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arXiv:2312.15117v2 [math.NT] 28 Dec 2023
ON PARTITIONS WITH BOUNDED LARGEST PART AND FIXED INTEGRAL
GBG-RANK MODULO PRIMES
ALEXANDER BERKOVICH AND ARITRAM DHAR
Dedicated to George E. Andrews on the occasion of his 85th birthday
ABS TR ACT. In 2009, Berkovich and Garvan introduced a new partition statistic called the
GBG-rank modulo twhich is a generalization of the well-known BG-rank. In this paper,
we use the Littlewood decomposition of partitions to study partitions with bounded largest
part and fixed integral value of GBG-rank modulo primes. As a consequence, we obtain new
elegant generating function formulas for unrestricted partitions, self-conjugate partitions, and
partitions whose parts repeat finite number of times.
1. INTRO DUCTI ON
A partition is a non-increasing finite sequence π= (λ1, λ2, λ3,...)of non-negative inte-
gers where λi’s are called the parts of π. We denote the number of parts of πby #(π)and
the largest part of πby l(π). The norm of π, denoted by |π|, is defined as
|π|=X
i≥1
λi.
We say that πis a partition of nif |π|=n. We may also write a partition πin terms of its
frequency of parts as
π= (1f1,2f2,3f3,...),
where fiis the frequency of the part i. Let Pdenote the set of all partitions.
The Young diagram of πis a convenient way of representing πgraphically wherein the
parts of πare depicted as rows of unit squares which are called cells. Given the Young di-
agram of π, we label a cell in the ith row and jth column by the least non-negative integer
≡j−i(mod t). The resulting diagram is called a t-residue diagram of π[8]. We can also
label cells in the infinite column 0and the infinite row 0in the same fashion and call the
resulting diagram the extended t-residue diagram of π[7]. For every partition πand positive
integer t, we can associate the t-dimensional vector
~r =~r(π, t) = (r0(π, t), r1(π, t),...,rt−1(π, t))
Date: December 29, 2023.
2020 Mathematics Subject Classification. 05A15, 05A17, 05A19, 11P81, 11P83, 11P84.
Key words and phrases. Littlewood decomposition, GBG-rank, partition, self-conjugate partition, t-core,
t-quotient, generating function, q-series.
1
2 ALEXANDER BERKOVICH AND ARITRAM DHAR
where
ri(π, t) = ri,0≤i≤t−1
is the number of cells labelled iin the t-residue diagram of π. For example, Figure 1 below
depicts the 3-residue diagram for the partition π= (10,7,4,3).
0 1 2 0 1 2 0 1 2 0
2012012
1 2 0 1
0 1 2
FIGUR E 1. 3-residue diagram of the partition π= (10,7,4,3)
We will now define the vector
~n =~n(π, t) = (n0, n1,...,nt−1)
where for 0≤i≤t−2
ni=ri−ri+1,
and
nt−1=rt−1−r0.
Note that
t−1
X
i=0
ni= 0.
In [2] and [3], Berkovich and Garvan defined a partition statistic of π
(1.1) GBG-rank(π, t) :=
t−1
X
j=0
rj(π, t)ωj
t,
where ωt:= e2πι
tis a tth root of unity and ι=√−1.
We call the statistic in (1.1) as the GBG-rank of πmod tand denote it by GBG(t)(π). For
example, in Figure 1 above, for the partition π= (10,7,4,3), GBG(3)(π) = r0+r1ω3+
3
r2ω2
3= 8 + 8ω3+ 8ω2
3= 0.
The special case t= 2 is called the BG-rank of a partition πdefined as
(1.2) BG-rank(π) := r0(π)−r1(π) = r0−r1.
It has been studied extensively in [2] and [5].
We will now recall the notions of rim hook and t-core [8]. If some cell of πshares a vertex
or edge with the rim of the diagram of π, we call this cell a rim cell of π. A connected col-
lection of rim cells of πis called a rim hook if (Young diagram of π)\(rim hook) represents a
legitimate partition. We then call a partition a t-core, denoted πt-core , if its Young diagram has
no rim hooks of length t[8]. Throughout the paper we will denote a t-core by πt-core . Any
partition πhas a uniquely determined t-core which we also denote by πt-core. This partition
πt-core is called the t-core of π. One can obtain πt-core from πby the successive removal of
rim hooks of length t. The t-core πt-core is independent of the way in which the hooks are
removed.
We now recall some definitions from [7]. A region rin the extended t-residue diagram of
πis the set of all cells (i, j)satisfying t(r−1) ≤j−i < tr. A cell of πis called exposed
if it is at the end of a row in the extended t-residue diagram of π. One can construct tinfi-
nite binary words W0, W1, . . . , Wt−1of two letters N, E as follows: The rth letter of Wiis
Eif there is an exposed cell labelled iin the region r, otherwise the rth letter of Wiis N.
It is then easy to see that the word set {W0, W1,...,Wt−1}fixes πuniquely. For example,
the three bi-infinite words W0, W1, W2for the partition (10,7,4,3) in Figure 1 are as follows:
Region : ······ −3−2−1012345······
W0:······ E E E N N N N E N ······
W1:······ E E E N E N N N N ······
W2:······ E E N E N E N N N ······
Let Pt-core denote the set of all t-cores. There is a well-known bijection
φ1:P −→ Pt-core × P × P × P × · · · × P
due to Littlewood [9]
φ1(π) = (πt-core,(ˆπ0,ˆπ1,...,ˆπt−1))
such that
4 ALEXANDER BERKOVICH AND ARITRAM DHAR
|π|=|πt-core|+t
t−1
X
i=0 |ˆπi|.
The vector partition (ˆπ0,ˆπ1,...,ˆπt−1)is called the t-quotient of πand is denoted by πt-quotient .
Let Pt-quotient be the set of all t-quotients. Moreover, it can be shown [7] that
|πt-core|=t
2~n ·~n +~
bt·~n,
where ~
bt:= (0,1,2,...,t−1).
For 0≤i≤t−1, define χi(π, t)to be the largest region in the extended t-residue diagram
of πwhere the cell labeled iis exposed. In [2], Berkovich and Garvan observed that
χi(π, t) = vi+ni(π, t),(1.3)
where viis the number of parts in the ith component of the t-quotient of πand ni(π, t)is the
ith component of ~n(π, t). For example, for the partition π= (10,7,4,3) in Figure 1, from its
three bi-infinite words W0,W1, and W2as shown above, we have χ0(π, 3) = 4,χ1(π, 3) = 1,
and χ2(π, 3) = 2. Note that ~n(π) = (0,0,0). Therefore, using (1.3), we observe that the
0-component of π3-quotient has 4parts, 1-component of π3-quotient has 1part, and 2-component
of π3-quotient has 2parts.
The conjugate of a partition π, denoted by π′, is associated to the Young diagram obtained
by reflecting the diagram for πacross the main diagonal. We say that πis self-conjugate if
π=π′. Let SCP denote the set of all self-conjugate partitions.
Note that if π′is the conjugate of π∈ P whose Littlewood decomposition is
φ1(π) = (πt-core, πt-quotient )
where
πt-quotient = (ˆπ0,ˆπ1,...,ˆπt−1),
then under conjugation, we have
φ1(π′) = (π′
t-core, π′
t-quotient)
where π′
t-core is the conjugate of πt-core and
π′
t-quotient = (ˆπ′
t−1,ˆπ′
t−2,...,ˆπ′
0)
5
is the conjugate of πt-quotient = (ˆπ0,ˆπ1,...,ˆπt−1). Also, observe that under conjugation,
~n(π) = (n0, n1,...,nt−2, nt−1)
becomes
~
˜n=~n(π′) = (−nt−1,−nt−2,...,−n1,−n0).
Now, suppose π∈ SCP. Then, we have
φ1(π) = (πt-core, πt-quotient )
where
πt-core =π′
t-core
and
πt-quotient = (ˆπ0,ˆπ1,ˆπ2,...,ˆπt−1)
with
ˆπi= ˆπ′
t−1−i,0≤i≤t−1.
Note that for tbeing odd,
ˆπt−1
2= ˆπ′
t−1
2.
Throughout the remainder of the paper, ⌊x⌋denotes the floor function,i.e, the greatest in-
teger less than or equal to x,⌈x⌉denotes the ceil function,i.e., the least integer greater than
or equal to x, and δi,j denotes the Kronecker delta function,i.e., δi,j is equal to 1if i=j
otherwise 0.
Let L, m, n be non-negative integers. We now recall some notations from the theory of
q-series that can be found in [1].
(a)L= (a;q)L:=
L−1
Y
k=0
(1 −aqk),
(a)∞= (a;q)∞:= lim
L→∞(a)Lwhere |q|<1.
6 ALEXANDER BERKOVICH AND ARITRAM DHAR
We define the q-binomial (Gaussian) coefficient as
m
nq
:= ((q)m
(q)n(q)m−n
for m≥n≥0,
0otherwise.
Remark 1.For m, n ≥0,m+n
nq
is the generating function for partitions into at most n
parts each of size at most m(see [1, Chapter 3]). Also, note that
lim
m−→∞ m+n
nq
=1
(q)n
,
where 1/(q)nis the generating function for partitions into at most nparts.
Remark 2.(−q;q2)Lis the generating function for partitions into district odd parts having
largest part at most 2L−1. Since self-conjugate partitions are in bijection with distinct odd
part partitions, (−q;q2)Lis also the generating function for self-conjugate partitions with
number of parts at most L.
If ˜
BN(k, q)denotes the generating function for the number of partitions into parts less
than or equal to Nwith BG-rank equal to k, then Berkovich and Uncu [4, Theorem 3.2]
showed that for any non-negative integer Nand any integer k,
(1.4) ˜
B2N+ν(k, q) = q2k2−k
(q2;q2)N+k(q2;q2)N+ν−k
,
where ν∈ {0,1}.
Dhar and Mukhopadhyay [6] ask for direct combinatorial proof of (1.4). In this paper, we
answer their question and generalize (1.4) too for any odd prime t.
The rest of the paper is organized as follows. In Section 2, we present explicit formulas for
the generating functions of the number of unrestricted partitions, self-conjugate partitions,
and partitions whose parts repeat no more than t−1times having bounded largest part and
fixed integral GBG-rank mod tvalue for any prime t. In Section 3, we prove the formulas
of Section 2 using Littlewood decomposition of partitions. In Section 4, we conclude with
some interesting observations.
7
2. MA IN RE SU LTS
In this Section, we present the statements of the main generating functions.
Let GN,t (k, q)denote the generating function for the number of partitions into parts less
than or equal to Nwith GBG-rank mod tequal to k, i.e.,
GN,t(k, q) := X
π∈P
GBG(t)(π)=k
l(π)≤N
q|π|.
We then have the following new result.
Theorem 2.1. For any prime t, a non-negative integer N, and any integer k, we have
(2.1) GtN+ν,t(k, q) = qtk2−(t−1)k
(˜q)N+⌈ν
t⌉−k(˜q)N+⌈ν−1
t⌉(˜q)N+⌈ν−2
t⌉...(˜q)N+⌈ν−(t−2)
t⌉(˜q)N+k
where ˜q:= qt, and ν∈ {0,1,2,...,t−1}.
Now, let GSCN,t (k, q)denote the generating function for the number of self-conjugate
partitions into parts less than or equal to Nwith GBG-rank mod tequal to k, i.e.,
GSCN,t(k, q) := X
π∈SC P
GBG(t)(π)=k
l(π)≤N
q|π|.
We then have the following new results.
Theorem 2.2. For any non-negative integer Nand any integer k, we have
(2.2) GSC2N+ν,2(k, q) = q2k2−k2N+ν
N+kq4
where ν∈ {0,1}.
Theorem 2.3. For any odd prime t, a non-negative integer N, and any integer k, we have
(2.3)
GSCtN+ν,t (k, q) = qtk2−(t−1)k(−qt;q2t)N+ν−t−1
2
t2N+⌈ν
t⌉
N+kq2t
t−3
2
Y
i=1 2N+⌈ν−i
t⌉+⌊ν+i
t⌋
N+⌊ν+i
t⌋q2t
where ν∈ {0,1,2,...,t−1}.
8 ALEXANDER BERKOVICH AND ARITRAM DHAR
Now, let ˜
GN,t(k, q)denote the generating function for the number of partitions into parts
repeating no more than t−1times and less than or equal to Nwith GBG-rank mod tequal
to k, i.e.,
˜
GN,t (k, q) := X
π=(1f1,2f2,...,NfN)∈P
fi≤t−1,1≤i≤N
GBG(t)(π)=k
q|π|.
We then have the following new result.
Theorem 2.4. For any prime t, a non-negative integer N, and any integer k, we have
(2.4) ˜
GtN+ν,t(k, q) = qtk2−(t−1)k(˜q)tN +ν
(˜q)N+⌈ν
t⌉−k(˜q)N+⌈ν−1
t⌉(˜q)N+⌈ν−2
t⌉...(˜q)N+⌈ν−(t−2)
t⌉(˜q)N+k
where ˜q:= qt, and ν∈ {0,1,2,...,t−1}.
3. PRO OF S O F THE MAIN RESULTS
In this Section, we provide new combinatorial proofs of the main generating functions of
Section 2.
3.1. Proof of Theorem 2.1. Let π= (λ1, λ2, λ3,...)∈ P be a partition with λ1≤tN +ν
where 0≤ν≤t−1and consider the Littlewood decomposition of π
φ1(π) = (πt-core, πt-quotient )
where πt-quotient = (ˆπ0,ˆπ1, . . . , ˆπt−1).
Now, let tbe any prime and GBG(t)(π) = kbe an integer. One can then easily verify that
GBG(t)(π) = kis an integer if and only if
r0(π) = k+r
and
ri(π) = r
for any r∈Z+∪ {0}and 1≤i≤t−1. We leave it as an exercise to the reader.
Therefore, we have
~n(π, t) = (k, 0,0,...,0,0,−k),
where n0(π) = k,nt−1(π) = −k, and ni(π) = 0 for 1≤i≤t−2.
9
Thus, we have
|πt-core|=t
2·2k2−(t−1) ·k
=tk2−(t−1)k.(3.1)
Now, we consider two cases regarding νwhich are as follows:
•Case I: ν > 0
Observe that it is easy to check that the cell labeled imay be exposed in the region
marked N+ 1 for 0≤i≤ν−1and is not exposed in the region marked N+ 1 for
ν≤i≤t−1. Thus, it follows that
N+ 1 ≥χi(π, t),0≤i≤ν−1,
and
N≥χi(π, t),ν≤i≤t−1.
Equation (1.3) then implies that
(3.2) N+ 1 ≥vi+kδi,0,0≤i≤ν−1,
and
(3.3) N≥vi−kδi,t−1,ν≤i≤t−1.
•Case II: ν= 0
Here, the cell labeled iis not exposed in the region marked N+1 for 0≤i≤t−1.
Thus, it follows that
N≥χi(π, t),0≤i≤t−1.
Equation (1.3)then implies that
(3.4) N≥vi+kδi,0−kδi,t−1,0≤i≤t−1.
Hence, combining (3.2), (3.3), and (3.4), we have
(3.5) vi≤N+lv−i
tm−kδi,0+kδi,t−1,0≤i≤t−1,
10 ALEXANDER BERKOVICH AND ARITRAM DHAR
where viis the number of parts in the ith component of the t-quotient of π.
Hence, the required generating function follows from Remark 1, (3.1), and (3.5) which
completes the proof of Theorem 2.1.
Remark 3.Observe that (1.4) is a special case of Theorem 2.1 where t= 2 and so, for t= 2,
the proof of Theorem 2.1 provides a direct combinatorial proof of (1.4) as asked by Dhar and
Mukhopadhyay [6].
3.2. Proof of Theorem 2.2. Suppose π∈ SCP be a self-conjugate partition having l(π)≤
2N+ 1 and consider the Littlewood decomposition of π
φ1(π) = (π2-core, π2-quotient )
where π2-quotient = (ˆπ0,ˆπ1)and ˆπ1= ˆπ′
0.
Now, let BG(π) = k∈Z. It is again easy to verify that BG(π) = kif and only if
r0(π) = k+r
and
r1(π) = r
for any r∈Z+∪ {0}which implies that
~n(π) = (k, −k).
Thus, we have
|π2-core|=2
2·2k2−(2 −1) ·k
= 2k2−k.(3.6)
Now, observe that the cell labeled 0may be exposed in the region marked N+ 1 and
the cell labeled 1is not exposed in the region marked N+ 1. Thus, again using (1.3), the
following equations hold:
(3.7) N+ 1 ≥k+v0,
which implies that
(3.8) v0≤N+ 1 −k,
11
and
(3.9) N≥ −k+v1,
which implies that
(3.10) v1≤N+k.
Hence, from (3.8) and (3.10), we have l(ˆπ0)≤N+kand #(ˆπ0)≤N+1 −kwhich when
combined with Remark 1 and (3.6) gives us the required generating function in Theorem 2.2
for ν= 1. Analogously, one can prove the required generating function for ν= 0 which
gives us the complete proof of Theorem 2.2.
3.3. Proof of Theorem 2.3. Let π= (λ1, λ2, λ3,...)∈ SCP be a partition with λ1≤
tN +νwhere 0≤ν≤t−1and consider the Littlewood decomposition of π
φ1(π) = (πt-core, πt-quotient )
where πt-quotient = (ˆπ0,ˆπ1, . . . , ˆπt−1).
Now, let tbe any odd prime and GBG(t)(π) = kbe an integer. It is then easy to verify that
GBG(t)(π) = kis an integer if and only if
r0(π) = k+r
and
ri(π) = r
for any r∈Z+∪ {0}and 1≤i≤t−1.
Therefore, we have
~n(π) = (k, 0,0,...,0,0,−k),
where n0(π) = k,nt−1(π) = −k, and ni(π) = 0 for 1≤i≤t−2.
Thus, we have
|πt-core|=t
2·2k2−(t−1) ·k
=tk2−(t−1)k.(3.11)
12 ALEXANDER BERKOVICH AND ARITRAM DHAR
Now, let us consider the pair of partitions (ˆπ0,ˆπt−1)from πt-quotient and observe that ˆπ0=
ˆπ′
t−1. Therefore, this pair contributes to the generating function as
qt(|ˆπ0|+|ˆπt−1|)=q2t|ˆπ0|.(3.12)
Now, by (3.5), we have #(ˆπ0)≤N−k+⌈ν
t⌉and #(ˆπt−1)≤N+kwhich is the same
as l(ˆπ0)≤N+k. So, using Remark 1 and (3.12), we get the term
2N+⌈ν
t⌉
N+kq2t
(3.13)
in the required generating function.
Next, for 1≤i≤(t−3)/2, we consider the pair of partitions (ˆπi,ˆπt−1−i)from πt-quotient
and observe that ˆπi= ˆπ′
t−1−i. Therefore, this pair contributes to the generating function as
qt(|ˆπi|+|ˆπt−1−i|)=q2t|ˆπi|.(3.14)
Now, by (3.5), for 1≤i≤(t−3)/2, we have #(ˆπi)≤N+⌈ν−i
t⌉and #(ˆπt−1−i)≤
N+⌈ν−(t−1−i)
t⌉=N+⌊ν+i
t⌋which is the same as l(ˆπi)≤N+⌊ν+i
t⌋. So, using Remark
1 and (3.14) followed by taking product over all values of i, we get the term
t−3
2
Y
i=1 2N+⌈ν−i
t⌉+⌊ν+i
t⌋
N+⌊ν+i
t⌋q2t
(3.15)
in the required generating function.
Finally, observe that ˆπt−1
2= ˆπ′
t−1
2
and similarly, by (3.5), we have #(ˆπt−1
2)≤N+
ν−t−1
2
t. Thus, using Remark 2, we get the term
(−qt;q2t)N+ν−t−1
2
t
(3.16)
in the required generating function.
Hence, combining (3.11), (3.13), (3.15), and (3.16), we have the desired generating func-
tion which completes the proof of Theorem 2.3.
13
3.4. Proof of Theorem 2.4. The proof of Theorem 2.4 comes from the combinatorial bijec-
tion of extracting tidentical rows in succession from the t-residue diagram of a partition to
form a new partition whose parts repeat no more than t−1times.
Let
π= (1tq1+f1,2tq2+f2,3tq3+f3...,NtqN+fN),
where 0≤fj≤t−1for 1≤j≤N.
Then, it can be seen that πis in one-one correspondence with the pair of partitions (π1, π2)
where
π1= (1tq1,2tq2,3tq3,...,NtqN)
and
π2= (1f1,2f2,3f3,...,NfN).
Now, note that
X
π1
q|π1|=1
(qt;qt)N
,(3.17)
and so, we have
X
π
q|π|=X
π1
q|π1|·X
π2
q|π2|
=1
(qt;qt)NX
π2
q|π2|,(3.18)
where (3.18) follows from (3.17).
Hence, from (3.18), we have
X
π2
q|π2|= (qt;qt)NX
π
q|π|,(3.19)
where π2is a partition whose parts repeat no more than t−1times.
Also, observe that GBG(t)(π) = GBG(t)(π2) = ksince removal of tidentical rows in
succession keeps the GBG-rank mod tvalue of πinvariant.
14 ALEXANDER BERKOVICH AND ARITRAM DHAR
Therefore, using the above observation and replacing Nby tN +νin (3.19) for 0≤ν≤
t−1alongwith Theorem 2.1 gives us the desired generating function ˜
GtN+ν,t(k, q).
Remark 4.Note that, for t= 2, we get [4, eq. (3.1)] for distinct part partitions.
4. FURTHER OBSERVATION S
In this Section, we will see that Theorem 2.1 can be easily extended to the case where
partitions have GBG-rank mod tvalue equal to kωj
tfor any odd prime t, any integer kand
1≤j≤t−1.
Let π= (λ1, λ2, λ3,...)∈ P be a partition with λ1≤tN +νwhere 0≤ν≤t−1and
consider the Littlewood decomposition of π
φ1(π) = (πt-core, πt-quotient )
where πt-quotient = (ˆπ0,ˆπ1, . . . , ˆπt−1).
Now, let tbe any odd prime and GBG(t)(π) = kωj
twhere kis an integer and 1≤j≤t−1.
One can then easily verify that GBG(t)(π) = kωj
tif and only if
ri(π) = r+kδi,j
for any r∈Z+∪ {0},0≤i≤t−1, and 1≤j≤t−1.
Therefore, we have
~n(π, t) = (n0, n1,...,nt−1),
where ni(π) = −kδi,j −1+kδi,j for 0≤i≤t−1.
Thus, we have
|πt-core|=tk2+k.(4.1)
We then have the following new result.
15
Theorem 4.1. For any odd prime t, a non-negative integer N, and any integer k, we have
(4.2) GtN+ν,t(kωj
t, q) = qtk2+k
t−1
Y
i=0
(˜q)N+kδi,j −1−kδi,j +⌈ν−i
t⌉
,
where 1≤j≤t−1,˜q:= qt, and ν∈ {0,1,2,...,t−1}.
The rest of the proof of Theorem 4.1 is analogous to the proof of Theorem 2.1.
5. ACKNOWLEDGMENT S
The authors would like to thank Frank Garvan for helping them with his QSERIES and
TCORE packages in Maple. The authors would also like to thank George Andrews and Ali
K. Uncu for their kind interest.
REFEREN CES
1. G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998.
2. A. Berkovich and F. G. Garvan, The BG-rank of a partition and its appications, Adv. in Appl. Math. 40 (3)
(2008) 377–400.
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DEPARTMEN T OF MATH EMATI CS , UN IV ER SI TY O F FLORIDA , GA IN ES VI LL E FL 32 611, USA
Email address:alexb@ufl.edu
DEPARTMEN T OF MATH EMATI CS , UN IV ER SI TY O F FLORIDA , GA IN ES VI LL E FL 32 611, USA
Email address:aritramdhar@ufl.edu
