Divisibility Patterns within Pascal Divisibility Networks

Abstract

The Pascal triangle is so simple and rich that it has always attracted the interest of professional and amateur mathematicians. Their coefficients satisfy a myriad of properties. Inspired by the work of Shekatkar et al., we study the divisibility patterns within the elements of the Pascal triangle, through its decomposition into Pascal's matrices, from the perspective of network science. Applying Kolmogorov-Smirnov test, we determine that the degree distribution of the resulting network follows a power-law distribution. We also study degrees, global and local clustering coefficients, stretching graph, averaged path length and the mixing assortative.

Divisibility Networks of the Rational Numbers in the Unit Interval

“…These degree distributions are heavy tailed, similarly as it was shown in [12,13] for other divisibility networks. These distributions show a plateau for the frequencies of high-order degrees, Ref.…”

“…We have represented the stretching separating the cases in which the numerators and denominators are prime or not. Unlike the results of [12,13], we have plotted the local clustering separating the cases in which numerators (denominators) are prime or not. The nodes are following the order in which they appear following the diagonal argument in their respective matrices.…”

“…Recently, network science has been used to study mathematical properties from the point of view of complex systems apart from graph theory itself. Some interesting networks have arisen when studying mathematical structures of numbers sets: divisibility networks of natural numbers following the increasing sequential order, [12], divisibility networks of natural numbers according to its arrangement within the Pascal Triangle [13] or networks of prime numbers [14]. Yan et al studied congruence relations through multiplex networks and studied the multiplex congruence network.…”

Sets of numbers from complex networks perspective

“…These degree distributions are heavy tailed, similarly as it was shown in [SBA15,SHMPBC20] for other divisibility networks. These distributions show a plateau for the frequencies of high-order degrees, [B + 16, Ch.…”

“…Recently, network science has been used to study mathematical properties from the point of view of complex systems apart from graph theory itself. Some interesting networks have arisen when studying mathematical structures of numbers sets: divisibility networks of natural numbers following the increasing sequential order, [SBA15], divisibility networks of natural numbers according to its arrangement within the Pascal Triangle [SHMPBC20] or networks of prime numbers [CD05]. Yan et al studied congruence relations through multiplex networks and studied the multiplex congruence network.…”