Bn

Bn is a notation used in several disciplines to denote different objects, so there is no single universal meaning. The exact interpretation depends on context, and in many texts the meaning is clarified by surrounding symbols or by the field of study.

In mathematics, Bn commonly denotes multiple distinct structures. The braid group on n strands, Bn, is a fundamental object in geometric and algebraic topology. It is generated by elementary braids with relations that encode how strands interact. The hyperoctahedral group of type Bn, often called the group of signed permutations, is the symmetry group of the n-dimensional cube and has order 2^n n!. In combinatorics, Bn denotes the nth Bell number, which counts the number of partitions of an n-element set. In topology and analysis, B^n is the standard notation for the n-dimensional unit ball, consisting of all points in R^n with norm at most 1; its boundary is the (n−1)-sphere S^{n−1}.

Beyond these, Bn or related forms appear in other mathematical contexts, such as different notational conventions for groups, Coxeter systems, or sequences. The precise meaning is typically inferred from the surrounding notation and the field of study.

In practical use, when encountering Bn or B^n, readers should consult the definition supplied in the relevant text to determine which object is being referenced. The distinction between subscript Bn and superscript B^n is often crucial, as they denote different families of objects in the same general area.