On Two Families of Generalizations of Pascal's Triangle
Michael A. Allen, Kenneth Edwards

TL;DR
This paper introduces two new families of Pascal-like triangles with unique properties, explores their combinatorial and algebraic structures, and establishes conditions under which they relate to Riordan arrays.
Contribution
It presents novel generalizations of Pascal's triangle, connects them to Riordan arrays, and provides combinatorial interpretations and properties of these new structures.
Findings
The first family obeys Pascal's recurrence inside the triangle.
The second family counts tilings with specific fences and squares.
Antidiagonals relate to Fibonacci polynomial products.
Abstract
We consider two families of Pascal-like triangles that have all ones on the left side and ones separated by zeros on the right side. The cases are Pascal's triangle and the two families also coincide when . Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the -th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the Riordan array. Properties of this family of triangles can be obtained quickly as a result. The -th entry in the -th member of the second family of triangles is the number of tilings of an board that use -fences and unit squares. A -fence is composed of two unit square sub-tiles separated by a gap of width . We show that the entries in the antidiagonals of these triangles are…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · semigroups and automata theory · Quasicrystal Structures and Properties
