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We consider various forms of Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey's theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey's theorem (these are similar to generalizations shown in [P....
Here we will derive formulas for expressing any polynomial as linear combinations of two kinds of higherorder Daehee polynomial basis. Then we will apply these formulas to certain polynomials in order to get new and interesting identities involving higher-order Daehee polynomials of the first kind and of the second kind.
Increasing integer sequences include many instances of interesting
sequences and combinatorial structures, ranging from tournaments to addition
chains, from permutations to sequences having the Goldbach property that any integer greater than 1 can be obtained as the sum of two elements
in the sequence. The paper introduces and compares several of these classes
of sequences, discussing recurrence relations, enumerative problems and
questions concerning shortest sequences.
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