On Two Families of Generalizations of Pascal's Triangle

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 $m-1$ zeros on the right side. The $m=1$ cases are Pascal's triangle and the two families also coincide when $m=2$. Members of the first family obey Pascal's recurrence everywhere inside the triangle. We show that the $m$-th triangle can also be obtained by reversing the elements up to and including the main diagonal in each row of the $(1/(1-x^m),x/(1-x))$ Riordan array. Properties of this family of triangles can be obtained quickly as a result. The $(n,k)$-th entry in the $m$-th member of the second family of triangles is the number of tilings of an $(n+k)\times1$ board that use $k$ $(1,m-1)$-fences and $n-k$ unit squares. A $(1,g)$-fence is composed of two unit square sub-tiles separated by a gap of width $g$. We show that the entries in the antidiagonals of these triangles are coefficients of products of powers of two consecutive Fibonacci polynomials and give a bijective proof that these coefficients give the number of $k$-subsets of $\{1,2,\ldots,n-m\}$ such that no two elements of a subset differ by $m$. Other properties of the second family of triangles are also obtained via a combinatorial approach. Finally, we give necessary and sufficient conditions for any Pascal-like triangle (or its row-reversed version) derived from tiling $(n\times1)$-boards to be a Riordan array.