Equivalent subsets of a colored set

Let X be a finite set. Let φ be a function from X to the set of positive integers N. A pair (X,φ) is called a colored set. Two colored sets (X1,φ1) and (X2,φ2) are called equivalent if there exists a permutation σ of N such that |φ1−1(y)|=|φ2−1(σ(y))| for any y∈N. We say that a colored set (X,φ) has a (k;l)-partition if there exists a partition X=X0∪X1∪⋯∪Xl such that |Xi|=k for 1≤i≤l, and (Xi,φ|Xi) and (Xj,φ|Xj) are equivalent for 1≤i