Binomisk

Binomisk is a term used in combinatorics and algebra to denote a generalized family of coefficient structures that extend the classical binomial coefficients. The word combines binomial with the suffix -isk and is used to discuss objects that share a Pascal-like recurrence without committing to a single canonical definition.

Definition and scope. For each nonnegative integer n there is a sequence binomisk(n,0), binomisk(n,1), ..., binomisk(n,n). The ordinary binomial coefficients form the canonical example: binomisk(n,k) = C(n,k). A binomisk is often defined by a Pascal-type recurrence binomisk(n+1,k) = binomisk(n,k) + binomisk(n,k-1) with boundary values binomisk(n,-1) = binomisk(n,n+1) = 0. Variants introduce parameters or weights, producing families such as q-binomial analogues or other deformations, while preserving a similar recursive structure.

Properties. Binomisk sequences are typically symmetric: binomisk(n,k) = binomisk(n,n-k). They form triangular arrays akin to Pascal’s triangle and can count ordinary subsets or weighted structures depending on the variant. The canonical row n reproduces the binomial coefficients, while deformations yield alternative combinatorial interpretations.

Relation and use. Binomisk serves as a flexible framework for studying binomial identities, generating functions, and generalized binomial theorems. It relates to q-binomial coefficients, multinomial coefficients, and various combinatorial interpretations, and is mainly used as a teaching and research shorthand for generalized coefficient tables.