pat·tern / ˈpatərn/ • n. 1. a repeated decorative design: a neat blue herringbone pattern. ∎ an arrangement or sequence regularly found in comparable objects or events: the house had been built on the usual pattern. ∎ a regular and intelligible form or sequence discernible in certain actions or situations: a complicating factor is the change in working patterns.2. a model or design used as a guide in needlework and other crafts. ∎ a set of instructions to be followed in making a sewn or knitted item. ∎ a wooden or metal model from which a mold is made for a casting. ∎ an example for others to follow: he set the pattern for subsequent study. ∎ a sample of cloth or wallpaper.• v. [tr.] 1. [usu. as adj.] (patterned) decorate with a recurring design: rosebud patterned wallpapers violet-tinged flowers patterned the grassy banks.2. give a regular or intelligible form to: the brain not only receives information, but interprets and patterns it. ∎ (pattern something on/after) give something a form based on that of (something else): the clothing is patterned on athletes' wear.
be a set of functions mapping elements from some domain D into some set A, which can be regarded as an alphabet. With each function f in F is associated a weight w(f), defined as the formal multiplication of all the images f(x) under f. In effect w(f) describes the number of occurrences of the different images in A.
An equivalence relation can then be defined between two functions of F in such a way that equivalent functions have equivalent weights, though the reverse is not in general true. The patterns of F are the equivalence classes that emerge from this equivalence relation.
The weight of a pattern is just the weight of any member of that pattern; the weight of the equivalence class [f] containing f is just w(f). The formal sum of the weights w(f) taken over all the equivalence classes in F gives the pattern inventory of the set F. An important theorem due mainly to George Pólya indicates the close link between pattern inventory and cycle index polynomial.
These ideas are often applied in combinatorics and switching theory. For example, a pattern inventory can indicate the number of essentially different wiring diagrams or logic circuits needed to realize the different possible logic functions.