N1×N2××Nk

N1×N2×...×Nk is a standard notation used across mathematics and computer science to describe either the Cartesian product of k finite sets or the shape of a k-dimensional object with side lengths N1, N2, ..., Nk. When the factors represent positive integers, the usual interpretation is that each Ni denotes the size of a corresponding index set, such as {1, 2, ..., Ni}.

If each Ni is a positive integer and the product denotes the Cartesian product of the index sets, the resulting set consists of all k-tuples (n1, n2, ..., nk) with 1 ≤ ni ≤ Ni for each i. The cardinality of this set is the product ∏ Ni. This construction underpins multi-index notation and the enumeration of multi-dimensional grids.

In practical contexts, N1×N2×...×Nk is also used to specify the shape of a tensor or multi-dimensional array. Here the i-th dimension has length Ni, so the total number of elements in the object is again the product ∏ Ni. This usage is common in linear algebra, scientific computing, and data architectures.

In algebraic settings, if the factors are groups of orders Ni and one takes their direct product, the resulting group has order ∏ Ni and elements represented as tuples with one coordinate from each factor.

Notes: the exact meaning depends on context—sometimes it refers to a size or shape, sometimes to a concrete Cartesian product of sets. The notation is often condensed as ∏_{i=1}^k Ni. Examples include 3×4×2 (24 elements) and 5×5 (25 elements). See also Cartesian product, tensor shapes, and multi-index notation.