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A general upper bound in extremal theory of sequences

Martin Klazar (1992)

Commentationes Mathematicae Universitatis Carolinae

We investigate the extremal function f ( u , n ) which, for a given finite sequence u over k symbols, is defined as the maximum length m of a sequence v = a 1 a 2 . . . a m of integers such that 1) 1 a i n , 2) a i = a j , i j implies | i - j | k and 3) v contains no subsequence of the type u . We prove that f ( u , n ) is very near to be linear in n for any fixed u of length greater than 4, namely that f ( u , n ) = O ( n 2 O ( α ( n ) | u | - 4 ) ) . Here | u | is the length of u and α ( n ) is the inverse to the Ackermann function and goes to infinity very slowly. This result extends the estimates in [S] and [ASS] which...

A generalization of Pascal’s triangle using powers of base numbers

Gábor Kallós (2006)

Annales mathématiques Blaise Pascal

In this paper we generalize the Pascal triangle and examine the connections among the generalized triangles and powering integers respectively polynomials. We emphasize the relationship between the new triangles and the Pascal pyramids, moreover we present connections with the binomial and multinomial theorems.

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