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We construct a formal power series on several variables that encodes many statistics on non-decreasing Dyck paths. In particular, we use this formal power series to count peaks, pyramid weights, and indexed sums of pyramid weights for all non-decreasing Dyck paths of length 2n. We also show that an indexed sum on pyramid weights depends only on the size and… (More)
The Hosoya polynomial triangle is a triangular arrangement of polynomials where each entry is a product of two polynomials. The geometry of this triangle is a good 1 tool to study the algebraic properties of polynomial products. In particular, we find closed formulas for the alternating sum of products of polynomials such as Fibonacci polynomials, Chebyshev… (More)
We study a relation between factorials and their additive analog, the triangular numbers. We show that there is a positive integer k such that n! = 2kT where T is a product of triangular numbers. We discuss the primality of T±1 and the primality of |T − p| where p is either the smallest prime greater than T or the greatest prime less than T .
In a social network individuals have prominent centrality if they are intermediaries between the communication of others. The betweenness centrality of a vertex measures the number of intersecting geodesics between two other vertices. Formally, the betweenness centrality of a vertex v is the ratio of the number of shortest paths between two other vertices u… (More)
We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power q, find a minimal non-orientable submatroid of the projective plane over the q-element field.
The generalized Hosoya triangle is an arrangement of numbers where each entry is a product of two generalized Fibonacci numbers. We define a discrete convolution 1 C based on the entries of the generalized Hosoya triangle. We use C and generating functions to prove that the sum of every k-th entry in the n-th row or diagonal of generalized Hosoya triangle,… (More)