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A new solution to Riordan's problem of combinatorial identities classification is presented. An algebgraic characterization of pairs of inverse relations of the Riordan type is given. The use of the integral representation approach for generating new types of combinatorial identities is demonstrated. (2000): combinatorics, algebra. Mathematics Subject Classifications

We give new definitions for the determinant over commutative ring K, noncommutative ring K, noncommutative ring K with associative powers, over noncommutative nonassociative ring K, and study their properties.Let K be a commutative ring, K a noncommutative associative ring, K a noncommutative ring with associative powers (one-monomial associativity), and K be a noncommutative nonassociative ring; let each ring be with division by integers.Here we obtain a new family of polynomial identities for determinant over the ring K, which allows us to give new definitions for determinants over rings K, K, K, and to study their properties. These definitions are closely related to the definition of symmetrized Barvinok's determinant sdet(A) [2] different from the well-known determinant of Dieudonné over a division ring, the quasideterminant [6], and other well-known determinants over noncommutative associative rings [1]. We also estimate the computational complexity of the obtained formulas for the determinants.Let A = (a ij ) be an n×n matrix with elements from the ring K. Let S n be the set of all permutations σ = (σ (1) , . . . , σ (n)) of the set {1, . . . , n}, τ (σ) be the number of inversions in σ. Let S