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A Recursive Relation for Bipartition Numbers

Yen-Chi Roger Lin Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan yclin@math.ntnu.edu.tw  and  Shu-Yen Pan Department of Mathematics, National Tsing Hua University, Hsinchu 300, Taiwan sypan@math.nthu.edu.tw
Abstract.

We establish a recursive relation for the bipartition number p2​(n)p_{2}(n) which might be regarded as an analogue of Euler’s recursive relation for the partition number p​(n)p(n). Two proofs of the main result are proved in this article. The first one is using the generating function, and the second one is using combinatoric objects (called “symbols”) created by Lusztig for studying representation theory of finite classical groups.

Key words and phrases:
bipartition number, partition number, generating function, Lusztig’s symbol
2020 Mathematics Subject Classification:
Primary: 05A17, 11P87; Secondary: 20C33

1. Introduction

For a positive integer nn, let p​(n)p(n) (resp. p2​(n)p_{2}(n)) denote the number of partitions (resp. bipartitions) of nn. By convention, we define p​(0)=p2​(0)=1p(0)=p_{2}(0)=1, and p​(x)=p2​(x)=0p(x)=p_{2}(x)=0 if xx is not a non-negative integer. For n≥1n\geq 1, Euler obtained the following recursive relation for p​(n)p(n):

p​(n)=∑k∈ℤ∖{0}(−1)k−1​p​(n−12​k​(3​k−1))p(n)=\sum_{k\in{\mathbb{Z}}\smallsetminus\{0\}}(-1)^{k-1}p\bigl{(}n-\tfrac{1}{2}k(3k-1)\bigr{)}

(cf. [And76] corollary 1.8). Our result in this article is the following recursive relation for p2​(n)p_{2}(n):

Theorem 1.1.

For n≥0n\geq 0, we have

p2​(n)=p​(n2)+∑k∈ℤ∖{0}(−1)k−1​p2​(n−k2).p_{2}(n)=p\bigl{(}\tfrac{n}{2}\bigr{)}+\sum_{k\in{\mathbb{Z}}\smallsetminus\{0\}}(-1)^{k-1}p_{2}\bigl{(}n-k^{2}\bigr{)}.

We will provide two proofs for the above recursive formula from different point of views. In Section 2, the theorem is first proved by manipulating the respective generating functions. In Section 3, we will derive the recursive relation from Lusztig’s theory of “symbols” and the binomial theorem.

It is well known that the irreducible characters of the Coxeter group of type BnB_{n} is parametrized by the set of bipartitions of nn (cf. [GP00] §5.5), just like the set of irreducible characters of the Coxeter group of type AnA_{n} (i.e., the symmetric group SnS_{n}) is parametrized by the set of partitions of nn. It is not surprising that Theorem 1.1 has applications to the representation theory of finite classical groups. Two consequences are given in Remark 3.9 and Remark 3.10.

In the appendix, we provide a new proof of a congruence relation for the bipartition number p2​(n)p_{2}(n) as a reference.

2. Generating Functions

2.1. Partitions and bipartitions

For a nonnegative integer nn, let 𝒫​(n){\mathcal{P}}(n) denote the set of partitions of nn. By convention, we define 𝒫​(0){\mathcal{P}}(0) to be the set of “the empty partition”, and 𝒫​(x){\mathcal{P}}(x) to be the empty set if xx is not a nonnegative integer. Let p​(n)=|𝒫​(n)|p(n)=|{\mathcal{P}}(n)| where |X||X| denotes the number of elements of a finite set XX.

Let 𝒫2​(n){\mathcal{P}}_{2}(n) denote the set of bi-partitions of nn, i.e.,

𝒫2​(n)={[λμ]∣|λ|+|μ|=n}\textstyle{\mathcal{P}}_{2}(n)=\left\{\,\genfrac{[}{]}{0.0pt}{}{\lambda}{\mu}\mid|\lambda|+|\mu|=n\,\right\}

where |λ||\lambda| means the sum of parts of λ\lambda. Let p2​(n)=|𝒫2​(n)|p_{2}(n)=|{\mathcal{P}}_{2}(n)|. We know that p2​(0)=1p_{2}(0)=1, and p2​(x)=0p_{2}(x)=0 if xx is not a nonnegative integer. For a bipartition Σ=[λμ]\Sigma=\genfrac{[}{]}{0.0pt}{}{\lambda}{\mu}, its transpose is given by Σt=[μλ]\Sigma^{\rm t}=\genfrac{[}{]}{0.0pt}{}{\mu}{\lambda}. A bipartition Σ\Sigma is called degenerate if Σt=Σ\Sigma^{\rm t}=\Sigma, it is called non-degenerate, otherwise. If Σ∈𝒫2​(n)\Sigma\in{\mathcal{P}}_{2}(n) is degenerate, then Σ=[λλ]\Sigma=\genfrac{[}{]}{0.0pt}{}{\lambda}{\lambda} for some λ∈𝒫​(n2)\lambda\in{\mathcal{P}}(\tfrac{n}{2}). Thus, the number of degenerate bipartitions in 𝒫2​(n){\mathcal{P}}_{2}(n) is equal to p​(n2)p(\tfrac{n}{2}).

2.2. First proof of Theorem 1.1

Lemma 2.1.

We have

∑n∈ℤ(−1)n​zn2=∏k=1∞(1−zk)21−z2​k.\sum_{n\in{\mathbb{Z}}}(-1)^{n}z^{n^{2}}=\prod_{k=1}^{\infty}\frac{(1-z^{k})^{2}}{1-z^{2k}}.
Proof.

We start with the Jacobi’s triple product identity (cf. [And76] theorem 2.8)

(2.2) ∑n∈ℤqn​(n+1)2​zn=∏k=1∞(1+z​qk)​(1+z−1​qk−1)​(1−qk).\sum_{n\in{\mathbb{Z}}}q^{\frac{n(n+1)}{2}}z^{n}=\prod_{k=1}^{\infty}(1+zq^{k})(1+z^{-1}q^{k-1})(1-q^{k}).

In both sides of (2.2), we replace zz by z−1z^{-1}, and replace qq by z2z^{2} to obtain

∑n∈ℤzn2=∏k=1∞(1+z2​k−1)2​(1−z2​k).\sum_{n\in{\mathbb{Z}}}z^{n^{2}}=\prod_{k=1}^{\infty}(1+z^{2k-1})^{2}(1-z^{2k}).

Then we replace zz by −z-z to get

∑n∈ℤ(−1)n​zn2=∏k=1∞(1−z2​k−1)2​(1−z2​k)=∏k=1∞(1−z2​k−1)​(1−zk).\sum_{n\in{\mathbb{Z}}}(-1)^{n}z^{n^{2}}=\prod_{k=1}^{\infty}(1-z^{2k-1})^{2}(1-z^{2k})=\prod_{k=1}^{\infty}(1-z^{2k-1})(1-z^{k}).

Now for the right-hand side, we need the Euler’s identity: pd​(n)=po​(n)p_{\rm d}(n)=p_{\rm o}(n) where pd​(n)p_{\rm d}(n) is the number of partitions into distinct parts of nn, and po​(n)p_{\rm o}(n) is the number of partitions into odd parts. In terms of generating functions, we have

∏k=1∞(1+zk)=∑n=0∞pd​(n)​zn=∑n=0∞po​(n)​zn=∏k=1∞11−z2​k−1.\prod_{k=1}^{\infty}(1+z^{k})=\sum_{n=0}^{\infty}p_{\rm d}(n)z^{n}=\sum_{n=0}^{\infty}p_{\rm o}(n)z^{n}=\prod_{k=1}^{\infty}\frac{1}{1-z^{2k-1}}.

Therefore,

∏k=1∞(1−z2​k−1)​(1−zk)=∏k=1∞1−zk1+zk\displaystyle\prod_{k=1}^{\infty}(1-z^{2k-1})(1-z^{k})=\prod_{k=1}^{\infty}\frac{1-z^{k}}{1+z^{k}} =∏k=1∞(1−zk)2(1+zk)​(1−zk)\displaystyle=\prod_{k=1}^{\infty}\frac{(1-z^{k})^{2}}{(1+z^{k})(1-z^{k})}
=∏k=1∞(1−zk)21−z2​k,\displaystyle=\prod_{k=1}^{\infty}\frac{(1-z^{k})^{2}}{1-z^{2k}},

and the lemma is proved. ∎

First Proof of Theorem 1.1.

Recall that

∑n=0∞p​(n)​zn=∏k=1∞11−zkand∑n=0∞p2​(n)​zn=∏k=1∞1(1−zk)2.\sum_{n=0}^{\infty}p(n)z^{n}=\prod_{k=1}^{\infty}\frac{1}{1-z^{k}}\qquad\text{and}\qquad\sum_{n=0}^{\infty}p_{2}(n)z^{n}=\prod_{k=1}^{\infty}\frac{1}{(1-z^{k})^{2}}.

By Lemma 2.1, we have

(∑n=0∞p2​(n)​zn)​(∑k∈ℤ(−1)k​zk2)=∑n=0∞p​(n)​z2​n,\left(\sum_{n=0}^{\infty}p_{2}(n)z^{n}\right)\left(\sum_{k\in{\mathbb{Z}}}(-1)^{k}z^{k^{2}}\right)=\sum_{n=0}^{\infty}p(n)z^{2n},

which implies that

∑k∈ℤ(−1)k​p2​(n−k2)=p​(n2),\sum_{k\in{\mathbb{Z}}}(-1)^{k}p_{2}(n-k^{2})=p\bigl{(}\tfrac{n}{2}\bigr{)},

i.e., the theorem is proved. ∎

3. Lusztig’s Symbols

3.1. Definition of the symbols

In this subsection and the next subsection, we recall the notions of “symbols” by Lusztig. The materials are modified from [Lus77] §2, §3 (see also [Pan24] §2).

A symbol is an ordered pair Λ=(AB)=(a1,a2,…,am1b1,b2,…,bm2)\Lambda=\binom{A}{B}=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}} of two finite subsets A,BA,B of nonnegative integers. The elements in each of AA and BB are written in decreasing order, i.e., a1>a2>⋯>am1a_{1}>a_{2}>\cdots>a_{m_{1}} and b1>b2>⋯>bm2b_{1}>b_{2}>\cdots>b_{m_{2}}. The sets A,BA,B in Λ\Lambda are also denoted by Λ∗,Λ∗\Lambda^{*},\Lambda_{*}, respectively, and called the first row and the second row of Λ\Lambda. An element in Λ∗\Lambda^{*} or Λ∗\Lambda_{*} is called an entry of Λ\Lambda. A symbol Λ′\Lambda^{\prime} is called a subsymbol of Λ\Lambda and denoted by Λ′⊂Λ\Lambda^{\prime}\subset\Lambda if Λ′⁣∗⊂Λ∗\Lambda^{\prime*}\subset\Lambda^{*} and Λ∗′⊂Λ∗\Lambda^{\prime}_{*}\subset\Lambda_{*}. For a subsymbol Λ′⊂Λ\Lambda^{\prime}\subset\Lambda, we define the substraction Λ∖Λ′\Lambda\smallsetminus\Lambda^{\prime} to be (Λ∗∖Λ′⁣∗Λ∗∖Λ∗′)\binom{\Lambda^{*}\smallsetminus\Lambda^{\prime*}}{\Lambda_{*}\smallsetminus\Lambda^{\prime}_{*}}. For two symbols Λ,Λ′\Lambda,\Lambda^{\prime}, their union is defined by Λ∪Λ′=(Λ∗∪Λ′⁣∗Λ∗∪Λ∗′)\Lambda\cup\Lambda^{\prime}=\binom{\Lambda^{*}\cup\Lambda^{\prime*}}{\Lambda_{*}\cup\Lambda^{\prime}_{*}}.

For a symbol Λ=(AB)=(a1,a2,…,am1b1,b2,…,bm2)\Lambda=\binom{A}{B}=\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}, its rank rk​(Λ){\rm rk}(\Lambda), defect def​(Λ){\rm def}(\Lambda) and transpose Λt\Lambda^{\rm t} are defined by

rk​(Λ)\displaystyle{\rm rk}(\Lambda) =∑i=1m1ai+∑j=1m2bj−⌊(|A|+|B|−12)2⌋,\displaystyle=\sum_{i=1}^{m_{1}}a_{i}+\sum_{j=1}^{m_{2}}b_{j}-\biggl{\lfloor}\biggl{(}\frac{|A|+|B|-1}{2}\biggr{)}^{2}\biggr{\rfloor},
def​(Λ)\displaystyle{\rm def}(\Lambda) =|A|−|B|,\displaystyle=|A|-|B|,
Λt\displaystyle\Lambda^{\rm t} =(BA)\displaystyle=\binom{B}{A}

where ⌊x⌋\lfloor x\rfloor denotes the floor function of a real number xx. It is not difficult to check that rk​(Λ)≥⌊(def​(Λ)2)2⌋{\rm rk}(\Lambda)\geq\bigl{\lfloor}\bigl{(}\frac{{\rm def}(\Lambda)}{2}\bigr{)}^{2}\bigr{\rfloor} for any symbol Λ\Lambda. A symbol Λ\Lambda is called degenerate if Λt=Λ\Lambda^{\rm t}=\Lambda; it is called non-degenerate, otherwise.

On the set of symbols , we define an equivalence relation generated by

(a1,a2,…,am1b1,b2,…,bm2)∼(a1+1,a2+1,…,am1+1,0b1+1,b2+1,…,bm2+1,0).\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\sim\binom{a_{1}+1,a_{2}+1,\ldots,a_{m_{1}}+1,0}{b_{1}+1,b_{2}+1,\ldots,b_{m_{2}}+1,0}.

It is easy to see that the ranks and the defects are invariant in a similarity class. Let Φn,d\Phi_{n,d} denote the set of similarity classes of symbols of rank nn and defect dd. Note that Φn,d=∅\Phi_{n,d}=\emptyset if ⌊(d2)2⌋>n\bigl{\lfloor}(\frac{d}{2})^{2}\bigr{\rfloor}>n.

We define a mapping

(a1,a2,…,am1b1,b2,…,bm2)↦[a1−(m1−1),a2−(m1−2),…,am1−1−1,am1b1−(m2−1),b2−(m2−1),…,bm2−1−1,bm2]\binom{a_{1},a_{2},\ldots,a_{m_{1}}}{b_{1},b_{2},\ldots,b_{m_{2}}}\mapsto\genfrac{[}{]}{0.0pt}{}{a_{1}-(m_{1}-1),a_{2}-(m_{1}-2),\ldots,a_{m_{1}-1}-1,a_{m_{1}}}{b_{1}-(m_{2}-1),b_{2}-(m_{2}-1),\ldots,b_{m_{2}-1}-1,b_{m_{2}}}

from symbols to bipartitions. It is easy to see that two symbols in the same similarity class has the same image. So the above mapping will be regarded as a mapping from the set of similarity classes of symbols to the set of bipartitions. Moreover, it is not difficult to check that the above mapping gives a bijection

(3.1) Υ:Φn,d⟶𝒫2​(n−⌊(d2)2⌋).\Upsilon\colon\Phi_{n,d}\longrightarrow{\mathcal{P}}_{2}\bigl{(}n-\bigl{\lfloor}\bigl{(}\tfrac{d}{2}\bigr{)}^{2}\bigr{\rfloor}\bigr{)}.

In particular, |Φn,d|=p2​(n−(d2)2)|\Phi_{n,d}|=p_{2}\bigl{(}n-(\tfrac{d}{2})^{2}\bigr{)} if dd is even. Note that if a symbol Λ\Lambda of rank nn is degenerate, then def​(Λ)=0{\rm def}(\Lambda)=0 and Υ​(Λ)\Upsilon(\Lambda) is a degenerate bipartition in 𝒫2​(n){\mathcal{P}}_{2}(n). Finally, we define

(3.2) Φn+=⋃d≡0(mod4)Φn,d,Φn−=⋃d≡2(mod4)Φn,d,Φn=Φn+∪Φn−=⋃d​ evenΦn,d.\Phi^{+}_{n}=\bigcup_{d\equiv 0\pmod{4}}\Phi_{n,d},\qquad\Phi^{-}_{n}=\bigcup_{d\equiv 2\pmod{4}}\Phi_{n,d},\qquad\Phi_{n}=\Phi^{+}_{n}\cup\Phi^{-}_{n}=\bigcup_{d\text{ even}}\Phi_{n,d}.
Example 3.3.

For n=4n=4 and dd even, we have the following

Φ4,0\displaystyle\Phi_{4,0} ={Λ,Λt∣Λ=(40),(31),(4,11,0),(3,21,0),(3,12,0),(3,02,1),(4,2,12,1,0),(3,2,13,1,0),(4,3,2,13,2,1,0)}∪{(22),(2,12,1)},\displaystyle=\textstyle\bigl{\{}\,\Lambda,\Lambda^{\rm t}\mid\Lambda=\binom{4}{0},\binom{3}{1},\binom{4,1}{1,0},\binom{3,2}{1,0},\binom{3,1}{2,0},\binom{3,0}{2,1},\binom{4,2,1}{2,1,0},\binom{3,2,1}{3,1,0},\binom{4,3,2,1}{3,2,1,0}\bigr{\}}\cup\bigl{\{}\binom{2}{2},\binom{2,1}{2,1}\,\bigr{\}},
Φ4,2\displaystyle\Phi_{4,2} ={(4,0−),(3,1−),(3,2,10),(4,1,01),(3,2,01),(3,1,02),(2,1,03),(4,2,1,02,1),(3,2,1,03,1),(4,3,2,1,03,2,1)},\displaystyle=\textstyle\bigl{\{}\binom{4,0}{-},\binom{3,1}{-},\binom{3,2,1}{0},\binom{4,1,0}{1},\binom{3,2,0}{1},\binom{3,1,0}{2},\binom{2,1,0}{3},\binom{4,2,1,0}{2,1},\binom{3,2,1,0}{3,1},\binom{4,3,2,1,0}{3,2,1}\bigr{\}},
Φ4,4\displaystyle\Phi_{4,4} ={(3,2,1,0−)},\displaystyle=\textstyle\bigl{\{}\binom{3,2,1,0}{-}\bigr{\}},
Φ4,−d\displaystyle\Phi_{4,-d} ={Λt∣Λ∈Φ4,d},\displaystyle=\textstyle\{\,\Lambda^{\rm t}\mid\Lambda\in\Phi_{4,d}\,\},
Φ4+\displaystyle\Phi^{+}_{4} =Φ4,0∪Φ4,4∪Φ4,−4,\displaystyle=\Phi_{4,0}\cup\Phi_{4,4}\cup\Phi_{4,-4},
Φ4−\displaystyle\Phi^{-}_{4} =Φ4,2∪Φ4,−2.\displaystyle=\Phi_{4,2}\cup\Phi_{4,-2}.

Note that |Φ4,0|=20=p2​(4)|\Phi_{4,0}|=20=p_{2}(4), |Φ4,2|=|Φ4,−2|=10=p2​(3)|\Phi_{4,2}|=|\Phi_{4,-2}|=10=p_{2}(3), and |Φ4,4|=|Φ4,−4|=1=p2​(0)|\Phi_{4,4}|=|\Phi_{4,-4}|=1=p_{2}(0).

3.2. Families associated to special symbols

A symbol Z=(a1,a2,…,amb1,b2,…,bm)∈Φn,0Z=\binom{a_{1},a_{2},\ldots,a_{m}}{b_{1},b_{2},\ldots,b_{m}}\in\Phi_{n,0} is called special if a1≥b1≥a2≥b2≥⋯≥am≥bma_{1}\geq b_{1}\geq a_{2}\geq b_{2}\geq\cdots\geq a_{m}\geq b_{m}. For a special symbol ZZ, let ZI=Z∖(Z∗∩Z∗Z∗∩Z∗)Z_{\rm I}=Z\smallsetminus\binom{Z^{*}\cap Z_{*}}{Z^{*}\cap Z_{*}} be the subsymbol of “singles” in ZZ. Clearly, def​(ZI)=0{\rm def}(Z_{\rm I})=0. The degree deg⁡(Z)\deg(Z) is defined to be |(ZI)∗|=|(ZI)∗||(Z_{\rm I})^{*}|=|(Z_{\rm I})_{*}|. Note that a degenerate symbol is always special. Moreover, a special symbol ZZ with deg⁡(Z)=0\deg(Z)=0 means that ZZ is degenerate. For a subsymbol M⊂ZIM\subset Z_{\rm I}, we define

ΛM=(Z∖M)∪Mt,\Lambda_{M}=(Z\smallsetminus M)\cup M^{\rm t},

i.e., ΛM\Lambda_{M} is obtained from ZZ by switching the row position of each entry of MM. It is clear that def​(ΛM)=−2​def​(M){\rm def}(\Lambda_{M})=-2\,{\rm def}(M). For a special symbol Z∈Φn,0Z\in\Phi_{n,0}, we define

ΦZ={ΛM∣M⊂ZI},\Phi_{Z}=\{\,\Lambda_{M}\mid M\subset Z_{\rm I}\,\},

i.e., ΦZ\Phi_{Z} is the set of symbols of exactly the same entries of ZZ. It is clear that |ΦZ|=22​deg⁡(Z)|\Phi_{Z}|=2^{2\deg(Z)}. Moreover, we have

(3.4) ΦZ∩Φn,−2​d={ΛM∣M⊂ZI,def​(M)=d},Φn=⋃Z∈Φn,0, specialΦZ.\displaystyle\begin{split}\Phi_{Z}\cap\Phi_{n,-2d}&=\{\,\Lambda_{M}\mid M\subset Z_{\rm I},\ {\rm def}(M)=d\,\},\\ \Phi_{n}&=\bigcup_{Z\in\Phi_{n,0},\ \text{ special}}\Phi_{Z}.\end{split}
Example 3.5.

We have the following special symbols ZZ in Φ4,0\Phi_{4,0}:

ZZ (3,12,0)\binom{3,1}{2,0} (40)\binom{4}{0} (31)\binom{3}{1} (4,11,0)\binom{4,1}{1,0} (4,2,12,1,0)\binom{4,2,1}{2,1,0} (3,2,13,1,0)\binom{3,2,1}{3,1,0} (4,3,2,13,2,1,0)\binom{4,3,2,1}{3,2,1,0} (22)\binom{2}{2} (2,12,1)\binom{2,1}{2,1}
ZIZ_{\rm I} (3,12,0)\binom{3,1}{2,0} (40)\binom{4}{0} (31)\binom{3}{1} (40)\binom{4}{0} (40)\binom{4}{0} (20)\binom{2}{0} (40)\binom{4}{0} (−−)\binom{-}{-} (−−)\binom{-}{-}
|ΦZ||\Phi_{Z}| 1616 44 44 44 44 44 44 11 11
Example 3.6.

Let Z=(3,12,0)∈Φ4,0Z=\binom{3,1}{2,0}\in\Phi_{4,0}. Then deg⁡(Z)=2\deg(Z)=2 and we have the following 1616 symbols ΛM\Lambda_{M} in ΦZ\Phi_{Z}:

MM (−−)\binom{-}{-} (3−)\binom{3}{-} (1−)\binom{1}{-} (−2)\binom{-}{2} (−0)\binom{-}{0} (3,1−)\binom{3,1}{-} (−2,0)\binom{-}{2,0} (32)\binom{3}{2}
ΛM\Lambda_{M} (3,12,0)\binom{3,1}{2,0} (13,2,0)\binom{1}{3,2,0} (32,1,0)\binom{3}{2,1,0} (3,2,10)\binom{3,2,1}{0} (3,1,02)\binom{3,1,0}{2} (−3,2,1,0)\binom{-}{3,2,1,0} (3,2,1,0−)\binom{3,2,1,0}{-} (2,13,0)\binom{2,1}{3,0}
MM (10)\binom{1}{0} (30)\binom{3}{0} (12)\binom{1}{2} (3,12)\binom{3,1}{2} (3,10)\binom{3,1}{0} (32,0)\binom{3}{2,0} (12,0)\binom{1}{2,0} (3,12,0)\binom{3,1}{2,0}
ΛM\Lambda_{M} (3,02,1)\binom{3,0}{2,1} (1,03,2)\binom{1,0}{3,2} (3,21,0)\binom{3,2}{1,0} (23,1,0)\binom{2}{3,1,0} (03,2,1)\binom{0}{3,2,1} (2,1,03)\binom{2,1,0}{3} (3,2,01)\binom{3,2,0}{1} (2,03,1)\binom{2,0}{3,1}

3.3. Second proof of Theorem 1.1

Lemma 3.7.

Let Z∈Φn,0Z\in\Phi_{n,0} be a special symbol, and let δ=deg⁡(Z)\delta=\deg(Z). Then

|ΦZ∩(⋃d≡0(mod4)Φn,d)|−|ΦZ∩(⋃d≡2(mod4)Φn,d)|={1,if δ=0;0,if δ≥1.\left|\Phi_{Z}\cap\left(\bigcup_{d\equiv 0\pmod{4}}\Phi_{n,d}\right)\right|-\left|\Phi_{Z}\cap\left(\bigcup_{d\equiv 2\pmod{4}}\Phi_{n,d}\right)\right|=\begin{cases}1,&\text{if $\delta=0$};\\ 0,&\text{if $\delta\geq 1$}.\end{cases}
Proof.

Note that |(ZI)∗|+|(ZI)∗|=2​δ|(Z_{\rm I})^{*}|+|(Z_{\rm I})_{*}|=2\delta, and def​(M)≡|M|(mod2){\rm def}(M)\equiv|M|\pmod{2}. Then from (3.4), we have

|ΦZ∩(⋃d≡0(mod4)Φn,d)|\displaystyle\left|\Phi_{Z}\cap\left(\bigcup_{d\equiv 0\pmod{4}}\Phi_{n,d}\right)\right| =|{M⊂ZI∣|M∗|+|M∗|​ even}|=∑k≥0, evenCk2​δ,\displaystyle=|\{\,M\subset Z_{\rm I}\mid|M^{*}|+|M_{*}|\text{ even}\,\}|=\sum_{k\geq 0,\text{ even}}C^{2\delta}_{k},
|ΦZ∩(⋃d≡2(mod4)Φn,d)|\displaystyle\left|\Phi_{Z}\cap\left(\bigcup_{d\equiv 2\pmod{4}}\Phi_{n,d}\right)\right| =|{M⊂ZI∣|M∗|+|M∗|​ odd}|=∑k≥1, oddCk2​δ\displaystyle=|\{\,M\subset Z_{\rm I}\mid|M^{*}|+|M_{*}|\text{ odd}\,\}|=\sum_{k\geq 1,\text{ odd}}C^{2\delta}_{k}

where Cml=l!m!​(l−m)!C^{l}_{m}=\frac{l!}{m!(l-m)!} is a binomial coefficient. Note that by convention Cml=0C^{l}_{m}=0 if l,m∈ℕ∪{0}l,m\in{\mathbb{N}}\cup\{0\} and l<ml<m. It is well known from the binomial theorem that

∑k≥0, evenCk2​δ−∑k≥1, oddCk2​δ={1,if δ=0;0,if δ≥1.\sum_{k\geq 0,\text{ even}}C^{2\delta}_{k}-\sum_{k\geq 1,\text{ odd}}C^{2\delta}_{k}=\begin{cases}1,&\text{if $\delta=0$};\\ 0,&\text{if $\delta\geq 1$}.\end{cases}

Then the lemma is proved. ∎

Corollary 3.8.

Let nn be a nonnegative integer. Then |Φn+|−|Φn−|=p​(n2)|\Phi_{n}^{+}|-|\Phi^{-}_{n}|=p(\tfrac{n}{2}).

Proof.

From (3.2) and (3.4), we have

Φn+=⋃Z∈Φn,0, special(ΦZ∩⋃d≡0(mod4)Φn,d),\displaystyle\Phi^{+}_{n}=\bigcup_{Z\in\Phi_{n,0},\text{ special}}\left(\Phi_{Z}\cap\bigcup_{d\equiv 0\pmod{4}}\Phi_{n,d}\right),
Φn−=⋃Z∈Φn,0, special(ΦZ∩⋃d≡2(mod4)Φn,d).\displaystyle\Phi^{-}_{n}=\bigcup_{Z\in\Phi_{n,0},\text{ special}}\left(\Phi_{Z}\cap\bigcup_{d\equiv 2\pmod{4}}\Phi_{n,d}\right).

Then by Lemma 3.7, we have |Φn+|−|Φn−|=∑Z∈Φn,0,deg⁡(Z)=01=p​(n2)|\Phi_{n}^{+}|-|\Phi^{-}_{n}|=\sum_{Z\in\Phi_{n,0},\ \deg(Z)=0}1=p(\tfrac{n}{2}). ∎

Second Proof of Theorem 1.1.

From (3.1), we know that |Φn,d|=p2​(n−(d2)2)|\Phi_{n,d}|=p_{2}\bigl{(}n-(\tfrac{d}{2})^{2}\bigr{)}, and from (3.2), we have

|Φn+|=∑k∈ℤ, evenp2​(n−k2),|Φn−|=∑k∈ℤ, oddp2​(n−k2).|\Phi^{+}_{n}|=\sum_{k\in{\mathbb{Z}},\text{ even}}p_{2}\bigl{(}n-k^{2}\bigr{)},\qquad|\Phi^{-}_{n}|=\sum_{k\in{\mathbb{Z}},\text{ odd}}p_{2}\bigl{(}n-k^{2}\bigr{)}.

Then the theorem follows from Corollary 3.8 immediately. ∎

Remark 3.9.

Let 𝐅q{\mathbf{F}}_{q} denote the finite field of qq elements where qq is a power of an odd prime. From Lusztig’s theory, for ϵ∈{+,−}\epsilon\in\{+,-\}, the set ℰ​(O2​nϵ​(𝐅q),1){\mathcal{E}}({\rm O}^{\epsilon}_{2n}({\mathbf{F}}_{q}),1) of unipotent characters of the finite even orthogonal group O2​nϵ​(𝐅q){\rm O}^{\epsilon}_{2n}({\mathbf{F}}_{q}) is parametrized by the set Φnϵ\Phi^{\epsilon}_{n} (cf. [Lus77] theorem 8.2, see also [Pan24] §3.3). Hence Corollary 3.8 means that O2​n+​(𝐅q){\rm O}^{+}_{2n}({\mathbf{F}}_{q}) has p​(n2)p(\frac{n}{2}) more unipotent characters than O2​n−​(𝐅q){\rm O}^{-}_{2n}({\mathbf{F}}_{q}).

Remark 3.10.

Another consequence of Theorem 1.1 is that the number of quadratic unipotent conjugacy classes of O2​nϵ​(𝐅q){\rm O}^{\epsilon}_{2n}({\mathbf{F}}_{q}) is equal to the number of quadratic unipotent characters of O2​nϵ​(𝐅q){\rm O}^{\epsilon}_{2n}({\mathbf{F}}_{q}) for ϵ∈{+,−}\epsilon\in\{+,-\}. The detail will appear in a paper in preparation by the second author.

Appendix A A Congruence Relation for p2​(n)p_{2}(n)

The following theorem on a congruence relation for p2​(n)p_{2}(n) has been proved in [HL04] and [CJW06]. Below we provide another proof along the approach taken by Michael Hirschhorn [Hir00].

Theorem A.1.

We have

p2​(5​n+2)≡p2​(5​n+3)≡p2​(5​n+4)≡0(mod5)p_{2}(5n+2)\equiv p_{2}(5n+3)\equiv p_{2}(5n+4)\equiv 0\pmod{5}

for any nonnegative integer nn.

Before the proof, we first set up some notations. Let

(a;q)=∏k=0∞(1−a​qk),(a;q)=\prod_{k=0}^{\infty}(1-aq^{k}),

and define

R​(q)=(q2;q5)​(q3;q5)(q;q5)​(q4;q5)andc=R​(q5).R(q)=\frac{(q^{2};q^{5})(q^{3};q^{5})}{(q;q^{5})(q^{4};q^{5})}\qquad\text{and}\qquad c=R(q^{5}).

Then Hirschhorn obtained (see also [Aig07] p.140)

(A.2) ∑n=0∞p​(n)​qn=(q25;q25)5(q5;q5)6​f​(q),\sum_{n=0}^{\infty}p(n)q^{n}=\frac{(q^{25};q^{25})^{5}}{(q^{5};q^{5})^{6}}f(q),

where

f​(q)=c4+c3​q+2​c2​q2+3​c​q3+5​q4−3​c−1​q5+2​c−2​q6−c−3​q7+c−4​q8.f(q)=c^{4}+c^{3}q+2c^{2}q^{2}+3cq^{3}+5q^{4}-3c^{-1}q^{5}+2c^{-2}q^{6}-c^{-3}q^{7}+c^{-4}q^{8}.

From (A.2) and the fact that the expansion of cc only involves powers of q5q^{5}, Hirschhorn obtained

∑n=0∞p​(5​n+4)​q5​n+4=5​(q25;q25)5(q5;q5)6​q4\sum_{n=0}^{\infty}p(5n+4)q^{5n+4}=5\frac{(q^{25};q^{25})^{5}}{(q^{5};q^{5})^{6}}q^{4}

and concluded the famous congruence by Ramanujan: p​(5​n+4)≡0(mod5)p(5n+4)\equiv 0\pmod{5}.

Proof of Theorem A.1.

We square both sides of (A.2) and reach

(A.3) ∑n=0∞p2​(n)​qn=(∑n⩾0p​(n)​qn)2=(q25;q25)10(q5;q5)12​f​(q)2,\sum_{n=0}^{\infty}p_{2}(n)q^{n}=\left(\sum_{n\geqslant 0}p(n)q^{n}\right)^{2}=\frac{(q^{25};q^{25})^{10}}{(q^{5};q^{5})^{12}}f(q)^{2},

and

f​(q)2=c8+2​c7​q+5​c6​q2+10​c5​q3+20​c4​q4+16​c3​q5+27​c2​q6+20​c​q7+15​q8−20​c−1​q9+27​c−2​q10−16​c−3​q11+20​c−4​q12−10​c−5​q13+5​c−6​q14−2​c−7​q15+c−8​q16.f(q)^{2}=c^{8}+2c^{7}q+5c^{6}q^{2}+10c^{5}q^{3}+20c^{4}q^{4}+16c^{3}q^{5}\\ +27c^{2}q^{6}+20cq^{7}+15q^{8}-20c^{-1}q^{9}+27c^{-2}q^{10}-16c^{-3}q^{11}\\ +20c^{-4}q^{12}-10c^{-5}q^{13}+5c^{-6}q^{14}-2c^{-7}q^{15}+c^{-8}q^{16}.

Using the fact again that the expansion of cc only involves powers of q5q^{5}, we obtain

∑n=0∞p2​(5​n+2)​q5​n+2\displaystyle\sum_{n=0}^{\infty}p_{2}(5n+2)q^{5n+2} =5​(q25;q25)10(q5;q5)12​(c6​q2+4​c​q7+4​c−4​q12),\displaystyle=5\frac{(q^{25};q^{25})^{10}}{(q^{5};q^{5})^{12}}\left(c^{6}q^{2}+4cq^{7}+4c^{-4}q^{12}\right),
∑n=0∞p2​(5​n+3)​q5​n+3\displaystyle\sum_{n=0}^{\infty}p_{2}(5n+3)q^{5n+3} =5​(q25;q25)10(q5;q5)12​(2​c5​q3+3​q8−2​c−5​q13),\displaystyle=5\frac{(q^{25};q^{25})^{10}}{(q^{5};q^{5})^{12}}\left(2c^{5}q^{3}+3q^{8}-2c^{-5}q^{13}\right),
∑n=0∞p2​(5​n+4)​q5​n+4\displaystyle\sum_{n=0}^{\infty}p_{2}(5n+4)q^{5n+4} =5​(q25;q25)10(q5;q5)12​(4​c4​q4−4​c−1​q9+c−6​q14).\displaystyle=5\frac{(q^{25};q^{25})^{10}}{(q^{5};q^{5})^{12}}\left(4c^{4}q^{4}-4c^{-1}q^{9}+c^{-6}q^{14}\right).

The theorem now follows from these identities immediately. ∎

References

  • [Aig07] Martin Aigner, A Course in Enumeration, Grad. Texts in Math., vol. 238, Springer, Berlin, 2007.
  • [And76] George E. Andrews, The Theory of Partitions, Encyclopedia Math. Appl., vol. 2, Addison-Wesley Publishing Co., Massachusetts, 1976.
  • [CJW06] William Y. C. Chen, Kathy Q. Ji, and Herbert S. Wilf, BG-ranks and 22-cores, Electron. J. Combin. 13 (2006), Note 18, 5 pp.
  • [GP00] Meinolf Geck and Götz Pfeiffer, Characters of Finite Coxeter Groups and Iwahori-Hecke algebras, London Math. Soc. Monogr. (N.S.), no. 21, The Clarendon Press, Oxford University Press, New York, 2000.
  • [Hir00] Michael D. Hirschhorn, An identity of Ramanujan, and applications, qq-series from a Contemporary Perspective, Contemp. Math., no. 254, American Mathematical Society, Providence, RI, 2000, pp. 229–234.
  • [HL04] Paul Hammond and Richard Lewis, Congruences in ordered pairs of partitions, Int. J. Math. Math. Sci. (2004), no. 45–48, 2509–2512.
  • [Lus77] George Lusztig, Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125–175.
  • [Pan24] S.-Y. Pan, Howe correspondence of unipotent characters for a finite symplectic/even-orthogonal dual pair, Amer. J. Math. 146 (2024), no. 3, 813–869.