On a Two-Parameter Family of Generalizations of Pascal's Triangle

TL;DR

This paper introduces a two-parameter family of generalized Pascal's triangles based on tilings with comb-like tiles, linking their entries to generalized Fibonacci polynomials and combinatorial subset problems.

Contribution

It presents a novel two-parameter generalization of Pascal's triangle, establishes a connection with generalized Fibonacci polynomials, and provides combinatorial proofs and identities related to these structures.

Findings

Entries are coefficients of products of generalized Fibonacci polynomials.

A bijection relates tilings to subsets with no two elements differing by a multiple of m.

Derived recursion relations and identities for specific parameter instances.

Abstract

We consider a two-parameter family of triangles whose $(n,k)$-th entry (counting the initial entry as the $(0,0)$-th entry) is the number of tilings of $N$-boards (which are linear arrays of $N$ unit square cells for any nonnegative integer $N$) with unit squares and $(1,m-1;t)$-combs for some fixed $m=1,2,\dots$ and $t=2,3,\dots$ that use $n$ tiles in total of which $k$ are combs. A $(1,m-1;t)$-comb is a tile composed of $t$ unit square sub-tiles (referred to as teeth) placed so that each tooth is separated from the next by a gap of width $m-1$. We show that the entries in the triangle are coefficients of the product of two consecutive generalized Fibonacci polynomials each raised to some nonnegative integer power. We also present a bijection between the tiling of an $(n+(t-1)m)$-board with $k$ $(1,m-1;t)$-combs with the remaining cells filled with squares and the $k$-subsets of $\{1,\ldots,n\}$ such that no two elements of the subset differ by a multiple of $m$ up to $(t-1)m$. We can therefore give a combinatorial proof of how the number of such $k$-subsets is related to the coefficient of a polynomial. We also derive a recursion relation for the number of closed walks from a particular node on a class of directed pseudographs and apply it obtain an identity concerning the $m=2$, $t=5$ instance of the family of triangles. Further identities of the triangles are also established mostly via combinatorial proof.