Elliptic and $q$-Analogs of the Fibonomial Numbers

Elliptic and $\boldsymbol{q}$-Analogs of the Fibonomial Numbers††This paper is
a contribution to the Special Issue on Elliptic Integrable Systems, Special
Functions and Quantum Field Theory. The full collection is available at
https://www.emis.de/journals/SIGMA/elliptic-integrable-systems.html

Nantel BERGERON †, Cesar CEBALLOS ‡ and Josef KÜSTNER §

N. Bergeron, C. Ceballos and J. Küstner

† Department of Mathematics and Statistics, York University, Toronto, Canada
<EMAIL_ADDRESS>http://www.math.yorku.ca/bergeron/

‡ Institute of Geometry, TU Graz, Graz, Austria<EMAIL_ADDRESS>http://www.geometrie.tugraz.at/ceballos/

§ Faculty of Mathematics, University of Vienna, Vienna, Austria
<EMAIL_ADDRESS>http://homepage.univie.ac.at/josef.kuestner/

Received March 14, 2020, in final form July 29, 2020; Published online August
13, 2020

In 2009, Sagan and Savage introduced a combinatorial model for the Fibonomial
numbers, integer numbers that are obtained from the binomial coefficients by
replacing each term by its corresponding Fibonacci number. In this paper, we
present a combinatorial description for the $q$-analog and elliptic analog of
the Fibonomial numbers. This is achieved by introducing some $q$-weights and
elliptic weights to a slight modification of the combinatorial model of Sagan
and Savage.

Fibonomial; Fibonacci; $q$-analog; elliptic analog; weighted enumeration

11B39; 05A30; 05A10

##  1 Introduction

The Fibonacci sequence $0,1,1,2,3,5,8,13,21,34,\dots$ is one of the most
important and beautiful sequences in mathematics. It starts with the numbers
$F_{0}=0$ and $F_{1}=1$, and is recursively defined by the formula
$F_{n}=F_{n-1}+F_{n-2}$.

Fibonacci analogs of famous numbers, such as the binomial coefficients and
Catalan numbers

${m+n\choose n}=\frac{(m+n)!}{m!\cdot
n!}\qquad{\text{and}}\qquad\frac{1}{n+1}{2n\choose n},$

have intrigued some mathematicians over the last few years [1, 4, 5, 10, 27,
34]. The _Fibonomial_ and _Fibo-Catalan_ numbers are defined, respectively, as

${{m+n}\choose n}_{\mathcal{F}}:=\frac{F_{m+n}^{!}}{F_{m}^{!}\cdot
F_{n}^{!}}\qquad{\text{and}}\qquad\frac{1}{F_{n+1}}{2n\choose
n}_{\mathcal{F}},$

where $F_{n}^{!}:=\prod\limits_{k=1}^{n}F_{k}$ is the Fibonacci analog of
$n!$. These rational expressions turn out to be positive integers. In [27],
Sagan and Savage introduced a combinatorial model to interpret the Fibonomial
numbers in terms of certain tilings of an $m\times n$ rectangle. A _path-
domino tiling_ of an $m\times n$ rectangle is a tiling with monominos and
dominos and a lattice path from $(0,0)$ to $(m,n)$ is specified, and such
that:

  * •

all tiles above the path are either monominos or horizontal dominos;

  * •

all tiles below the path are either monominos or vertical dominos; and

  * •

all tiles that touch the path from below are vertical dominos.

We call these last tiles touching the path from below _special vertical
dominos_ , and denote by $\mathcal{T}_{m,n}$ the collection of all path-domino
tilings of an $m\times n$ rectangle. An example is illustrated on the left of
Fig. 1. The following result is a special case of [27, Theorem 3].111The path-
domino tilings here are a slight modification of the tilings used in [27]. The
only difference is that in [27] the tiles below the path touching the bottom
of the rectangle are required to be vertical dominos, while here this
condition is required for the tiles below the path touching the path itself.
This modification is essential to make our combinatorial model work.

######  Theorem 1.1 ([27]).

The Fibonomial number ${{m+n}\choose n}_{\mathcal{F}}$ counts the number of
path-domino tilings of an $m\times n$ rectangle.

The main objective of this paper is to present a $q$-analog and an elliptic
analog generalization of this result. The resulting $q$-Fibonomial and
elliptic Fibonomial numbers count the number of path-domino tilings of an
$m\times n$ rectangle according to their $q$-weights and elliptic weights,
respectively.

##  2 $\boldsymbol{q}$-analog of the Fibonomial numbers

We denote by $\mathbb{N}:=\\{1,2,3,\dots\\}$ the set of natural numbers. The
_$q$ -analog of $n\in\mathbb{N}$_ is defined as

$[n]_{q}:=1+q+q^{2}+\dots+q^{n-1}.$

The evaluation of this polynomial at $q=1$ recovers the number $n$. To
simplify notation, we sometimes omit the subindex $q$ when it is clear from
the context. Before studying the $q$-analog of the Fibonomial numbers, let us
recall some useful and known straightforward lemmas.

######  Lemma 2.1.

For $m,n\in\mathbb{N}$, the following identities hold:

$\displaystyle[m+n]_{q}=[m]_{q}+q^{m}[n]_{q},$ (2.1) $\displaystyle[m\cdot
n]_{q}=[m]_{q}[n]_{q^{m}}.$

It is well known that the Fibonacci number $F_{n}$ counts the number of
tilings of an _$(n-1)$ -strip_ (a rectangle with diagonal endpoints $(0,0)$
and $(n-1,1)$) using dominos and monominos. Given such a tiling $T$, we define
the _weight $\omega(T)$_ of $T$ as the product of the weights of its tiles,
where a monomino has weight 1 and a domino whose top-right coordinate is
$(i,1)$ has weight $q^{F_{i}}$. The weight of a 0-strip is by definition equal
to 1.

######  Lemma 2.2 (cf. [27]).

For $n\in\mathbb{N}$, the $q$-analog of the Fibonacci numbers222The elliptic
and $q$-analogs of the Fibonacci numbers we use are different from the analogs
considered, e.g., in [32]. can be computed as

$[F_{n}]_{q}=\sum_{T}{\omega(T)},$

where $[F_{n}]_{q}=1+q+q^{2}+\dots+q^{F_{n}-1}$ and the sum ranges over all
tilings of an $(n-1)$-strip using dominos and monominos.

###### Proof.

The result is clearly true for $n=1,2$. Let $n>2$, applying equation (2.1)
from Lemma 2.1 we get

$\displaystyle[F_{n}]_{q}=[F_{n-1}+F_{n-2}]_{q}=[F_{n-1}]_{q}+q^{F_{n-1}}[F_{n-2}]_{q}.$

By induction, the first term of this sum corresponds to the tilings of an
$(n-1)$-strip that finish with a monomino, while the second term to the
tilings of an $(n-1)$-strip that finish with a domino. ∎

######  Lemma 2.3.

For $m,n\in\mathbb{N}$, the following identities hold:

$\displaystyle F_{m+n}=F_{n}F_{m+1}+F_{m}F_{n-1},$ (2.2)
$\displaystyle[F_{m+n}]_{q}=[F_{n}]_{q}[F_{m+1}]_{q^{F_{n}}}+q^{F_{n}F_{m+1}}[F_{m}]_{q}[F_{n-1}]_{q^{F_{m}}}.$
(2.3)

###### Proof.

Equation (2.2) is a well known identity for Fibonacci numbers; see for
instance [27, Lemma 1]. The Fibonacci number $F_{m+n}$ counts the number of
tilings of an $(m+n-1)$-strip with monominos and dominos. These tilings can be
subdivided into two types: those containing a domino which is cut in two by
the line $x=m$, and all other tilings. The first kind is counted by
$F_{m}F_{n-1}$ (the number of tilings of an $(m-1)$-strip times the number of
tilings of an $(n-2)$-strip), while the second kind is counted by
$F_{n}F_{m+1}$ (the number of tilings of an $m$-strip times the number of
tilings of an $(n-1)$-strip). Therefore, equation (2.2) follows.

Applying Lemma 2.1 to (2.2) leads to equation (2.3). ∎

For $m,n\in\mathbb{N}$, the _$q$ -analog of the Fibonomial number_ is defined
as

$\displaystyle\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}}:=\frac{[F_{m+n}]_{q}^{!}}{[F_{m}]_{q}^{!}\cdot[F_{n}]_{q}^{!}},$

where $[F_{n}]_{q}^{!}:=\prod\limits_{k=1}^{n}[F_{k}]_{q}$ is the
$q$-Fibonacci analog of $n!$. Surprisingly, this rational expression turns out
to be a polynomial. Our objective is to present a combinatorial model to
describe it. In order to achieve this, we will introduce some $q$-weights
associated to path-domino tilings of an $m\times n$ rectangle (with $m$
columns and $n$ rows).

Let $T\in\mathcal{T}_{m,n}$ be a path-domino tiling of an $m\times n$
rectangle. The _$q$ -weights_ of the possible tiles in $T$ are defined as
follows

$\omega\left(\,\leavevmode\hbox to18.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=1$, |  $\omega\left(\,\leavevmode\hbox to36.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 36.20006pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-36.00006pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-18.00003pt}{0.0pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-27.00005pt}{-9.00002pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@moveto{-24.50005pt}{-9.00002pt}\pgfsys@curveto{-24.50005pt}{-7.6193pt}{-25.61932pt}{-6.50002pt}{-27.00005pt}{-6.50002pt}\pgfsys@curveto{-28.38077pt}{-6.50002pt}{-29.50005pt}{-7.6193pt}{-29.50005pt}{-9.00002pt}\pgfsys@curveto{-29.50005pt}{-10.38074pt}{-28.38077pt}{-11.50002pt}{-27.00005pt}{-11.50002pt}\pgfsys@curveto{-25.61932pt}{-11.50002pt}{-24.50005pt}{-10.38074pt}{-24.50005pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=q^{F_{i}F_{j}}$, |  $\omega\left(\,\leavevmode\hbox to18.4pt{\vbox to36.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-36.20006pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-36.00006pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-9.00002pt}{-27.00005pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@moveto{-6.50002pt}{-27.00005pt}\pgfsys@curveto{-6.50002pt}{-25.61932pt}{-7.6193pt}{-24.50005pt}{-9.00002pt}{-24.50005pt}\pgfsys@curveto{-10.38074pt}{-24.50005pt}{-11.50002pt}{-25.61932pt}{-11.50002pt}{-27.00005pt}\pgfsys@curveto{-11.50002pt}{-28.38077pt}{-10.38074pt}{-29.50005pt}{-9.00002pt}{-29.50005pt}\pgfsys@curveto{-7.6193pt}{-29.50005pt}{-6.50002pt}{-28.38077pt}{-6.50002pt}{-27.00005pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=q^{F_{i}F_{j}}$, |  $\omega\left(\,\leavevmode\hbox to18.4pt{\vbox to36.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-36.20006pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{1,0.9,0.8}\pgfsys@color@rgb@fill{1}{0.9}{0.8}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-36.00006pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-9.00002pt}{-27.00005pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@moveto{-6.50002pt}{-27.00005pt}\pgfsys@curveto{-6.50002pt}{-25.61932pt}{-7.6193pt}{-24.50005pt}{-9.00002pt}{-24.50005pt}\pgfsys@curveto{-10.38074pt}{-24.50005pt}{-11.50002pt}{-25.61932pt}{-11.50002pt}{-27.00005pt}\pgfsys@curveto{-11.50002pt}{-28.38077pt}{-10.38074pt}{-29.50005pt}{-9.00002pt}{-29.50005pt}\pgfsys@curveto{-7.6193pt}{-29.50005pt}{-6.50002pt}{-28.38077pt}{-6.50002pt}{-27.00005pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=q^{F_{i+1}F_{j}}$,
---|---|---|---

where $(i,j)$ denotes the coordinate of the top-right corner of the tile, and
the shaded vertical domino represents a special vertical domino touching the
path from below. The _$q$ -weight_ of $T$ is defined as the product of the
weights of its tiles; see an example in Fig. 1. The following theorem is one
of our main results.

a

$0$$0$$0$$0$$0$$0$$0$$0$$0$$q^{0\cdot 0}$$q^{0\cdot 0}$$q^{0\cdot
0}$$q^{0\cdot 0}$$q^{0\cdot 0}$$q^{0\cdot 0}$$q^{0\cdot 0}$

Figure 1: A path-domino tiling $T$ of a $5\times 4$ rectangle (left), and the
$q$-Fibonacci weights of its tiles (right). The weight of the tiling is the
product of the weights of its tiles, ${\omega(T)=q^{1+2+6+3+10+24+5}=q^{51}}$.

######  Theorem 2.4.

For $m,n\in\mathbb{N}$, the $q$-analog of the Fibonomial number is a
polynomial in $q$ with non-negative integer coefficients. It can be computed
as

$\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}}=\sum_{T\in\mathcal{T}_{m,n}}\omega(T).$

###### Proof.

Let us start proving the result for the initial cases $m=1$ or $n=1$.

For $n=1$, we have ${\left[\\!\begin{smallmatrix}m+1\\\
1\end{smallmatrix}\\!\right]}_{\mathcal{F}}=[F_{m+1}]_{q}$. The collection
$\mathcal{T}_{m,1}$ coincides with the tilings of an $m$-strip with dominos
and monominos, since only the last step of the specified lattice path can be a
north step because of the special vertical domino condition. The weight of a
domino in a tiling, whose top-right corner has coordinate $(i,1)$, is
$q^{F_{i}F_{1}}=q^{F_{i}}$. Therefore, the result follows from Lemma 2.2.

For $m=1$, we have ${\left[\\!\begin{smallmatrix}1+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}}=[F_{n+1}]_{q}$. The collection
$\mathcal{T}_{1,n}$ can be identified with the collection of tilings of a
vertical $n$-strip with dominos and monominos, where the topmost domino has a
special weight. The weight of a usual vertical domino, whose top-right corner
has coordinate $(1,j)$, is $q^{F_{1}F_{j}}=q^{F_{j}}$, while the weight of a
special vertical domino located at the same place is
$q^{F_{1+1}F_{j}}=q^{F_{j}}$. Therefore, the result also follows from Lemma
2.2.

Now assume the result holds when $m=1$ or $n=1$. Letting $m,n>1$ and using
equation (2.3) in the following equation we obtain

$\displaystyle\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}}$
$\displaystyle=\frac{[F_{m+n}]_{q}[F_{m+n-1}]_{q}^{!}}{[F_{m}]_{q}^{!}\cdot[F_{n}]_{q}^{!}}$
$\displaystyle=[F_{m+1}]_{q^{F_{n}}}\frac{[F_{m+n-1}]_{q}^{!}}{[F_{m}]_{q}^{!}\cdot[F_{n-1}]_{q}^{!}}+q^{F_{n}F_{m+1}}[F_{n-1}]_{q^{F_{m}}}\frac{[F_{m+n-1}]_{q}^{!}}{[F_{m-1}]_{q}^{!}\cdot[F_{n}]_{q}^{!}}$
$\displaystyle=[F_{m+1}]_{q^{F_{n}}}\genfrac{[}{]}{0.0pt}{0}{m+n-1}{n-1}_{\mathcal{F}}+q^{F_{n}F_{m+1}}[F_{n-1}]_{q^{F_{m}}}\genfrac{[}{]}{0.0pt}{0}{m-1+n}{n}_{\mathcal{F}}.$

By induction (and using again Lemma 2.2), the first term of the sum is the
weighted counting of the path-domino tilings of the $m\times n$ rectangle
whose specified path ends with a north step, while the second term is the
weighted counting of those finishing with an east step. Indeed, the path-
domino tilings whose path ends with a north step have an extra contribution
$[F_{m+1}]_{q^{F_{n}}}$ corresponding to the tilings of the last row with
horizontal dominos and monominos. The path-domino tilings whose path ends with
an east step have an extra contribution
$q^{F_{n}F_{m+1}}[F_{n-1}]_{q^{F_{m}}}$; this corresponds to the weight of the
forced special vertical domino ($q^{F_{n}F_{m+1}}$) and the tilings of the
remaining $(n-2)$-strip in the last column ($[F_{n-1}]_{q^{F_{m}}}$). ∎

We have checked the following conjecture by computer for $m,n\leq 10$:

######  Conjecture 2.5.

The polynomials ${\left[\\!\begin{smallmatrix}m+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}}$ are unimodal.

$q$

$q$

$q^{2}$

$q^{2}$

$q^{3}$

Figure 2: The six path-domino tilings of a $2\times 2$ rectangle, and their
contribution to the $q$-Fibonomial ${\left[\\!\begin{smallmatrix}2+2\\\
2\end{smallmatrix}\\!\right]}_{\mathcal{F}}=\frac{[F_{4}][F_{3}]}{[F_{2}][F_{1}]}=\frac{[3][2]}{[1][1]}=1+2q+2q^{2}+q^{3}$
when computed as a generating function
$\sum\limits_{T\in\mathcal{T}_{2,2}}\omega(T)$.

######  Example 2.6.

Fig. 2 illustrates an example of the $q$-Fibonomial for $m=n=2$.

######  Example 2.7 ($n=2$).

Let $m\in\mathbb{N}$ and $n=2$. Theorem 2.4 leads to the identity

$\displaystyle[F_{m+2}][F_{m+1}]=\sum_{k=1}^{m+1}q^{c_{k}^{m}}[F_{k}]^{2},$
(2.4)

where $c_{k}^{m}=\sum\limits_{i=k}^{m}F_{i+1}$.

The left hand side comes from the equality
${\left[\\!\begin{smallmatrix}m+2\\\
2\end{smallmatrix}\\!\right]}_{\mathcal{F}}=[F_{m+2}][F_{m+1}]$. The right
hand side is the sum of the weights of all path-domino tilings of an $m\times
2$ rectangle. In fact, the term $q^{c_{k}^{m}}[F_{k}]^{2}$ indicates the sum
of the weights of the path-domino tilings whose specified path is
$E^{k-1}N^{2}E^{m-(k-1)}$: $q^{c_{k}^{m}}$ is the product of the weights of
the special vertical dominos, and $[F_{k}]^{2}$ is the weight of the two
horizontal rows above the path. Since there are no more possibilities for the
specified path due to the special vertical domino condition, the identity
(2.4) follows. The evaluation at $q=1$ recovers

$\displaystyle F_{m+2}F_{m+1}=\sum_{k=1}^{m+1}F_{k}^{2}.$ (2.5)

######  Remark 2.8.

Equation (2.5) is a well known identity due to its relation with the golden
ratio and golden spirals in nature, see for instance [12, 23]. The left hand
side of the equation is the area of a $F_{m+2}\times F_{m+1}$ rectangle, which
can be subdivided into a sequence of squares, with side lengths
$F_{1},F_{2},\dots,F_{m+1}$, forming a spiral as illustrated in Fig. 3 (left).
This Fibonacci spiral is an approximation of the golden spiral, a special case
of logarithmic spirals which describe the shape of various natural phenomena
such as galaxies, nautilus shells and hurricanes. On the other hand, equation
(2.4) also has a natural geometric interpretation. The left hand side
represents the weighted area of a $F_{m+2}\times F_{m+1}$ rectangle, where a
unit square whose bottom-left corner is located at $(i,j)$ has weight
$q^{i+j}$. This rectangle can be subdivided into a sequence of squares, with
side lengths $F_{1},F_{2},\dots,F_{m+1}$, in the north-east direction as
illustrated in Fig. 3 (right). The sum of their weighted areas is exactly the
right hand side of equation (2.4). This sum can also be interpreted as the
“mass” of the rectangle, where the $F_{k}$-square has density
$d(F_{k})=q^{c_{k}^{m}}$. This density increases according to the ratio
$\frac{d(F_{k})}{d(F_{k+1})}=q^{F_{k+1}}$, satisfying the initial condition
$d(F_{m+1})=1$. It would be interesting to assign these densities to the
squares giving rise to the Fibonacci spiral on Fig. 3 (left), and see if the
resulting equation has some physical meaning. Or even more interesting, to
have a continuous version of the equation representing the mass of the
Fibonacci spiral (or golden spiral), in order to describe some physical
phenomenon in nature. For instance, it is quite natural to think that the
density of galaxies grows exponentially as it approaches the center of the
spiral. In Example 3.6, we additionally provide a generalization of equation
(2.4) using elliptic weight functions.

Figure 3: Geometric interpretation of equations (2.4) and (2.5).

##  3 Elliptic analog of the Fibonomial numbers

As in the previous section, the elliptic analog of the Fibonomial number is
obtained by replacing each term in the binomial coefficient by its
corresponding Fibonacci “elliptic number”. The elliptic number used here is a
slight modification of the elliptic number introduced by Schlosser and Yoo in
[31], motivated by work of Schlosser on elliptic binomial coefficients [29].
The elliptic number is an elliptic function that generalizes the $q$-analog of
a number, and plays an important role in the theory of hypergeometric series
and special functions. Over the last years, elliptic enumeration of
combinatorial models and elliptic generalization of special combinatorial
numbers is becoming an active research area due to its natural appearance in
the theory of hypergeometric functions, see, e.g., [9, 28, 31]. In this
section, we contribute to this general program by studying an elliptic
generalization of the Fibonomial number. Before defining the elliptic analog,
we introduce some basic notations and definitions concerning elliptic
functions.

An _elliptic function_ is a function defined over the complex numbers that is
meromorphic and doubly periodic. It is well known (cf., e.g., [26, 35]) that
elliptic functions can be obtained as quotients of _modified Jacobi theta
functions_. These are defined as

$\theta(x;p):=\prod_{j\geq
0}\left(\big{(}1-p^{j}x\big{)}\left(1-\frac{p^{j+1}}{x}\right)\right),\qquad\theta(x_{1},\dots,x_{\ell};p)=\prod_{k=1}^{\ell}\theta(x_{k};p),$

where $x,x_{1},\dots,x_{\ell}\neq 0$ and $|p|<1$. The _elliptic analog_ of a
natural number $n\in\mathbb{N}$ (or simply _elliptic number_) is defined as

$\displaystyle[n]_{a,b;q,p}:=\frac{\theta\big{(}q^{n},aq^{n},bq,\frac{a}{b}q;p\big{)}}{\theta\big{(}q,aq,bq^{n},\frac{a}{b}q^{n};p\big{)}}.$

Here, $a$, $b$ are two additional parameters. This definition follows the
definition in [30] and corresponds to the definition in [31] subject to the
substitution $b\mapsto bq^{-1}$.333We are grateful to an anonymous referee for
suggesting this substitution. This edition made many of our formulas look
simpler and more symmetric. The elliptic number is indeed an elliptic function
in its parameters [31, Remark 4]. Taking the limit $p\rightarrow 0$, then
$a\rightarrow 0$ and then $b\rightarrow 0$, one recovers the $q$-analog
$[n]_{q}$. We simply define the _elliptic analog of the Fibonacci number_
$F_{n}$ by $[F_{n}]_{a,b;q,p}$.

For $m,n\in\mathbb{N}$, the _elliptic analog of the Fibonomial number_ is
defined as

$\displaystyle\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}_{a,b;q,p}}:=\frac{[F_{m+n}]_{a,b;q,p}^{!}}{[F_{m}]_{a,b;q,p}^{!}\cdot[F_{n}]_{a,b;q,p}^{!}},$

where $[F_{n}]_{a,b;q,p}^{!}:=\prod\limits_{k=1}^{n}[F_{k}]_{a,b;q,p}$ is the
elliptic Fibonacci analog of $n!$.

Similarly as before, the elliptic Fibonomial number counts path-domino tilings
of an $m\times n$ rectangle according to certain elliptic weights. For
$T\in\mathcal{T}_{m,n}$, the _elliptic weights_ of the possible tiles in $T$
are defined as follows

${\widetilde{\omega}}\left(\,\leavevmode\hbox to18.4pt{\vbox
to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-18.20003pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=1$,
${\widetilde{\omega}}\left(\,\leavevmode\hbox to36.4pt{\vbox
to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 36.20006pt\lower-18.20003pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-36.00006pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-18.00003pt}{0.0pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-27.00005pt}{-9.00002pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@moveto{-24.50005pt}{-9.00002pt}\pgfsys@curveto{-24.50005pt}{-7.6193pt}{-25.61932pt}{-6.50002pt}{-27.00005pt}{-6.50002pt}\pgfsys@curveto{-28.38077pt}{-6.50002pt}{-29.50005pt}{-7.6193pt}{-29.50005pt}{-9.00002pt}\pgfsys@curveto{-29.50005pt}{-10.38074pt}{-28.38077pt}{-11.50002pt}{-27.00005pt}{-11.50002pt}\pgfsys@curveto{-25.61932pt}{-11.50002pt}{-24.50005pt}{-10.38074pt}{-24.50005pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=\omega_{1}(i,j)$,
${\widetilde{\omega}}\left(\,\leavevmode\hbox to18.4pt{\vbox
to36.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-36.20006pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-36.00006pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-9.00002pt}{-27.00005pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@moveto{-6.50002pt}{-27.00005pt}\pgfsys@curveto{-6.50002pt}{-25.61932pt}{-7.6193pt}{-24.50005pt}{-9.00002pt}{-24.50005pt}\pgfsys@curveto{-10.38074pt}{-24.50005pt}{-11.50002pt}{-25.61932pt}{-11.50002pt}{-27.00005pt}\pgfsys@curveto{-11.50002pt}{-28.38077pt}{-10.38074pt}{-29.50005pt}{-9.00002pt}{-29.50005pt}\pgfsys@curveto{-7.6193pt}{-29.50005pt}{-6.50002pt}{-28.38077pt}{-6.50002pt}{-27.00005pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=\omega_{1}(j,i)$,
${\widetilde{\omega}}\left(\,\leavevmode\hbox to18.4pt{\vbox
to36.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-36.20006pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{}
{{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{1,0.9,0.8}\pgfsys@color@rgb@fill{1}{0.9}{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{-36.00006pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-36.00006pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{
} {}{{}}{}{{}{}}{{}}{}
{{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-9.00002pt}{-27.00005pt}\pgfsys@stroke\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{
}
{}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@moveto{-6.50002pt}{-27.00005pt}\pgfsys@curveto{-6.50002pt}{-25.61932pt}{-7.6193pt}{-24.50005pt}{-9.00002pt}{-24.50005pt}\pgfsys@curveto{-10.38074pt}{-24.50005pt}{-11.50002pt}{-25.61932pt}{-11.50002pt}{-27.00005pt}\pgfsys@curveto{-11.50002pt}{-28.38077pt}{-10.38074pt}{-29.50005pt}{-9.00002pt}{-29.50005pt}\pgfsys@curveto{-7.6193pt}{-29.50005pt}{-6.50002pt}{-28.38077pt}{-6.50002pt}{-27.00005pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-27.00005pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=\omega_{2}(i,j)$,

where $(i,j)$ denotes the coordinate of the top-right corner of the tile, the
shaded vertical domino represents a special vertical domino touching the path
from below, and

$\displaystyle\omega_{1}(i,j):=v_{a,b;q^{F_{j}},p}(F_{i},F_{i-1}),$
$\displaystyle\omega_{2}(i,j):=v_{a,b;q,p}(F_{i+1}F_{j},F_{i}F_{j-1})$

are defined in terms of the following expression

$\displaystyle
v_{a,b;q,p}(m,n):=\frac{\theta\big{(}aq^{2m+n},b,bq^{n},\frac{a}{b}q^{n},\frac{a}{b};p\big{)}}{\theta\big{(}aq^{n},bq^{m},bq^{m+n},\frac{a}{b}q^{m},\frac{a}{b}q^{m+n};p\big{)}}q^{m}.$

The elliptic weight function $v_{a,b;q,p}(m,n)$ comes from equation (3.2)
below and is the missing factor in that equation. Note that the weight of a
“regular” vertical domino is evaluated at $(j,i)$ instead of $(i,j)$. This
transposition does not make any difference for the $q$-analog of the
Fibonomial numbers, but it does for the elliptic case. The _elliptic weight
${\widetilde{\omega}}(T)$_ of $T$ is defined as the product of the weights of
its tiles; see an example in Fig. 4. The elliptic weight is a generalization
of the $q$-weight, since we obtain the $q$-weight by taking the limit
$p\rightarrow 0$, $a\rightarrow 0$ and $b\rightarrow 0$ in this order.

$1$$2$$3$$4$$5$$1$$2$$3$$4$$\omega_{1}(3,4)$$\omega_{1}(2,3)$$\omega_{1}(2,2)$$\omega_{1}(2,5)$$\omega_{2}(5,4)$$\omega_{2}(4,3)$$\omega_{2}(3,2)$
Figure 4: The path-domino tiling $T$ from Fig. 1 with the elliptic Fibonacci
weights of its tiles. The weight of the tiling is the product of the weights
of its tiles.

######  Theorem 3.1.

For $m,n\in\mathbb{N}$, the elliptic analog of the Fibonomial number can be
computed as

$\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}_{a,b;q,p}}=\sum_{T\in\mathcal{T}_{m,n}}{\widetilde{\omega}}(T).$

The proof of this theorem follows the same steps as the proof of Theorem 2.4.
The proofs of the technical lemmas and examples use some basic properties of
theta functions summarized in the following proposition, which are essential
in the theory of elliptic hypergeometric series.

######  Proposition 3.2 (cf. [35, p. 451, Example 5]).

The theta function satisfies the following basic properties

$\displaystyle\theta(x;0)=1-x,$
$\displaystyle\theta\big{(}\tfrac{1}{x};p\big{)}=-\tfrac{1}{x}\theta(x;p),$
$\displaystyle\theta(px;p)=-\tfrac{1}{x}\theta(x;p),$
$\displaystyle\theta\big{(}xy,\tfrac{x}{y},uz,\tfrac{u}{z};p\big{)}=\theta\big{(}uy,\tfrac{u}{y},xz,\tfrac{x}{z};p\big{)}+\tfrac{x}{z}\theta\big{(}zy,\tfrac{z}{y},ux,\tfrac{u}{x};p\big{)}.$
(3.1)

Before proving Theorem 3.1, let us again prove some straightforward lemmas:

######  Lemma 3.3.

For $m,n\in\mathbb{N}$, the following identities hold

$\displaystyle[m+n]_{a,b;q,p}=[m]_{a,b;q,p}+v_{a,b;q,p}(m,n)[n]_{a,b;q,p},$
(3.2) $\displaystyle[m\cdot n]_{a,b;q,p}=[m]_{a,b;q,p}[n]_{a,b;q^{m},p}.$
(3.3)

###### Proof.

Rearranging the left hand side of equation (3.2) and applying equation (3.1)
(by taking $x=a^{\frac{1}{2}}q^{m}$, $y=a^{\frac{1}{2}}b^{-1}$,
$u=a^{\frac{1}{2}}q^{m+n}$ and $z=a^{\frac{1}{2}}$) yields

$\displaystyle[m+n]_{a,b;q,p}=\frac{\theta\big{(}bq,\frac{a}{b}q;p\big{)}}{\theta\big{(}q,aq,bq^{m+n},\frac{a}{b}q^{m+n};p\big{)}}\cdot\frac{\theta\big{(}\frac{a}{b}q^{m},bq^{m},aq^{m+n},q^{m+n};p\big{)}}{\theta\big{(}\frac{a}{b}q^{m},bq^{m};p\big{)}}$
$\displaystyle\hphantom{[m+n]_{a,b;q,p}}{}=\frac{\theta\big{(}bq,\frac{a}{b}q;p\big{)}}{\theta\big{(}q,aq,bq^{m+n},\frac{a}{b}q^{m+n};p\big{)}}$
$\displaystyle\hphantom{[m+n]_{a,b;q,p}=}{}\times\frac{\theta\big{(}\frac{a}{b}q^{m+n},bq^{m+n},aq^{m},q^{m};p\big{)}+q^{m}\theta\big{(}\frac{a}{b},b,aq^{2m+n},q^{n};p\big{)}}{\theta(\frac{a}{b}q^{m},bq^{m};p)}$
$\displaystyle\hphantom{[m+n]_{a,b;q,p}}{}=[m]_{a,b;q,p}+\frac{\theta\big{(}aq^{2m+n},b,bq^{n},\frac{a}{b}q^{n},\frac{a}{b};p\big{)}}{\theta\big{(}aq^{n},bq^{m},bq^{m+n},\frac{a}{b}q^{m},\frac{a}{b}q^{m+n};p\big{)}}q^{m}[n]_{a,b;q,p}.$

Equation (3.3) follows from simple cancellations. ∎

Using the same arguments as in the $q$-case and replacing the weight of a
domino whose top-right coordinate is $(i,1)$ by $\omega_{1}(i,1)$, we obtain
the following lemmas.

######  Lemma 3.4.

For $n\in\mathbb{N}$, the elliptic analog of the Fibonacci numbers can be
computed as

$[F_{n}]_{a,b;q,p}=\sum_{T}{{\widetilde{\omega}}(T)},$

where the sum ranges over all tilings of an $(n-1)$-strip using dominos and
monominos.

###### Proof.

The result is clearly true for $n=1,2$. Let $n>2$, applying equation (3.2)
from Lemma 3.3 we obtain

$\displaystyle[F_{n}]_{a,b;q,p}$
$\displaystyle=[F_{n-1}+F_{n-2}]_{a,b;q,p}=[F_{n-1}]_{a,b;q,p}+v_{a,b;q,p}(F_{n-1},F_{n-2})[F_{n-2}]_{a,b;q,p}$
$\displaystyle=[F_{n-1}]_{a,b;q,p}+\omega_{1}(n-1,1)[F_{n-2}]_{a,b;q,p}.$

By induction, the first term of this sum corresponds to the tilings of an
$(n-1)$-strip that finish with a monomino, while the second term to the
tilings of an $(n-1)$-strip that finish with a domino. ∎

######  Lemma 3.5.

For $m,n\in\mathbb{N}$, the following identity holds

$\displaystyle[F_{m+n}]_{a,b;q,p}$
$\displaystyle=[F_{n}]_{a,b;q,p}[F_{m+1}]_{a,b;q^{F_{n}},p}+\omega_{2}(m,n)[F_{m}]_{a,b;q,p}[F_{n-1}]_{a,b;q^{F_{m}},p}.$
(3.4)

###### Proof.

Applying Lemma 3.3 to equation (2.2) leads to equation (3.4). ∎

Now, we have all tools to prove the main theorem in this section.

###### Proof of Theorem 3.1.

For $n=1$, we have ${\left[\\!\begin{smallmatrix}m+1\\\
1\end{smallmatrix}\\!\right]}_{\mathcal{F}_{a,b;q,p}}=[F_{m+1}]_{a,b;q,p}$.
The collection $\mathcal{T}_{m,1}$ coincides with the tilings of an $m$-strip
with dominos and monominos, since only the last step of the specified lattice
path can be a north step because of the special vertical domino condition.
Therefore, the result follows from Lemma 3.4.

For $m=1$, we have ${\left[\\!\begin{smallmatrix}1+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}_{a,b;q,p}}=[F_{n+1}]_{a,b;q,p}$.
The collection $\mathcal{T}_{1,n}$ can be identified with the collection of
tilings of a vertical $n$-strip with dominos and monominos, where the topmost
domino has a special weight. The weight of a usual vertical domino, whose top-
right corner has coordinate $(1,j)$, is $\omega_{1}(j,1)$ while the weight of
a special vertical domino located at the same place is $\omega_{2}(1,j)$.
Since $\omega_{2}(1,j)=\omega_{1}(j,1)$, the result also follows from Lemma
3.4.

Now assume the result holds when $m=1$ or $n=1$. Letting $m,n>1$ and using
equation (3.4) in the following equation we obtain

$\displaystyle\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}_{a,b;q,p}}$
$\displaystyle=\frac{[F_{m+n}]_{a,b;q,p}[F_{m+n-1}]_{a,b;q,p}^{!}}{[F_{m}]_{a,b;q,p}^{!}\cdot[F_{n}]_{a,b;q,p}^{!}}=[F_{m+1}]_{a,b;q^{F_{n}},p}\genfrac{[}{]}{0.0pt}{0}{m+n-1}{n-1}_{\mathcal{F}_{a,b;q,p}}$
$\displaystyle\quad+\omega_{2}(m,n)[F_{n-1}]_{a,b;q^{F_{m}},p}\genfrac{[}{]}{0.0pt}{0}{m-1+n}{n}_{\mathcal{F}_{a,b;q,p}}.$

By induction (and using again Lemma 3.4), the first term of the sum is the
weighted counting of the path-domino tilings of the $m\times n$ rectangle
whose specified path ends with a north step, while the second term is the
weighted counting of those finishing with an east step.

Indeed, the path-domino tilings whose path ends with a north step have an
extra contribution $[F_{m+1}]_{a,b;q^{F_{n}},p}$. This corresponds to the
weighted enumeration of the tilings of the last row with horizontal dominos
and monominos. This follows from the fact that both quantities satisfy the
same initial conditions and recurrence relation, which is obtained by applying
equation (3.2) to $F_{m}+F_{m-1}$

$[F_{m+1}]_{a,b;q^{F_{n}},p}=[F_{m}]_{a,b;q^{F_{n}},p}+\omega_{1}(m,n)[F_{m-1}]_{a,b;q^{F_{n}},p}.$

The path-domino tilings whose path ends with an east step have an extra
contribution

$\omega_{2}(m,n)[F_{n-1}]_{a,b;q^{F_{m}},p}.$

This corresponds to the weight of the forced special vertical domino
($\omega_{2}(m,n)$) and the tilings of the remaining $(n-2)$-strip in the last
column ($[F_{n-1}]_{a,b;q^{F_{m}},p}$). ∎

######  Example 3.6 ($n=2$).

The identity (2.4) in Example 2.7 for $m\in\mathbb{N}$ and $n=2$ generalizes
in the elliptic case to

$\displaystyle[F_{m+2}]_{a,b;q,p}[F_{m+1}]_{a,b;q,p}=\sum_{k=1}^{m+1}\Omega_{k}^{m}[F_{k}]_{a,b;q,p}^{2},$

where $\Omega_{k}^{m}=\prod\limits_{i=k}^{m}\omega_{2}(i,2)$ is the product of
the weights of the special vertical dominos, and
$[F_{k}]_{a,b;q,p}^{2}=[F_{k}]_{a,b;q,p}\cdot[F_{k}]_{a,b;q^{F_{2}},p}$ is the
weight of the two horizontal rows above the path.

######  Example 3.7 ($a,b;p\to 0$).

By computing the limits $p\to 0$, $a\to 0$ and $b\to 0$ (in this order) of
${\widetilde{\omega}}(T)$ and ${\left[\\!\begin{smallmatrix}m+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}_{a,b;q,p}}$ we obtain Theorem 2.4.

######  Remark 3.8.

The elliptic analog of the Fibonomial number is a Fibonacci analog of the
“regular” elliptic binomial coefficient

$\displaystyle\genfrac{[}{]}{0.0pt}{0}{n}{k}_{a,b;q,p}:=\frac{[n]_{a,b;q,p}!}{[k]_{a,b;q,p}!\cdot[n-k]_{a,b;q,p}!},$

where $[n]_{a,b;q,p}!=\prod\limits_{i=1}^{n}[i]_{a,b;q,p}$. This can be
expressed as

$\displaystyle\genfrac{[}{]}{0.0pt}{0}{n}{k}_{a,b;q,p}=\frac{(q^{n-k+1},aq^{n-k+1},bq,\frac{a}{b}q;q,p)_{k}}{(q,aq,bq^{n-k+1},\frac{a}{b}q^{n-k+1};q,p)_{k}},$
(3.5)

where the theta shifted factorial is defined as
$(a;q,p)_{k}=\prod\limits_{i=0}^{k-1}\theta(aq^{i};p)$ for $k>0$,
$(a;q,p)_{0}=1$ and
$(a_{1},a_{2},\dots,a_{l};q,p)_{k}=\prod\limits_{i=1}^{l}(a_{i};q,p)_{k}$.

Several elliptic versions of the binomial coefficient have been already
considered in the literature, see for instance [24] or [28]. By substituting
$b\mapsto bq^{k}$ in equation (3.5), we obtain the elliptic binomial
coefficient defined by Schlosser in [28]

$\genfrac{[}{]}{0.0pt}{0}{n}{k}_{a,b;q,p}^{\prime}:=\frac{\big{(}q^{n-k+1},aq^{n-k+1},bq^{1+k},\frac{a}{b}q^{1-k};q,p\big{)}_{k}}{\big{(}q,aq,bq^{n+1},\frac{a}{b}q^{n-2k+1};q,p\big{)}_{k}}.$

One can also check that our regular elliptic binomial coefficient is different
to one defined by Rains in [24, Definition 11], even in the simplest case
where both partitions are single columns of respective lengths $n$ and $k$,
cf. [31, equation (3.12)].

An interesting task for future works on the elliptic Fibonomials could be to
generalize Fibonomial identities to ellitpic identities, see, e.g., [4, 25]
for potential candidates. For instance, we generalize a proof by Reiland [25]
for the identity

${{m+n}\choose
n}_{\mathcal{F}}=\sum_{j=0}^{n}F_{m+1}^{j}F_{n-j-1}{{m-1+n-j}\choose
n-j}_{\mathcal{F}}$

in the following corollary.

######  Corollary 3.9.

For $m,n\geq 1$, the elliptic Fibonomial ${\left[\\!\begin{smallmatrix}m+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}_{a,b;q,p}}$ is equal to

$\sum_{j=0}^{n}\left(\prod_{i=0}^{j-1}[F_{m+1}]_{a,b;q^{F_{n-i}},p}\right)[F_{n-1-j}]_{a,b;q^{F_{m}},p}w_{2}(m,n-j)\genfrac{[}{]}{0.0pt}{0}{m-1+n-j}{n-j}_{\mathcal{F}_{a,b;q,p}}.$

###### Proof.

For $0\leq j\leq n$, consider all weighted path-domino tilings
$\mathcal{T}_{m,n}^{j}$ of an $m\times n$ rectangle for $m,n\geq 1$ where the
last horizontal step of the path ends at position $(m,n-j)$. To obtain the sum
of the weights
$\sum\limits_{T\in\mathcal{T}_{m,n}^{j}}{\widetilde{\omega}}(T)$, we subdivide
the tilings (as illustrated in Fig. 5): The collection $\mathcal{T}_{m,n}^{j}$
can be identified with the collection of

  * •

a path domino tiling of an $(m-1)\times(n-j)$ rectangle which adds weight
${\left[\\!\begin{smallmatrix}m-1+n-j\\\
n-j\end{smallmatrix}\\!\right]}_{\mathcal{F}_{a,b;q,p}}$,

  * •

$j$ full rows on top which add weights $[F_{m+1}]_{a,b;q^{F_{n-i}},p}$ for
$0\leq i\leq j-1$ or $1$ if $j=0$,

  * •

the tile below the last horizontal step, which is forced to be a special
vertical domino with weight $\omega_{2}(m,n-j)$,

  * •

and the column below the special vertical domino which adds weight
$[F_{n-1-j}]_{a,b;q^{F_{m}},p}$

to the sum. Summing over all $j$, we obtain the required convolution formula.
Note that in the case $j=n-1$ the number of tilings is 0 since there has to be
a vertical domino below the last horizontal step and indeed, there appears the
factor $[F_{0}]_{a,b;q^{F_{m}},p}=0$ in the corresponding summand. For $j=n$
there is no special vertical domino and no column below the last east step of
the path but $\omega_{2}(m,0)~{}=~{}1$ and $[F_{-1}]_{a,b;q^{F_{m}},p}=1$ (if
we define $F_{-1}$ by $F_{-1}+F_{0}=F_{1}$ to be $1$). ∎

$[F_{m+1}]_{a,b;q^{F_{n}},p}$$[F_{m+1}]_{a,b;q^{F_{n-1}},p}$$[F_{m+1}]_{a,b;q^{F_{n-(j-1)}},p}$$\omega_{2}(m,n-j)$$[F_{n-1-j}]_{a,b;q^{F_{m}},p}$$\genfrac{[}{]}{0.0pt}{0}{m-1+n-j}{n-j}_{\mathcal{F}_{a,b;q,p}}$
Figure 5: The parts of one summand of the right hand side of Corollary 3.9 for
$m=6$, $n=7$ and $j=3$.

##  4 The $\boldsymbol{q}$-Fibonacci analog of the rational Catalan numbers

Given a pair of relatively prime numbers $m,n\in\mathbb{N}$ (that is, such
that their greatest common divisor is $(m,n):=\gcd\\{m,n\\}=1$), the _$m,n$
-Catalan number_ is defined as

$\operatorname{Cat}_{m,n}:=\frac{1}{m+n}{m+n\choose n}.$

This number is equal to the number of lattice paths from $(0,0)$ to $(m,n)$
that stay weakly above the main diagonal of the $m\times n$ rectangle. The
study of these numbers (also in the non-coprime case), and their $q$-analog
and $q,t$-analog generalizations, is connected to numerous relevant topics
including rectangular diagonal harmonics and Macdonald polynomials [3, 6, 7,
8, 22], Shi hyperplane arrangements and affine Weyl groups [2, 16, 33], affine
Springer fibers [16, 19], affine Hecke algebras [11], knot theory [17, 18],
and representation theory of Cherednik algebras [13, 14, 15].

The _$q$ -Fibonacci analog of the $m,n$-Catalan number_ is defined as

$[\operatorname{FCat}_{m,n}]:=\frac{1}{[F_{m+n}]}\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}}=\frac{[F_{m+n-1}]_{q}^{!}}{[F_{m}]_{q}^{!}[F_{n}]_{q}^{!}}.$

Surprisingly, this rational expression also turns out to be a polynomial when
the greatest common divisor $(m,n)\in\\{1,2\\}$. Before proving this, we need
the following lemma.

######  Lemma 4.1 ([20]).

For $m,n\in\mathbb{N}$, we have $(F_{m},F_{n})=F_{(m,n)}$.

######  Proposition 4.2 (S.X. Li [21]).

If the greatest common divisor $(m,n)\in\\{1,2\\}$, then
$[\operatorname{FCat}_{m,n}]$ is a polynomial in $q$ with integer
coefficients.

###### Proof.

First note that

$[F_{n}]\genfrac{[}{]}{0.0pt}{0}{m+n}{n}_{\mathcal{F}}=[F_{m+n}]\genfrac{[}{]}{0.0pt}{0}{m+n-1}{n-1}_{\mathcal{F}}.$

Since all the terms involved in this identity are polynomials, we know
$[F_{m+n}]$ divides $[F_{n}]{\left[\\!\begin{smallmatrix}m+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}}$. But

$(F_{m+n},F_{n})=F_{(m+n,n)}=F_{(m,n)}=1$

whenever $(m,n)$ is equal to 1 or 2. Since the roots of $[n]$ are the $n$-th
roots of unity that are different to $1$, the polynomials $[F_{m+n}]$ and
$[F_{n}]$ have no roots in common. Thus, $[F_{m+n}]$ divides
${\left[\\!\begin{smallmatrix}m+n\\\
n\end{smallmatrix}\\!\right]}_{\mathcal{F}}$. ∎

Computational experimentation for $m,n\leq 15$ suggests that the coefficients
of these polynomials are non-negative integers. However, we do not have a
proof nor a combinatorial model to describe them.

######  Open Problem 4.3.

Find a combinatorial interpretation of the $q$-Fibonacci analog of the
rational Catalan numbers.

######  Remark 4.4.

The model of Sagan and Savage in [27] gives a combinatorial interpretation of
the Lucas analog of the binomial coefficients. The _Lucas polynomials_
$\\{n\\}$ generalize the Fibonacci numbers, and are defined by the initial
conditions $\\{0\\}=0$, $\\{1\\}=1$ and the recurrence
$\\{n\\}=s\\{n-1\\}+t\\{n-2\\}$ for some variables $s$, $t$. It was proven in
[5, Section 6.2], that the Lucas analog of the rational $m,n$-Catalan numbers
is also polynomial in $s$, $t$ with integer coefficients.444Their proof is
reproduced from the proof of Proposition 4.2 with our permission. However, to
the best of our knowledge, no $q$-analogs of the Lucas-binomial coefficients
have been studied in the literature. The proof of Proposition 4.2 was
originally found by our colleague S.X. Li, during discussions about this topic
in the Algebraic Combinatorics Seminar at the Fields Institute in 2015. At
this Seminar, our colleague F. Aliniaeifard showed that Conjecture 2.5 implies
the positivity of the coefficients.

######  Remark 4.5.

Given a crystallographic Coxeter group $W$ with Coxeter exponents
$e_{1}<e_{2}<\dots<e_{n}$, the rational $W$-Catalan number is defined as
$C_{W}(a)=\prod\limits_{i=1}^{n}\frac{a+e_{i}}{e_{i}+1}$, and this is an
integer when $a$ is relatively prime to $e_{n}+1$. The Coxeter exponents for
the crystallographic Coxeter groups are

$\begin{array}[]{c|l}\text{type of $W$}&e_{1},e_{2},\ldots,e_{n}\\\ \hline\cr
A_{n}&1,2,3,\ldots,n\\\ B_{n}&1,3,5,\ldots,2n-1\\\
D_{n}&n-1,1,3,5,\ldots,2n-3\\\ E_{6}&1,4,5,7,8,11\\\ E_{7}&1,5,7,9,11,13,17\\\
E_{8}&1,7,11,13,17,19,23,29\\\ F_{4}&1,5,7,11\\\ G_{2}&1,5\\\ \end{array}$

The classical Catalan number corresponds to type $A_{n}$. We can now define a
$q$-Fibonacci analog as follows

$C_{W,{\mathcal{F}}}(a)=\prod_{i=1}^{n}\frac{[F_{a+e_{i}}]}{[F_{e_{i}+1}]}.$

We have computationally checked that this is a polynomial with positive
integer coefficients when $a$ and $e_{n}+1$ are relatively prime, for each
type and various values of $a$. It is interesting to note that although in
type $A_{n}$ we have shown that it is a polynomial as long as
$(F_{a},F_{e_{n}+1})=1$, for other types we must have the stronger condition
$(a,e_{n}+1)=1$. For example $C_{F_{4},{\mathcal{F}}}(2)$ is not a polynomial.

##  5 The model of Bennett, Carrillo, Machacek, and Sagan

Recently, Bennett, Carrillo, Machacek, and Sagan [5] gave another
combinatorial interpretation of the Lucas analog of the Binomial coefficient.
A special case of their interpretation is that the Fibonomial number
$\binom{n}{k}_{\mathcal{F}}=\frac{F_{n}^{!}}{F_{k}^{!}\cdot F_{n-k}^{!}}$
counts the number of certain partial tilings of a Young diagram of staircase
shape of size $n$. Similarly as above, we can also assign $q$-weights and
elliptic weights to this model, giving rise to other combinatorial
interpretations for the $q$-analog and elliptic analog of the Fibonomial
number, respectively.

We start by briefly recalling the Bennett–Carrillo–Machacek–Sagan model. For
$n\in\mathbb{N}$, consider the Young diagram of staircase shape of size $n$
(in French notation) associated to the partition $(n-1,n-2,\dots,2,1)$: the
position of the south-west corner is placed at the origin of the Cartesian
coordinate system and the number of boxes in row $i$ (counted from bottom to
top) is $n-i$. We also include the lines from $(0,n-1)$ to $(0,n)$ and from
$(n-1,0)$ to $(n,0)$ for convenience. An example of a Young diagram of
staircase shape for $n=9$ is illustrated on the left of Fig. 6.

Given two natural numbers $n\geq k$, an _$(n,k)$ -tiling_ is a partial tiling
of a staircase shape of size $n$ with monominos and dominos where one lattice
path is specified such that:

  * •

the lattice path goes from $(k,0)$ to $(0,n)$ using steps $(0,1)$ and
$(-1,0)$. It stays inside the Young diagram and every west step $(-1,0)$ must
be followed by a north step $(0,1)$.

  * •

if a north step is not forced by a previous west step, the boxes that are on
its left in the same row are tiled with monominos and horizontal dominos.

  * •

if a north step is forced by a previous west step, the boxes that are on its
right in the same row are tiled with monominos and horizontal dominos. The
tile touching the north step is forced to be a domino (we call it a special
domino).

We denote by $\mathcal{S}_{n,k}$ the collection of all $(n,k)$-tilings. An
example for $n=9$ and $k=5$ is shown on the right of Fig. 6.

Figure 6: A Young diagram of staircase shape of size $9$ (left) and a
$(5,9)$-tiling (right).  floorheight-1height-1floor Figure 7: An illustration
of the height and floor statistics of one domino on the left and one domino on
the right of the path. The domino on the top row has floor $3$ and height $4$,
while the domino on the bottom row has floor $2$ and height $5$.

The tilings in the Bennett–Carrillo–Machacek–Sagan model are in natural
bijection with the tilings of the Sagan–Savage model: the tilings on the right
(resp. left) of the path in the staircase become the tilings of the columns
below (resp. rows above) the path in the rectangle (see last paragraph before
Section 3 in [5]). We have chosen our example to match that of Fig. 1.

Using this bijection, one can easily reinterpret the $q$-weights and elliptic
weights described in Sections 2 and 3 for this new model. Let $T$ be an
$(n,k)$-tiling. For a tile $t$ (monomino or domino) on the left of the path,
we define the _floor_ of $t$ as the number of boxes in the same row from the
western border of the Young diagram to the eastern border of $t$, and the
_height_ as 1 plus the number of boxes in the same row from the eastern border
of the Young diagram to the north step to the east of $t$. For a tile $t$ on
the right of the path the statistics are mirrored as illustrated in Fig. 7.

The $q$-weights of the tiles are defined as follows

$\nu\left(\,\leavevmode\hbox to18.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=1$, |  $\nu\left(\,\leavevmode\hbox to36.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 36.20006pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-36.00006pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-18.00003pt}{0.0pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-27.00005pt}{-9.00002pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@moveto{-24.50005pt}{-9.00002pt}\pgfsys@curveto{-24.50005pt}{-7.6193pt}{-25.61932pt}{-6.50002pt}{-27.00005pt}{-6.50002pt}\pgfsys@curveto{-28.38077pt}{-6.50002pt}{-29.50005pt}{-7.6193pt}{-29.50005pt}{-9.00002pt}\pgfsys@curveto{-29.50005pt}{-10.38074pt}{-28.38077pt}{-11.50002pt}{-27.00005pt}{-11.50002pt}\pgfsys@curveto{-25.61932pt}{-11.50002pt}{-24.50005pt}{-10.38074pt}{-24.50005pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=q^{F_{\text{floor}}F_{\text{height}}}$, |  $\nu\left(\,\leavevmode\hbox to36.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 36.20006pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{1,0.9,0.8}\pgfsys@color@rgb@fill{1}{0.9}{0.8}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-36.00006pt}{-18.00003pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-18.00003pt}{0.0pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-27.00005pt}{-9.00002pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@moveto{-24.50005pt}{-9.00002pt}\pgfsys@curveto{-24.50005pt}{-7.6193pt}{-25.61932pt}{-6.50002pt}{-27.00005pt}{-6.50002pt}\pgfsys@curveto{-28.38077pt}{-6.50002pt}{-29.50005pt}{-7.6193pt}{-29.50005pt}{-9.00002pt}\pgfsys@curveto{-29.50005pt}{-10.38074pt}{-28.38077pt}{-11.50002pt}{-27.00005pt}{-11.50002pt}\pgfsys@curveto{-25.61932pt}{-11.50002pt}{-24.50005pt}{-10.38074pt}{-24.50005pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=q^{F_{\text{floor}}F_{\text{height}+1}}$,
---|---|---

where the shaded domino represents a special domino forced by a west step of
the path. In the elliptic case the weights are defined as

$\widetilde{\nu}\left(\,\leavevmode\hbox to18.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 18.20003pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@lineto{-18.00003pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=1$, |  $\widetilde{\nu}\left(\,\leavevmode\hbox to36.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 36.20006pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-36.00006pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-18.00003pt}{0.0pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-27.00005pt}{-9.00002pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@moveto{-24.50005pt}{-9.00002pt}\pgfsys@curveto{-24.50005pt}{-7.6193pt}{-25.61932pt}{-6.50002pt}{-27.00005pt}{-6.50002pt}\pgfsys@curveto{-28.38077pt}{-6.50002pt}{-29.50005pt}{-7.6193pt}{-29.50005pt}{-9.00002pt}\pgfsys@curveto{-29.50005pt}{-10.38074pt}{-28.38077pt}{-11.50002pt}{-27.00005pt}{-11.50002pt}\pgfsys@curveto{-25.61932pt}{-11.50002pt}{-24.50005pt}{-10.38074pt}{-24.50005pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=\omega_{1}(\text{floor},\text{height})$, |  $\widetilde{\nu}\left(\,\leavevmode\hbox to36.4pt{\vbox to18.4pt{\pgfpicture\makeatletter\hbox{\hskip 36.20006pt\lower-18.20003pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{} {}{{}}{} {{}{}}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{1,0.9,0.8}\pgfsys@color@rgb@fill{1}{0.9}{0.8}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{-18.00003pt}\pgfsys@lineto{-36.00006pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-36.00006pt}{-18.00003pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-18.00003pt}{0.0pt}\pgfsys@lineto{-18.00003pt}{-18.00003pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{} {{}{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@lineto{-27.00005pt}{-9.00002pt}\pgfsys@stroke\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@moveto{-6.50002pt}{-9.00002pt}\pgfsys@curveto{-6.50002pt}{-7.6193pt}{-7.6193pt}{-6.50002pt}{-9.00002pt}{-6.50002pt}\pgfsys@curveto{-10.38074pt}{-6.50002pt}{-11.50002pt}{-7.6193pt}{-11.50002pt}{-9.00002pt}\pgfsys@curveto{-11.50002pt}{-10.38074pt}{-10.38074pt}{-11.50002pt}{-9.00002pt}{-11.50002pt}\pgfsys@curveto{-7.6193pt}{-11.50002pt}{-6.50002pt}{-10.38074pt}{-6.50002pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-9.00002pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } {}{{}}{}{{}{}}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@moveto{-24.50005pt}{-9.00002pt}\pgfsys@curveto{-24.50005pt}{-7.6193pt}{-25.61932pt}{-6.50002pt}{-27.00005pt}{-6.50002pt}\pgfsys@curveto{-28.38077pt}{-6.50002pt}{-29.50005pt}{-7.6193pt}{-29.50005pt}{-9.00002pt}\pgfsys@curveto{-29.50005pt}{-10.38074pt}{-28.38077pt}{-11.50002pt}{-27.00005pt}{-11.50002pt}\pgfsys@curveto{-25.61932pt}{-11.50002pt}{-24.50005pt}{-10.38074pt}{-24.50005pt}{-9.00002pt}\pgfsys@closepath\pgfsys@moveto{-27.00005pt}{-9.00002pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\right)=\omega_{2}(\text{floor},\text{height})$.
---|---|---

The $q$\- and elliptic weights of a tiling $T$ are defined as the product of
the weights of its tiles.

As consequences of Theorem 2.4 and Theorem 3.1 we get the following two
results.

######  Corollary 5.1.

For $n\geq k$, the $q$-analog of the Fibonomial number can be computed as

$\genfrac{[}{]}{0.0pt}{0}{n}{k}_{\mathcal{F}}=\sum_{T\in\mathcal{S}_{n,k}}\nu(T).$

######  Corollary 5.2.

For $n\geq k$, the elliptic analog of the Fibonomial number can be computed as

$\genfrac{[}{]}{0.0pt}{0}{n}{k}_{\mathcal{F}_{a,b;q,p}}=\sum_{T\in\mathcal{S}_{n,k}}\widetilde{\nu}(T).$

An example of Corollary 5.1, for $n=4$ and $k=2$, is given in Fig. 8. We have
chosen the order of the tilings to match the order of the tilings for the
corresponding example in Fig. 2.

$1$

$q$

$q$

$q^{2}$

$q^{2}$

$q^{3}$

Figure 8: The six $(4,2)$-tilings of a staircase shape of size 4, and their
contribution to the $q$-Fibonomial ${\left[\\!\begin{smallmatrix}4\\\
2\end{smallmatrix}\\!\right]}_{\mathcal{F}}=\frac{[F_{4}][F_{3}]}{[F_{2}][F_{1}]}=\frac{[3][2]}{[1][1]}=1+2q+2q^{2}+q^{3}$
when computed as a generating function
$\sum\limits_{T\in\mathcal{S}_{4,2}}\nu(T)$.

######  Remark 5.3.

The Bennett–Carrillo–Machacek–Sagan model also gives rise to combinatorial
interpretations of the Lucas analog of the (ordinary) Catalan numbers and
Fuss–Catalan numbers described in [5]. They define a Catalan partial tiling
$C$ of a Young diagram of staircase shape of size $2n$ to be a
$(2n,n-1)$-tiling where the first row (at the bottom) stays blank except for
the forced special domino if the first step is a west step.

The Lucas analog of the Catalan numbers

$C_{\\{n\\}}:=\frac{1}{\\{n+1\\}}\frac{\\{2n\\}!}{\\{n\\}!\\{n\\}!},$

where $\\{n\\}!=\prod\limits_{k=1}^{n}\\{k\\}$, is the sum over the weights of
all Catalan partial tilings $C$ of size $2n$ if all dominos have weight $t$,
all monominos have weight $s$, and the weight of the tiling $C$ is defined as
the product of the weights of its tiles [5, Corollary 4.2.].

This model does not translate to the $q$-Fibonacci analog of the Catalan
numbers

$[\operatorname{FCat}_{n}]:=\frac{1}{[F_{n+1}]_{q}}\genfrac{[}{]}{0.0pt}{0}{2n+n}{n}_{\mathcal{F}}=\frac{[F_{2n}]_{q}^{!}}{[F_{n+1}]_{q}^{!}[F_{n}]_{q}^{!}}.$

For example, $[\operatorname{FCat}_{3}]$ is not equal to the sum of the
weights of all Catalan partial tilings $C$ of size $6$ with the corresponding
$q$-weights defined above.

Although the Bennett–Carrillo–Machacek–Sagan model does not seem to trivially
solve the following problem, it still might be easier than Open Problem 4.3.

######  Open Problem 5.4.

Find a combinatorial interpretation of the $q$-Fibonacci analog of the Catalan
numbers (and the Fuss–Catalan numbers).

### Acknowledgements

The authors are grateful to Farid Aliniaeifard, Tom Denton, Shu Xiao Li, Drew
Armstrong, Bruce Sagan, Michael Schlosser, and Mike Zabrocki for helpful and
inspiring discussions. We also thank anonymous referees for their useful
comments and suggestions. NB was supported by NSERC and a York Research Chair.
CC was supported by the Austrian Science Foundation FWF, grant F 5008-N15, in
the framework of the Special Research Program Algorithmic and Enumerative
Combinatorics. JK was supported by the Austrian Science Foundation FWF, grant
P 32305.

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