Binomial Coefficients of Multidimensional Arrays

Motivated by Parikh matrices of picture arrays introduced in combinatorial image analysis, we propose a generalization of binomial coefficients of words to multidimensional arrays. These coefficients recursively count prescribed patterns occurring in an array. The base case is the one of binomial coefficients of words.
With our definition we extend Pascal’s rule, the Chu–Vandermonde identity and therefore, the concept of Parikh matrices, in a natural way. We further present some more binomial-related identities and introduce (q, t)-deformations, i.e., multivariate polynomials whose evaluation at (q, t) = (1, 1) recovers the value of the classical coefficients. We explain the additional combinatorial information encoded in the coefficients of these (q, t)-polynomials compared to their integer-valued counterparts.