Binomial coefficients can be expressed in terms of multinomial coefficients as sums over integer partitions. This approach allows us to introduce new upper bounds for the number of partitions into a given number of parts.

A very special case of a Ramus’s identity is used in this note to derive an infinite family of inequalities involving finite sums with cosecant function. As a corollary of this result, we obtain the Wallis's formula.

The complete and elementary symmetric functions are specializations of Schur functions. In this paper, we use this fact to give two identities for the complete and elementary symmetric functions. This result can be used to proving and discovering some algebraic identities involving r-Whitney and other special numbers.

The Jacobi–Stirling numbers of both kinds are specializations of elementary and complete homogeneous symmetric functions. We use this fact to discover and prove some algebraic identities involving Jacobi–Stirling numbers. The Legendre–Stirling numbers are very special cases of the Jacobi–Stirling numbers. New connections between the Legendre–Stirling numbers and the central factorial numbers of odd indices are presented.

The nth-order determinant of a Toeplitz-Hessenberg matrix is expressed as a sum over the integer partitions of n. Many combinatorial identities involving integer partitions and multinomial coefficients can be generated using this formula.

A generalization for the symmetry between complete symmetric functions and elementary symmetric functions is given. As corollaries we derive the inverse of a triangular Toeplitz matrix and the expression of the Toeplitz-Hessenberg determinant. A very large variety of identities involving integer partitions and multinomial coefficients can be generated using this generalization. The partitioned binomial theorem and a new formula for the partition

The algorithms for generating the integer partitions of a positive integer have long been invented. Nevertheless, data structures for storing the integer partitions are not received due attention. In 2005 there were introduced diagram structures to store integer partitions. In this article, we present a recently obtained result in storing integer partitions: the binary diagrams.

In this paper we give a convolution identity for complete and elementary symmetric functions. This result can be used to prove and discover some combinatorial identities involving r-Stirling numbers, r-Whitney numbers and q-binomial coefficients. As a corollary we derive a generalization of the quantum Vandermonde's convolution identity.

In this note, using the multisection series method, we establish the formulas for various power sums of sine or cosine functions.