inversionsbasis

Inversionsbasis is a term used in algebraic combinatorics to denote a particular basis for a vector space of functions or polynomials that encode inversion statistics of permutations. The inversion statistic counts pairs (i, j) with i < j and π(i) > π(j) in a permutation π. An inversionsbasis consists of basis elements that correspond to elementary inversion patterns, enabling a unique linear decomposition of any inversion-structure function into a sum of pattern-shaped components.

Formally, let V_n be the vector space of real-valued functions on the symmetric group S_n, or equivalently a suitable polynomial space in n variables. An inversionsbasis is a set {b_β} indexed by inversion patterns β (often via Lehmer codes) such that every f in V_n can be written uniquely as f = sum_β c_β b_β, with coefficients determined by the inversion content of f. The basis is constructed to have support aligned with inversion-classes, which makes it convenient to isolate contributions from specific inversion configurations.

Properties commonly studied include linear independence, span of the target subspace, and behavior under natural group actions. It is closely related to other basis constructions in permutation statistics, such as descent bases and quasisymmetric function bases, and it is frequently used in the derivation of generating functions and q-analog identities for inversion-related statistics.

Usage and status: Inversionsbasis is a specialized construct found in a subset of theoretical treatments of permutation statistics. It is not universally standardized and may appear under varying definitions in different texts. Nevertheless, it provides a useful framework for organizing inversion data and for comparing different theories of permutation statistics.