Arndt and De Morgan Integer Compositions

In 2013, Joerg Arndt recorded that the Fibonacci numbers count integer compositions where the first part is greater than the second, the third part is greater than the fourth, etc. We provide a new combinatorial proof that verifies his observation using compositions with only odd parts as studied by De Morgan. We generalize the descent condition to establish families of recurrence relations related to two types of compositions: those made of any odd part and certain even parts, and those made of any even part and certain odd parts. These generalizations connect to compositions studied by Andrews and Viennot. New tools used in the combinatorial proofs include two permutations of compositions and a statistic based on the signed pairwise difference between parts.Comment: 13 pages, 1 figure, 11 table

Ties in Worst-Case Analysis of the Euclidean Algorithm

We determine all pairs of positive integers below a given bound that require the most division steps in the Euclidean algorithm. Also, we find asymptotic probabilities for a unique maximal pair or an even number of them. Our primary tools are continuant polynomials and the Zeckendorf representation using Fibonacci numbers