Graded Rational Fibonacci Numbers
Kirill Paramonov
Department of Mathematics, UC Davis
Introduction
Bigraded Fibonacci: s = 1 case.
Simultaneous (a, b)-cores are widely known for their correspondence with rational (a, b)-Dyck paths and connection with rational
Catalan numbers. It had been noticed that the number of simultaneous (a, a + 1)-cores with distinct parts Fa,a+1 is equal to the a-th
Fibonacci number ([Amd,15], [Str,16], [X,15]). Based on that observation, we construct generalizations of Fibonacci numbers and give a
natural grading to those numbers. We then obtain recursive relations, which hint on the connection of bigraded Fibonacci numbers with
Rogers-Ramanujan identities.
Simultaneous (a, b)-cores with distinct parts and nice sequences.
a-core is the partition without hook lengths equal to a. We are using Anderson’s
construction of abacus diagram with a runners to obtain a correspondence
a−1
between a-cores and nonnegative integer sequences {di}i=0 with d0 = 0.
a−1
I Cores with distinct parts correspond to nonnegative integer sequences {di}
i=0
with d0 = 0 satisfying the property di 6= 0 ⇒ di−1 = di+1 = 0. Such
sequences will be called a-nice.
I Simultaneous (a, as + 1)-cores with distinct parts correspond to a-nice
sequences with elements less or equal to s. Such sequences will be called
(a, s)-nice.
I The support of the nice sequence d will be called a nice set. That is,
B ⊂ [a − 1] is nice if it satisfies the property i ∈ B ⇒ i − 1, i + 1 6∈ B.
I Example. The partition λ = (6, 3, 2, 1) is a (4, 13)-core with distinct parts.
It’s corresponding (4, 3)-nice sequence is d = (0, 3, 0, 1). Note that λ is not a
(4, 9)-core since d is not (4, 2)-nice.
I
I
Fa,b(q, t) =
q
area(κ) sl0(κ)
t
,
2
3
2
3
Φ1(3; q, t) = 1 + qt + qt and Φ1(4; q, t) = 1 + qt + qt + qt(1 + qt ).
I
The first two terms of the sequence are Φ1(0; q, t) = Φ1(1; q, t) = 1 and the recursive
relation for Φ1(a; q, t) is
Φ1(a; q, t) = Φ1(a − 1; qt, t) + qtΦ1(a − 2; qt2, t).
When a → ∞, that gives the Andrews q-difference equation for Rogers-Ramanujan
identities
2
Φ1(∞; q, t) − Φ1(∞; qt, t) − qtΦ1(∞; qt , t) = 0.
Rogers-Ramanujan identity
Graded rational Catalan numbers are defined by Ca,b(q) =
the number of rows of κ.
I We define the graded rational Fibonacci numbers similarly:
I
P
area(κ)
q
Fa,b(q) =
, where the sum is taken over all (a, b)-cores κ and area(κ) is
X
qarea(κ),
where the sum is taken over all simultaneous (a, b)-cores κ with distinct parts.
P
I In terms of the corresponding sequence d, the area statistic is simply area(d) =
di.
I Denote Φs(a; q) = Fa,as+1(q). Then Φs(0; q) = Φs(1; q) = 1 and the recursion for Φs(a; q) is
1 − qs
Φs(a; q) = Φs(a − 1; q) + q
Φs(a − 2; q).
1−q
I When q = 1, Φs(a; 1) counts the number of (a, as + 1)-cores with distinct parts φs(a). The recursion is then
Φ1(a; q) = Φ1(a − 1; q) + qΦ1(a − 2; q).
When s → ∞, the recursion takes form
Φ∞(a; q) = Φ∞(a − 1; q) +
Φ∞(a − 2; q).
1−q
This recursion appears for the conjectured graded dimension for the lower level of Khovanov homology of the torus knot T(a, ∞)
[GOR,13].
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Department of Mathematics
[Amd,15] T. Amderhan, Theorems, problems and conjectures. Preprint, July 2015.
Available at: arxiv.org/abs/1207.4045v6.
I [Str,16] A. Straub, Core partitions into distinct parts and an analog of Euler’s theorem.
Preprint, Jan 16. Available at: arxiv.org/abs/1601.07161
I [X,15] H. Xiong, Core partitions with distinct parts. Preprint, Aug 15. Available at:
arxiv.org/abs/1508.07918
I [AHJ,14] D. Armstrong, C. Hanusa, and B. Jones, Results and conjectures on
simultaneous core partitions, Eur. J. Combin. 41 (2014), 205–220.
I [And,72] G. Andrews, On the Rogers-Ramanujan identities and partial q -difference
equations. Illinois J. Math. 16 (1972), no. 2, 270–275.
I [GOR,13] E.Gorsky, A. Oblomkov and J. Rasmussen, On stable Khovanov homology of
torus knots. Experimental Mathematics, 22 (2013), 265–281.
I
so Φ1(a, 1) is the sequence of the regular Fibonacci numbers.
I When s = 1, we get the natural q-generalization of Fibonacci numbers:
q
First Rogers-Ramanujan identity:
n2
X
Y
t
1
Φ1(∞; 1, t) =
=
.
(1 − t) . . . (1 − tn)
(1 − t5k+1)(1 − t5k+4)
I One can check directly that the left equality is satisfied. It follows from the bijection
between nice sets and partitions such that adjacent parts
have
difference
at
least
2.
P
P i
Therefore, the sum over all nice sets Φ1(∞; 1, t) =
t is the generating function of
such partitions.
I
References
φs(a) = φs(a − 1) + sφs(a − 2),
Kirill Paramonov
X
where the sum is over all (a, b)-cores with distinct parts and sl0(κ) is some statistic on
(a, b)-cores. Given the connections of Fibonacci numbers and Catalan numbers, we
expect that statistic to be similar to the skew length statistic on cores [AHJ,14].
I Let B = {i | di 6= 0} be the support of the (a, 1)-nice sequence d, which corresponds to
the (a, a +P1)-core κ with distinct parts. Then area(κ) = |B| and define
0
sl (κ) := i∈B i.
P |B| P i
I Denote Φ1(a; q, t) = Fa,a+1(q, t) =
, where the last sum is taken over all
Bq t
a-nice sets B.
I Example. Observing all (3, 4)- cores and (4, 5)-cores with distinct parts and calculating
area and sl0 statistics, one can obtain
Graded Fibonacci numbers
I
Bigraded Fibonacci numbers are defined as
–
UC Davis
–
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