Available via license: CC BY 4.0
Content may be subject to copyright.
arXiv:2308.13457v1 [math.CO] 25 Aug 2023
Super FiboCatalan Numbers and Generalized
FiboCatalan Numbers
Kendra Killpatrick
Pepperdine University
24255 Pacific Coast Hwy
Malibu, CA 90265
August 28, 2023
Abstract
Catalan observed in 1874 that the numbers S(m, n) = (2m)!(2n)!
m!n!(m+n)! , now
called the super Catalan numbers, are integers but there is still no known
combinatorial interpretation for them in general, although interpretations
have been given for the case m= 2 and for S(m, m +s) for 0 ≤s≤
3. In this paper, we define the super FiboCatalan numbers S(m, n)F=
F2m!F2n!
Fm!Fn!Fm+n!and prove they are integers for m= 1 and m= 2. In
addition, we prove that S(m, m +s)Fis an integer for 0 ≤s≤4.
1 Background and Definitions
The well-known Fibonacci sequence is defined recursively by Fn=Fn−1+Fn−2
with initial conditions F0= 0 and F1= 1. The nth Fibonacci number, Fn,
counts the number of tilings of a strip of length n−1 with squares of length 1
and dominoes of length 2.
The fibonomial coefficients, an analogue of the binomial coefficients, are
defined as
n
kF
=Fn!
Fk!Fn−k!
where Fn! = FnFn−1···F2F1.
In 2008, Benjamin and Plott [2] gave a combinatorial proof that the fibono-
mial coefficients are integers, using a notion of staggered tilings. In 2010, Sagan
and Savage [12] gave a combinatorial interpretation of the coefficients in terms
of tilings associated to paths in a kx (n−k) rectangle.
A second famous sequence, the Catalan sequence, is defined recursively by
Cn=C0Cn−1+C1Cn−2+···+Cn−2C1+Cn−1C0with initial conditions C0= 1
and C1= 1. The Catalan numbers also have an explicit formula given by
1
Cn=1
n+ 1 2n
n.
The FiboCatalan number Cn,F , first given by Lou Shapiro, is defined as
Cn,F =1
Fn+1 2n
nF
.
Shapiro posed the question about whether these numbers are integers and,
if so, whether there is a combinatorial interpretation for them. The numbers
are known to be integers since
Cn,F =2n−1
n−2F
+2n−1
n−1F
but there is still no known combinatorial interpretation for them.
Since the Catalan numbers,
(2n)!
n!(n+ 1)!
are integers, one might wonder if the numbers
(2n)!
n!(n+ 2)!
are integers. Interestingly, these numbers are not necessarily integers but the
numbers given by
6(2n)!
n!(n+ 2)!
do form an integer sequence. In 1992, Gessel [7] showed that, in fact, the
generalized Catalan numbers
Jr
(2n)!
n!(n+r+ 1)!
are integers when Jris chosen to be (2r+ 1)!/r!.
In 2005, Gessel and Xin [8] gave a combinatorial interpretation of these
numbers for r= 1 and proved
6(2n)!
n!(n+ 2)! = 4Cn−Cn+1.
E. Catalan [4] observed as far back as 1874 that the numbers
S(m, n) = (2m)!(2n)!
m!n!(m+n)!
are integers, but there is no known combinatorial interpretation for them in gen-
eral. Gessel [7] called these numbers the super Catalan numbers since S(1, n)/2
2
gives the Catalan number Cn. Note that S(2, n)/2 = 6 (2n)!
n!(n+2)! . Allen and Ghe-
orghiciuc [1] have given a combinatorial interpretation for S(m, n) in the case
m= 2 and Gheorghiciuc and Orelowitz have given a combinatorial interpreta-
tion for T(m, n) = 1
2S(m, n) for m= 3 and m= 4 [9]. Chen and Wang [5] have
given an interpretation for S(m, m +s) for 0 ≤s≤3.
In this paper, we define the super FiboCatalan numbers
S(m, n)F=F2m!F2n!
Fm!Fn!Fm+n!
and the generalized FiboCatalan numbers as
Jr,F
F2n!
Fn!Fn+r+1!
where Jr,F =F2r+1!/Fr!. Note the following relationship between super Fibo-
Catalan numbers and the generalized FiboCatalan numbers:
Jm−1,F
F2n!
Fn!Fn+m!=F2m−1!F2n!
Fm−1!Fn!Fn+m!=Fm
F2m
S(n, m)F.
This paper explores the generalized FiboCatalan numbers and the super
FiboCatalan numbers. In Section 2, we prove that the generalized FiboCatalan
numbers are integers for r= 1 (and trivially shows they are integers for r= 0).
In Section 3, we prove that the super FiboCatalan numbers are integers for
m= 1 and m= 2 and prove that S(m, m +s)Fare integers for 0 ≤s≤3. In
Section 4, we make several conjectures and state open problems in this area.
2 The generalized FiboCatalan numbers
The generalized FiboCatalan number for r= 0 is equal to S(1, n)Fis equal to
Cn,F :
J0,F
F2n!
Fn!Fn+0+1!=F1!
F0!
F2n!
Fn!Fn+1!=Cn,F =S(1, n)F.
The generalized FiboCatalan number for r= 1 is:
J1,F
F2n!
Fn!Fn+1+1!=F3!
F1!
F2n!
Fn!Fn+2!= 2 F2n!
Fn!Fn+2!=1
3S(2, n)F.
We will prove that these numbers are always integers.
Lemma 1.
F2nFn+2 −F2n+2Fn= (−1)nFn.
Proof. This is a fairly well-known result for the Fibonacci numbers and the proof
is a straightforward tail-swapping argument similar to those found in Section
1.2, Chapter 1 of Proofs That Really Count by Benjamin and Quinn [3]. For a
more algebraic argument, see Theorem 1.2 (with q= 1) in a paper by Garrett
[6].
3
In general, we have
Lemma 2.
FknFn+2 −Fkn+2Fn= (−1)nF(k−1)n.
Theorem 1.
F2n+1F2nCn,F −Fn+1 FnCn+1,F = (−1)nFnF2n+1
F2n!
Fn+2!Fn!.(1)
Proof.
F2n+1F2nCn,F −Fn+1 FnCn+1,F =F2n+1F2nF2n!
Fn+1Fn!Fn!−Fn+1 FnF2n+2!
Fn+2Fn+1 !Fn+1!
=F2n+1F2nFn+2
F2n!
Fn+2!Fn!−F2n+2 F2n+1Fn
F2n!
Fn+2!Fn!
=F2n+1[F2nFn+2 −F2n+2Fn]F2n!
Fn+2!Fn!
=F2n+1(−1)nFn
F2n!
Fn+2!Fn!
Corollary 1. For n≥1,
F2n+1
F2n!
Fn+2!Fn!=1
Fn+2 2n+ 1
nF
is an integer.
Given that the FiboCatalan number is defined as
Cn,F =1
Fn+1 2n
nF
,
the Corollary states that the odd FiboCatalan number
1
Fn+2 2n+ 1
nF
is always an integer. This is not true for the usual binomial expression
1
n+ 2 2n+ 1
n
since this number is a fraction when n= 2, for example.
Proof. It is well know that F2n=FnFn+1 +FnFn−1, thus the left side of
Equation (1) from Theorem 1 is equal to
F2n+1[FnFn+1 +FnFn−1]Cn,F −Fn+1 FnCn+1,F
4
and is therefore divisible by Fn. Using this expression as the left side and
dividing both sides of Equation (1) by Fngives
F2n+1Fn+1 Cn,F +F2n+1Fn−1Cn,F −Fn+1Cn+1,F = (−1)nF2n+1
F2n!
Fn+2!Fn!
= (−1)n1
Fn+2 2n+ 1
nF
.
Since the left side of this equation is clearly an integer, we have the result.
We can also rewrite the expression on the right side of Equation (1) as:
(−1)nF2n+1
F2n!
Fn+2!Fn!= (−1)nF2n+1
1
Fn+2
Cn,F .
A well-known fact of the Fibonacci numbers is that gcd(Fn, Fm) = Fgcd(m,n).
Thus gcd(F2n+1, Fn+2 ) = Fgcd(2n+1,n+2). The gcd(2n+ 1, n + 2) = 1 or 3. If
gcd(2n+ 1, n + 2) = 1, then gcd(F2n+1 , Fn+2) = F1= 1 and so Fn+2 divides
Cn,F . If gcd(2n+ 1, n + 2) = 3, then gcd(F2n+1, Fn+2) = F3= 2 and so Fn+2
divides 2Cn,F .
Corollary 2. For n≥1, the generalized FiboCatalan number for r= 1,
2F2n!
Fn+2!Fn!=1
Fn+2
2Cn,F
is an integer.
3 The super FiboCatalan numbers
When m= 1, the super FiboCatalan numbers reduce to the FiboCatalan num-
bers and are known to be integers.
S(1, n)F=F2!F(2n)!
F1!Fn!F(n+1)!=1
Fn+1
F2n!
Fn!Fn!=Cn,F
When m= 2, we have
S(2, n)F=F4!F2n!
F2!Fn!F(n+2)!=6F2n!
Fn!F(n+2)!.
Theorem 2. For n≥1, the super FiboCataln number is an integer for m= 2.
I.e.,
S(2, n)F=6F2n!
Fn!Fn+2!
is an integer.
5
Proof. From Corollary 2 in the preceding section , we have that
2F2n!
Fn!Fn+2!
is an integer, thus we have the result.
Theorem 3. S(m, m +s)Fis an integer for 0≤s≤5.
Proof. When m= 0 we have:
S(m, m)F=F2m!F2m!
Fm!Fm!F2m!=2m
mF
which is an integer.
When m= 1,
S(m, m + 1)F=F2m!F2m+2 !
Fm!Fm+1!F2m+1 !=F2m+2F2m!
Fm+1!Fm!=F2m+2 Cm,F
which is an integer.
When m= 2,
S(m, m + 2)F=F2m!F2m+4 !
Fm!Fm+2!F2m+2 !
=F2m!F2m+4F2m+3 F2m+2!
Fm+2Fm+1 Fm!Fm!F2m+2!
=1
Fm+1
F2m!
Fm!Fm!
F2m+4
Fm+2
F2m+3
=F2m+3Cm,F
F2(m+2)
Fm+2
.
Since F2n=FnFn−1+FnFn+1 then Fndivides F2nso Fm+2 divides F2(m+2).
Therefore,
F2m+3Cm,F
F2(m+2)
Fm+2
is an integer.
When m= 3,
S(m, m + 3)F=F2m!F2(m+3) !
Fm!Fm+3!F2m+3 !
=F2m!F2m+6F2m+5 F2m+4F2m+3!
Fm!Fm!Fm+1Fm+2 Fm+3F2m+3!
=Cm,F
F2m+6
Fm+3
F2m+4
Fm+2
F2m+5
which again is an integer since Fndivides F2n.
6
When m= 4,
S(m, m + 4)F=F2m!F2(m+4) !
Fm!Fm+4!F2m+4 !
=F2m!F2m+8F2m+7 F2m+6F2m+5F2m+4 !
Fm!Fm!Fm+1Fm+2 Fm+3Fm+4F2m+4 !
=1
Fm+2
Cm,F
F2m+8
Fm+4
F2m+6
Fm+3
F2m+7F2m+5 .
Since Fndivides F2n, we must show that
1
Fm+2
Cm,F F2m+7F2m+5
is an integer. From Corollary 2, we know that
1
Fm+2
2Cm,F
is an integer. If m≡0 mod 3, then Fm+2 is odd, so Fm+2 must divide Cm,F .
If m≡1 mod 3, then F2m+7 is even, thus Fm+2 divides Cm,F F2m+7. If m≡2
mod 3, then F2m+5 is even, thus Fm+2 divides Cm,F F2m+5.
4 Open Problems
It remains an open problem to determine if the super FiboCatalan numbers are
integers for all values of mand nand if the generalized FiboCatalan numbers
are integers for all values of nand r. The problem of finding a combinatorial
interpretation of the super FiboCatalan numbers remains an interesting open
problem, yet will likely prove challenging given that there is a combinatorial
interpretation for the super Catalan numbers in only a handful of cases.
In addition, the super Catalan numbers satisfy a number of interesting bi-
nomial identities, such as this identity of von Szily (1894), which can be found
in [7], Eq. (29), p. 11:
S(m, n) = X
k∈Z
(−1)k2m
m+k 2n
n+k.
Mikic [10] recently proved the following alternating convolution formula for the
super Catalan numbers:
2n
X
k=0
(−1)k2n
kS(k, l)S(2n−k , l) = S(n, l)S(n+l, n)
for all non-negative integers nand l. Mikic [11] also proved a similar identity
for the Catalan numbers:
2n
X
k=0
(−1)k2n
kCkC2n−k=Cn2n
n.
7
We conjecture that many of these identities have analogues for the super
FiboCatalan numbers and are interested in exploring these analogues in further
research.
References
[1] Allen, E. and Gheorghiciuc, I., A weighted interpretation for the super
Catalan numbers, Journal of Integer Sequences, Vol. 17, Article 14.10.7,
(2014).
[2] Benjamin, A. and Plott, S., A combinatorial approach to Fibonomial coef-
ficients, Fibonacci Quarterly, 46/47(1):7-9, 2008/09.
[3] Benjamin, A. and Quinn, J., Proofs That Really Count, Vol. 27 of The
Dolciani Mathematical Expositions, Mathematical Association of America,
Washington, DC, 2003.
[4] Catalan, E., Question 1135, Nouvelles Annales de Mathematiques: Journal
des Candidats aux Ecole Polytechnic et Normale, Series 2, 13, (1874), 207.
[5] Chen, X. and Wang, J., The super Catalan numbers S(m, m +s) for s≤4,
arXiv:1208.4196v1 [math.CO] (2012).
[6] Garrett, K., A Determinant Identity that Implies Rogers-Ramanujan, The
Electronic Journal of Combinatorics, 12, #R35 (2005).
[7] Gessel, I., Super ballot numbers, J. Symb. Comput. 14 (1992), 179-194.
[8] Gessel, I. and Xin, G., A combinatorial interpretation of the numbers
6(2n)!/n!(n+ 2)!, J. Integer Seq. 8, Article 15.2.3, (2005).
[9] Gheorghiciuc, I. and Orelowitz, G., Super-Catalan Numbers of the Third
and Fourth Kind, arXiv:2008.00133v1 [math.CO] (2020).
[10] Mikic, J., On a new alternating convolution formula for the super Catalan
numbers, arXiv:2110.04805v1 [math.CO] (2021).
[11] Mikic, J., Two new identities involving the Catalan numbers and sign-
reversing involutions., J. Integer Sequences 23, Article 20.1.6 (2020)
[12] Sagan, B. and Savage, C., Combinatorial interpretations of binomial coeffi-
cient analogues related to Lucas sequences. Integers, The Electronic Journal
of Combinatorial Number Theory, 10: A52, 697-703 (2010).
8