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\begin{document}

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\begin{center}
\vskip 1cm{\LARGE\bf
On a Two-Parameter Family of   
\vskip .08in
Generalizations of Pascal's Triangle}
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\large
Michael A. Allen\\
Physics Department\\
Faculty of Science\\
Mahidol University\\
Rama 6 Road\\
Bangkok 10400  \\
Thailand \\
\href{mailto:maa5652@gmail.com}{\tt maa5652@gmail.com} \\
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\begin{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.
\end{abstract}

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