We consider a two-parameter family of triangles whose
![$(n,k)$](img2.svg)
-th entry
(counting the initial entry as the
![$(0,0)$](img3.svg)
-th entry) is the number of
tilings of
![$N$](img4.svg)
-boards (which are linear arrays of
![$N$](img4.svg)
unit square cells for any
nonnegative integer
![$N$](img4.svg)
) with unit squares and
![$(1,m-1;t)$](img5.svg)
-combs for
some fixed
![$m=1,2,\dots$](img6.svg)
and
![$t=2,3,\dots$](img7.svg)
that use
![$n$](img8.svg)
tiles in total
of which
![$k$](img9.svg)
are combs. A
![$(1,m-1;t)$](img5.svg)
-comb is a tile composed of
![$t$](img10.svg)
unit square sub-tiles (referred to as teeth) placed so that each tooth
is separated from the next by a gap of width
![$m-1$](img11.svg)
. 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
![$(n+(t-1)m)$](img12.svg)
-board with
![$(1,m-1;t)$](img5.svg)
-combs with the
remaining cells filled with squares and the
![$k$](img9.svg)
-subsets of
![$\{1,\ldots,n\}$](img13.svg)
such that no two elements of the subset differ by a
multiple of
![$m$](img14.svg)
up to
![$(t-1)m$](img15.svg)
. We can therefore give a combinatorial
proof of how the number of such
![$k$](img9.svg)
-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
![$m=2$](img16.svg)
,
![$t=5$](img17.svg)
instance of the family of triangles. Further identities
of the triangles are also established mostly via combinatorial proof.