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  2. Integer partition - Wikipedia

    en.wikipedia.org/wiki/Integer_partition

    There is a natural partial order on partitions given by inclusion of Young diagrams. This partially ordered set is known as Young's lattice . The lattice was originally defined in the context of representation theory , where it is used to describe the irreducible representations of symmetric groups S n for all n , together with their branching ...

  3. Integer programming - Wikipedia

    en.wikipedia.org/wiki/Integer_programming

    An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints (other than the integer constraints) are linear.

  4. Catalan number - Wikipedia

    en.wikipedia.org/wiki/Catalan_number

    We call a sequence of X's and Y's dominating if, reading from left to right, the number of X's is always strictly greater than the number of Y's. The cycle lemma [ 13 ] states that any sequence of m {\displaystyle m} X's and n {\displaystyle n} Y's, where m > n {\displaystyle m>n} , has precisely m − n {\displaystyle m-n} dominating circular ...

  5. Ring of integers - Wikipedia

    en.wikipedia.org/wiki/Ring_of_integers

    The ring of integers O K is a finitely-generated Z-module.Indeed, it is a free Z-module, and thus has an integral basis, that is a basis b 1, ..., b n ∈ O K of the Q-vector space K such that each element x in O K can be uniquely represented as