley–Wilf conjecture by Klazar (2000).
In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets (posets). Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of the orders can be obtained from the other just by renaming of elements. Two strictly weaker notions that relate to order isomorphisms are order embeddings and Galois connections.
In combinatorial mathematics, a separable permutation is a permutation that can be obtained from the trivial permutation 1 by direct sums and skew sums. Separable permutations may be characterized by the forbidden permutation patterns 2413 and 3142; they are also the permutations whose permutation graphs are cographs and the permutations that realize the series-parallel partial orders. It is possible to test in polynomial time whether a given separable permutation is a pattern in a larger permutation, or to find the longest common subpattern of two separable permutations.
In the study of permutation patterns, there has been considerable interest in enumerating specific permutation classes, especially those with relatively few basis elements. This area of study has turned up unexpected instances of Wilf equivalence, where two seemingly-unrelated permutation classes have the same numbers of permutations of each length.
In music theory, equivalence class is an equality (=) or equivalence between sets (unordered) or twelve-tone rows. A relation rather than an operation, it may be contrasted with derivation. "It is not surprising that music theorists have different concepts of equivalence [from each other]..." "Indeed, an informal notion of equivalence has always been part of music theory and analysis. Pitch class set theory, however, has adhered to formal definitions of equivalence." Octave equivalency is assumed, while inversional, permutational, and transpositional equivalency are not considered.
In combinatorial mathematics, a Baxter permutation is a permutation which satisfies the following generalized pattern avoidance property:
In the study of permutations and permutation patterns, a permutation class is a set of permutations such that every pattern within a permutation in is also in . That is, it is a downset in the permutation pattern order. A permutation class may also be known as a pattern class, closed class, or simply class of permutations.
David Bevan is an English mathematician, computer scientist and software developer. He is known for Bevan's Theorem, which gives the asymptotic enumeration of grid classes of permutations and for his work on enumerating the class of permutations avoiding the pattern 1324. He is also known for devising weighted reference counting, an approach to computer memory management that is suitable for use in distributed systems.
In the theory of permutation patterns, a skew-merged permutation is a permutation that can be partitioned into an increasing sequence and a decreasing sequence. They were first studied by Stankova (1994) and given their name by Atkinson (1998).
In the mathematics of permutations, a layered permutation is a permutation that reverses contiguous blocks of elements. Equivalently, it is the direct sum of decreasing permutations.
