Polynomiality of some hook-content summations for doubled distinct and self-conjugate partitions

0.28 MB·

by  Guo-Niu Han

POLYNOMIALITY OF SOME HOOK-CONTENT SUMMATIONS
FOR DOUBLED DISTINCT AND SELF-CONJUGATE
6 PARTITIONS
1
0
2 GUO-NIUHANANDHUANXIONG

n
a Abstract. In 2009, the first author proved the Nekrasov-Okounkov formula
J onhooklengthsforintegerpartitionsbyusinganidentityofMacdonaldinthe
e
7 frameworkof type A affine rootsystems, and conjectured that somesumma-
1 tions overthesetofallpartitionsofsizenarealwayspolynomialsinn. This
conjecturewasgeneralizedandprovedbyStanley. Recently,P´etr´eollederived
e e
] two Nekrasov-Okounkov type formulas for C and Cˇwhich involve doubled
O
distinct and self-conjugate partitions. Inspired by all those previous works,
C weestablishthepolynomialityofsomehook-content summationsfordoubled
distinctandself-conjugatepartitions.
.
h
t
a
m
1. Introduction
[

1 The following so-called Nekrasov-Okounkovformula
v z
q|λ| 1− = (1−qk)z−1,
9 h2
6 λX∈P h∈YH(λ)(cid:0) (cid:1) kY≥1
3
where P is the set of all integer partitions λ with |λ| denoting the size of λ and
4
H(λ) the multiset of hook lengths associated with λ (see [6]), was discovered in-
0
. dependently several times: First, by Nekrasov and Okounkov in their study of the
1
theory of Seiberg-Witten on supersymmetric gauges in particle physics [16]; Then,
0
6 proved by Westbury using D’Arcais polynomials [28]; Finally, by the first author
1 using an identity of Macdonald [15] in the framework of type A affine root sys-
: tems [6]. Moreover,he asked to find Nekrasov-Okounkovtype formulas associated
v
i with other root systems [7, Problem 6.4], and conjectured that e
X
1
r n! h2k
a H(λ)2
|λX|=n h∈XH(λ)
is always a polynomial in n for any k ∈ N, where H(λ) = h. This con-
h∈H(λ)
jecture was proved by Stanley in a more general form. In particular, he showed
Q
that
1
n! F (h2 :h∈H(λ))F (c:c∈C(λ))
H(λ)2 1 2
|λX|=n
is a polynomial in n for any symmetric functions F and F , where C(λ) is the
1 2
multiset of contents associated with λ (see [24]). For some special functions F
1
Date:December25,2015.
2010 Mathematics Subject Classification. 05A15,05A17,05A19,05E05,05E10, 11P81.
Key words and phrases. strict partition, doubled distinct partition, self-conjugate partition,
hooklength,content, shiftedYoungtableau, differenceoperator.
1
2 GUO-NIUHANANDHUANXIONG
Figure 1. From strict partitions to doubled distinct partitions.
and F the latter polynomial has explicit expression, as shown by Fujii, Kanno,
2
Moriyama,Okada and Panova [4, 19].
A strict partition is a finite strict decreasing sequence of positive integers λ¯ =
(λ¯ ,λ¯ ,...,λ¯ ). Theinteger|λ¯|= λ¯ iscalledthesize andℓ(λ¯)=ℓiscalled
1 2 ℓ 1≤i≤ℓ i
thelengthofλ¯.Forconvenience,letλ¯ =0fori>ℓ(λ¯). Astrictpartitionλ¯couldbe
Pi
identicalwithitsshiftedYoungdiagram,whichmeansthatthei-throwoftheusual
Young diagram is shifted to the right by i boxes. We define the doubled distinct
partition of λ¯, denoted by λ¯λ¯, to be the usual partition whose Young diagram is
obtained by adding λ¯ boxes to the i-th column of the shifted Young diagram of λ¯
i
for 1≤ i≤ ℓ(λ¯) (see [5, 20, 21]). For example, (6,4,4,1,1) is the doubled distinct
partition of (5,2,1) (see Figure 1).
For each usual partition λ, let λ′ denote the conjugate partition of λ (see [5,
15, 20, 21]). A usual partition λ is called self-conjugate if λ = λ′. The set of all
doubled distinct partitions and the set of all self-conjugate partitions are denoted
by DD and SC respectively. For each positive integer t, let
H (λ)={h∈H(λ): h≡0 (mod t)}
t
be the multiset of hook lengths of multiples of t. Write H (λ)= h.
t h∈Ht(λ)
Recently, P´etr´eolle derived two Nekrasov-Okounkov type formulas for C and
Q
Cˇwhich involve doubled distinct and self-conjugate partitions. In particular, he
obtained the following two formulas [20, 21]. e
e
Theorem 1.1 (P´etr´eolle[20, 21]). For positive integers n and t we have
1 1
(1.1) = , if t is odd;
H (λ) (2t)nn!
t
λ∈DDX,|λ|=2nt
#Ht(λ)=2n
1 1
(1.2) = , if t is even.
H (λ) (2t)nn!
t
λ∈SCX,|λ|=2nt
#Ht(λ)=2n
Inspired by all those previous works, we establish the polynomiality of some
hook-content summations for doubled distinct and self-conjugate partitions. Our
main result is stated next.
Theorem 1.2. Let t be a given positive integer. The following two summations for
the positive integer n
F (h2 :h∈H(λ))F (c:c∈C(λ))
(1.3) (2t)nn! 1 2 (t odd)
H (λ)
t
λ∈DDX,|λ|=2nt
#Ht(λ)=2n
POLYNOMIALITY OF SOME HOOK-CONTENT SUMMATIONS 3
and
F (h2 :h∈H(λ))F (c:c∈C(λ))
(1.4) (2t)nn! 1 2 (t even)
H (λ)
t
λ∈SCX,|λ|=2nt
#Ht(λ)=2n
are polynomials in n for any symmetric functions F and F .
1 2
In fact, the degrees of the two polynomials in Theorem 1.2 can be estimated
explicitly in terms of F and F (see Corollary 4.8 and Theorem 5.3). When F
1 2 1
and F are two constant symmetric functions, we derive Theorem 1.1. Other spe-
2
cializations are listed as follows.
Corollary 1.3. We have
1 1
(1.5) (2t)nn! h2 =6t2n2+ (t2−6t+2)tn (t odd),
H (λ) 3
t
λ∈DDX,|λ|=2nt h∈XH(λ)
#Ht(λ)=2n
1 1
(1.6) (2t)nn! h2 =6t2n2+ (t2−6t−1)tn (t even),
H (λ) 3
t
λ∈SCX,|λ|=2nt h∈XH(λ)
#Ht(λ)=2n
1 1
(1.7) (2t)nn! c2 =2t2n2+ (t2−6t+2)tn (t odd),
H (λ) 3
t
λ∈DDX,|λ|=2nt c∈XC(λ)
#Ht(λ)=2n
1 1
(1.8) (2t)nn! c2 =2t2n2+ (t2−6t−1)tn (t even).
H (λ) 3
t
λ∈SCX,|λ|=2nt c∈XC(λ)
#Ht(λ)=2n
TherestofthepaperisessentiallydevotedtocompletetheproofofTheorem1.2.
The polynomiality of summations in (1.3) for t = 1 with F = 1 or F = 1 has an
1 2
equivalentstatementintermsofstrictpartitions,whoseproofisgiveninSection2.
AfterrecallingsomebasicdefinitionsandpropertiesofLittlewooddecompositionin
Section 3, the doubled distinct and self-conjugate cases of Theorem 1.2 are proved
in Sections 4 and 5 respectively. Finally, Corollary 1.3 is proved in Section 6.
2. Polynomiality for strict and doubled distinct partitions
In this section we prove an equivalent statement of the polynomiality of (1.3)
for t=1 with F =1 or F =1, which consists a summation over the set of strict
1 2
partitions. Let λ¯ = (λ¯ ,λ¯ ,...,λ¯ ) be a strict partition. Therefore the leftmost
1 2 ℓ
box in the i-th row of the shifted Young diagram of λ¯ has coordinate (i,i+1).
The hook length of the (i,j)-box, denoted by h , is defined to be the number of
(i,j)
boxesexactlytotheright,orexactlyabove,ortheboxitself,plusλ¯ . Forexample,
j
consider the box (cid:3) = (i,j) = (1,3) in the shifted Young diagram of the strict
partition (7,5,4,1). There are 1 and 5 boxes below and to the right of the box (cid:3)
respectively. Since λ¯ =4, the hook length of (cid:3) is equal to 1+5+1+4 =11, as
3
illustrated in Figure 1. The content of (cid:3) = (i,j) is defined to be c(cid:3) = j −i, so
that the leftmost box in each row has content 1. Also, let H(λ¯) be the multi-set of
hook lengths of boxes and H(λ¯) be the product of all hook lengths of boxes in λ¯.
The hook length and content multisets of the doubled distinct partition λ¯λ¯ can be
obtained from H(λ¯) and C(λ¯) by the following relations:
(2.1) H(λ¯λ¯)=H(λ¯)∪H(λ¯)∪{2λ¯ ,2λ¯ ,...,2λ¯ }\{λ¯ ,λ¯ ,...,λ¯ },
1 2 ℓ 1 2 ℓ
4 GUO-NIUHANANDHUANXIONG
1 1
5 4 2 1 1 2 3 4
9 6 5 3 2 1 2 3 4 5
1211 8 7 5 4 1 1 2 3 4 5 6 7
Figure 2. The shifted Young diagram, the hook lengths and the
contents of the strict partition (7,5,4,1).
Figure 3. The skew shifted Young diagram of the skew strict
partition (7,5,4,1)/(4,2,1).
(α ,β )
· 1 1
(α ,β )
· 2 2
·
·
(α ,β )
· m m
·
Figure 4. A strict partition and its corners. The outer corners
are labelled with (α ,β ) (i = 1,2,...,m). The inner corners are
i i
indicated by the dot symbol “·”.
(2.2) C(λ¯λ¯)=C(λ¯)∪{1−c|c∈C(λ¯)}.
For two strict partitions λ¯ and µ¯, we write λ¯ ⊇ µ¯ if λ¯ ≥ µ¯ for any i ≥ 1. In
i i
this case, the skew strict partition λ¯/µ¯ is identical with the skew shifted Young
diagram. For example, the skew strict partition (7,5,4,1)/(4,2,1) is represented
by the white boxes in Figure 2. Let fλ¯ (resp. fλ¯/µ¯) be the number of standard
shifted Young tableaux of shape λ¯ (resp. λ¯/µ¯). The following formulas for strict
partitions are well-known (see [2, 23, 27]):
(2.3) fλ¯ = H|λ¯(|λ¯!) and n1! 2n−ℓ(λ¯)fλ¯2 =1.
|λ¯X|=n
Identity (1.1) with t=1, obtained by P´etr´eolle,becomes
1 1
= ,
H(λ) 2nn!
λ∈DDX,|λ|=2n
which is equivalent to the second identity of (2.3) in view of (2.1).
POLYNOMIALITY OF SOME HOOK-CONTENT SUMMATIONS 5
For a strict partition λ¯, the outer corners (see [11]) are the boxes which can be
removed in such a way that after removal the resulting diagram is still a shifted
Young diagram of a strict partition. The coordinates of outer corners are denoted
by (α ,β ),...,(α ,β ) such that α > α > ··· > α . Let y := β − α
1 1 m m 1 2 m j j j
(1≤j ≤m) be the contents of outer corners. We set α =0, β =ℓ(λ¯)+1 and
m+1 0
call(α ,β ),(α ,β ),...,(α ,β )the inner corners ofλ¯. Letx =β −α be
1 0 2 1 m+1 m i i i+1
the contents of inner corners for 0 ≤ i ≤ m (see Figure 3). The following relation
of x and y are obvious.
i j
(2.4) x =1≤y <x <y <x <···<y <x .
0 1 1 2 2 m m
Notice that x0 = y1 = 1 iff λ¯ℓ(λ¯) = 1. Let λ¯i+ = λ¯ {(cid:3)i} such that c(cid:3)i = xi for
0≤i≤m. Here λ¯0+ does not exist if y =1. The set of contents of inner corners
1 S
andthesetofcontentsofoutercornersofλ¯aredenotedbyX(λ¯)={x ,x ,...,x }
0 1 m
andY(λ¯)={y ,y ,...,y }respectively. Thefollowingrelationsbetweenthehook
1 2 m
lengths of λ¯ and λ¯i+ are established in [11].
Theorem 2.1 (Theorem 3.1 of [11]). Let λ¯ be a strict partition with X(λ¯) =
{x ,x ,...,x } and Y(λ¯)={y ,y ,...,y }. For 1≤i≤m, we have
0 1 m 1 2 m
H(λ¯)∪{1,x ,2x −2}∪{|x −x |:1≤j ≤m,j 6=i}
i i i j
∪{x +x −1:1≤j ≤m,j 6=i}
i j
=H(λ¯i+)∪{|x −y |:1≤j ≤m}∪{x +y −1:1≤j ≤m}
i j i j
and
xi − yj
H(λ¯) 1 1≤j≤m 2 2
= · .
H(λ¯i+) 2 Q (cid:0)(cid:0)xi(cid:1)−(cid:0)xj(cid:1)(cid:1)
2 2
0≤j≤m
jQ6=i (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
If y >1, we have
1
H(λ¯)∪{1,x ,x −1,x ,x −1,··· ,x ,x −1}
1 1 2 2 m m
=H(λ¯0+) ∪{y ,y −1,y ,y −1,··· ,y ,y −1}
1 1 2 2 m m
and
x0 − yj
H(λ¯) 1≤j≤m 2 2
= .
H(λ¯0+) Q (cid:0)(cid:0)x0(cid:1)−(cid:0)xj(cid:1)(cid:1)
2 2
1≤j≤m
Q (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
Let k be a nonnegative integer, and ν = (ν ,ν ,...,ν ) be a usual partition.
1 2 ℓ(ν)
For arbitrary two finite alphabets A and B, the power sum of the alphabet A−B
is defined by [14, p.5]
(2.5) Ψk(A,B):= ak− bk,
a∈A b∈B
X X
ℓ(ν)
(2.6) Ψν(A,B):= Ψνj(A,B).
j=1
Y
Let λ¯ be a strict partition. We define
x y
(2.7) Φν(λ¯):=Ψν { i },{ i } .
2 2
(cid:16) (cid:18) (cid:19) (cid:18) (cid:19) (cid:17)
6 GUO-NIUHANANDHUANXIONG
Theorem 2.2 (Theorem 3.5 of [11]). Let k be a given nonnegative integer. Then,
there exist some ξ ∈Q such that
j
k−1 j
x
Φk(λ¯i+)−Φk(λ¯)= ξ i
j
2
j=0 (cid:18) (cid:19)
X
for every strict partition λ¯ and 0≤i≤m, where x ,x ,...,x are the contents of
0 1 m
inner corners of λ¯.
Lemma 2.3. Let k be agiven nonnegative integer. Then, thereexist some a such
ij
that
i j
x y
(x−y)2k+(x+y−1)2k = a
ij
2 2
i+j≤k (cid:18) (cid:19) (cid:18) (cid:19)
X
for every x,y ∈C.
Proof. The claim follows from
x y
(x−y)2+(x+y−1)2 =4 +4 +1
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
and
x y
(x−y)2(x+y−1)2 = 2 −2 2. (cid:3)
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
Lemma 2.4 (Theorem 3.2 of [11]). Let k be a nonnegative integer. Then, there
exist some ξ ∈Q indexed by usual partitions ν such that
ν
(a −b )
i j
1≤j≤m ak = ξ Ψν({a },{b })
Q (a −a ) i ν i i
i j
0≤Xi≤m0≤j≤m |νX|≤k
jQ6=i
for arbitrary complex numbers a <a <···<a and b <b <···<b .
0 1 m 1 2 m
We define the difference operator D¯ for strict partitions by
m
(2.8) D¯ g(λ¯) :=2 g(λ¯i+)+g(λ¯0+)− g(λ¯),
i=1
(cid:0) (cid:1) X
where λ¯ is a strict partition and g is a function of strict partitions. In the above
definition,thesymbolg(λ¯0+)takesthevalue0ifλ¯0+ doesnotexist,orequivalently
if λ¯ℓ(λ¯) =1. By Theorem 2.1, we have
1
(2.9) D¯ =0.
H(λ¯)
(cid:16) (cid:17)
Theorem 2.5 (Theorem2.3 of [11]). Let g be a function of strict partitions and µ¯
be a given strict partition. Then we have
n
n
(2.10) 2|λ¯|−|µ¯|−ℓ(λ¯)+ℓ(µ¯)fλ¯/µ¯g(λ¯)= k D¯kg(µ¯)
|λ¯/Xµ¯|=n Xk=0(cid:18) (cid:19)
and
n
n
(2.11) D¯ng(µ¯)= (−1)n+k k 2|λ¯|−|µ¯|−ℓ(λ¯)+ℓ(µ¯)fλ¯/µ¯g(λ¯).
Xk=0 (cid:18) (cid:19)|λ¯/Xµ¯|=k
POLYNOMIALITY OF SOME HOOK-CONTENT SUMMATIONS 7
In particular, if there exists some positive integer r such that D¯rg(λ¯) = 0 for
every strict partition λ¯, then the left-hand side of (2.10) is a polynomial of n with
degree at most r−1.
For each usual partition δ let
pδ(λ¯):=Ψδ({h2 :h∈H(λ¯λ¯)},∅).
By (2.1), we have
ℓ(λ¯)
pk(λ¯)= h2k =2 h2k+(4k−1) λ¯2k
i
h∈XH(λ¯λ¯) h∈XH(λ¯) Xi=1
for a nonnegative integer k.
Theorem 2.6. Suppose that ν and δ are two given usual partitions. Then,
pδ(λ¯)Φν(λ¯)
(2.12) D¯r =0
H(λ¯)
(cid:16) (cid:17)
for every strict partition λ¯, where r =|δ|+ℓ(δ)+|ν|+1. Consequently, for a given
strict partition µ¯,
(2.13) 2|λ¯|−ℓ(λ¯)fλ¯/µ¯pδ(λ¯)
H(λ¯)
|λ¯/Xµ¯|=n
is a polynomial in n of degree at most |δ|+ℓ(δ).
Proof. Let X(λ¯) = {x ,x ,...,x } and Y(λ¯) = {y ,y ,...,y }. First, we show
0 1 m 1 2 m
that the difference pk(λ¯i+)−pk(λ¯) can be written as the following form
k j
x
η (λ¯) i
j
2
j=0 (cid:18) (cid:19)
X
for 0≤i≤ m and a nonnegative integer k, where each coefficient η (λ¯) is a linear
j
combination of some Φτ(λ¯) for some usual partition τ of size |τ| ≤ k. Indeed, by
Lemma 2.3 and Theorem 2.1,
m m
pk(λ¯0+)−pk(λ¯)=2 (x2k+(x −1)2k)−2 (y2k+(y −1)2k)+22k+1
j j j j
j=1 j=1
X X
k x j
=η0(λ¯)= ηj(λ¯) 20 [ if i=0 and λ¯ℓ(λ¯) ≥2]
j=0 (cid:18) (cid:19)
X
and
pk(λ¯i+)−pk(λ¯)
m m
=2 ((x −x )2k+(x +x −1)2k)−2 (x −y )2k+(x +y −1)2k
i j i j i j i j
j=1 j=1
X X(cid:0) (cid:1)
+2x2k+2(2x −2)2k+2−2(2x −1)2k+(22k−1) x2k−(x −1)2k
i i i i i
k x j (cid:0) (cid:1)
= η (λ¯) i [ if 1≤i≤m ].
j
2
j=0 (cid:18) (cid:19)
X
8 GUO-NIUHANANDHUANXIONG
Next, let A=Φν(λ¯) and B =pδ(λ¯). We have
∆iA:=Φν(λ¯i+)−Φν(λ¯)= Φνs(λ¯) Φνs′(λ¯i+)−Φνs′(λ¯) ,
X(∗) sY∈U sY′∈V(cid:0) (cid:1)
∆iB :=pδ(λ¯i+)−pδ(λ¯)= pδs(λ¯) pδs′(λ¯i+)−pδs′(λ¯) ,
X(∗∗)sY∈U sY′∈V(cid:0) (cid:1)
where the sum (∗) (resp. (∗∗)) ranges over all pairs (U,V) of positive integer sets
such that U ∪V ={1,2,...,ℓ(ν)} (resp. U ∪V ={1,2,...,ℓ(δ)}), U ∩V =∅ and
V 6=∅.
Finally, it follows from (2.9) and Theorem 2.1 that
pδ(λ¯)Φν(λ¯)
H(λ¯)D¯
H(λ¯)
(cid:16) (cid:17)
H(λ¯)
= pδ(λ¯0+)Φν(λ¯0+)−pδ(λ¯)Φν(λ¯)
H(λ¯0+)
(cid:0)m H(λ¯) (cid:1)
+2 pδ(λ¯i+)Φν(λ¯i+)−pδ(λ¯)Φν(λ¯)
H(λ¯i+)
i=1
X (cid:0) (cid:1)
xi − yj
2 2
= 1≤j≤m pδ(λ¯i+)Φν(λ¯i+)−pδ(λ¯)Φν(λ¯)
Q (cid:0)(cid:0)xi(cid:1)−(cid:0)xj(cid:1)(cid:1)
0≤i≤m 2 2
X 0≤j≤m (cid:0) (cid:1)
jQ6=i (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
xi − yj
2 2
1≤j≤m
= A·∆ B+B·∆ A+∆ A·∆ B .
Q (cid:0)(cid:0)xi(cid:1)−(cid:0)xj(cid:1)(cid:1) i i i i
0≤i≤m 2 2
X 0≤j≤m (cid:0) (cid:1)
jQ6=i (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
ByTheorems2.4and2.2,eachoftheabovethreetermscouldbewrittenasalinear
combination of some pδ(λ¯)Φν(λ¯) satisfying |δ|+ℓ(δ)+|ν| ≤ |δ|+ℓ(δ)+|ν|−1.
Then the claim follows by induction on |δ|+ℓ(δ)+|ν|. (cid:3)
When µ¯=∅, the summation (2.13) in Theorem 2.6 becomes
2n−ℓ(λ¯)n!
(2.14) pδ(λ¯)
H(λ¯)2
|λ¯X|=n
or
1
(2.15) 2nn! Ψδ({h2 :h∈H(λ¯λ)},∅)
H(λ¯λ¯)
|λ¯λX¯|=2n
by (2.1). The above summation is a polynomial in n. Consequently, Theorem 1.2
is true when t=1 and F =1. Other specializations are listed as follows.
2
Theorem 2.7. Let µ¯ be a given strict partition. Then,
(2.16) 2|λ¯|−ℓ(λ¯)−|µ¯|+ℓ(µ¯)fλ¯/µ¯Hµ¯ p1(λ¯)−p1(µ¯) =12 n +(12|µ¯|+5)n.
|λ¯/Xµ¯|=n Hλ¯ (cid:0) (cid:1) (cid:18)2(cid:19)
POLYNOMIALITY OF SOME HOOK-CONTENT SUMMATIONS 9
Let µ¯ =∅. We obtain
1 n
(2.17) 2nn! h2 =12 +5n.
H(λ¯λ¯) 2
|λ¯λX¯|=2n h∈XH(λ¯λ¯) (cid:18) (cid:19)
Proof. We have
m m
p1(λ¯0+)−p1(λ¯)=2 (x2+(x −1)2)−2 (y2+(y −1)2)+22+1
j j j j
j=1 j=1
X X
=η0(λ¯)=8|λ¯|+5 [ if i=0 and λ¯ℓ(λ¯) ≥2]
and
p1(λ¯i+)−p1(λ¯)
m m
=2 ((x −x )2+(x +x −1)2)−2 ((x −y )2+(x +y −1)2)
i j i j i j i j
j=1 j=1
X X
+2x2+2(2x −2)2+2−2(2x −1)2+(22−1)(x2−(x −1)2)
i i i i i
x
=4 i +8|λ¯|+5 [ if 1≤i≤m ].
2
(cid:18) (cid:19)
So that
xi − yj
Hλ¯D(cid:16)pH1(λλ¯¯)(cid:17)=0≤Xi≤m01≤≤Qjj≤≤mm(cid:0)(cid:0)x22i(cid:1)−(cid:0)x22j(cid:1)(cid:1)(4(cid:18)x2i(cid:19)+8|λ¯|+5)
jQ6=i (cid:0)(cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
=4Φ1(λ¯)+8|λ¯|+5
=12|λ¯|+5.
Therefore we have
p1(λ¯)
Hλ¯D2 Hλ¯ =12,
(cid:16)p1(λ¯)(cid:17)
Hλ¯D3 Hλ¯ =0.
(cid:16) (cid:17)
Identity (2.16) follows from Theorem 2.5. By (2.1), we derive (2.17). (cid:3)
Recall the following results obtained in [11] involving the contents of strict par-
titions.
Theorem 2.8. Suppose that Q is a given symmetric function, and µ¯ is a given
strict partition. Then
2|λ¯|−|µ¯|−ℓ(λ¯)+ℓ(µ¯)fλ¯/µ¯Q c :c∈C(λ¯)
H(λ¯) 2
|λ¯/Xµ¯|=n (cid:16)(cid:18) (cid:19) (cid:17)
is a polynomial in n.
Theorem 2.9. Suppose that k is a given nonnegative integer. Then
2|λ¯|−ℓ(λ¯)fλ¯ c+k−1 = 2k n .
H(λ¯) 2k (k+1)! k+1
|λ¯X|=n c∈XC(λ¯)(cid:18) (cid:19) (cid:18) (cid:19)
10 GUO-NIUHANANDHUANXIONG
Theorem 2.10. Let µ¯ be a strict partition. Then,
2|λ¯|−ℓ(λ¯)−|µ¯|+ℓ(µ¯)fλ¯/µ¯Hµ¯ c c n
(2.18) − = +n|µ¯|.
H(λ¯) 2 2 2
|λ¯/Xµ¯|=n (cid:0)c∈XC(λ¯)(cid:18) (cid:19) c∈XC(µ¯)(cid:18) (cid:19)(cid:1) (cid:18) (cid:19)
The above results can be interpreted in terms of doubled distinct partitions. In
particular, we obtain Theorem 1.2 when t=1 and F =1.
1
Theorem 2.11. For each usual partition δ, the summation
1
(2.19) 2nn! Ψδ(C(λ¯λ¯),∅)
H(λ¯λ¯)
|λ¯λX¯|=2n
is a polynomial in n.
Proof. Since c+(1−c) = 1 and c(1−c) = −2 c , there exists some a such that
2 i
ck+(1−c)k = s a c i. By (2.2), we obtain
i=1 i 2 (cid:0) (cid:1)
s i
P (cid:0) (cid:1) c
ck = ck+(1−c)k = a .
i
2
c∈XC(λ¯λ¯) c∈XC(λ¯)(cid:0) (cid:1) Xi=1 c∈XC(λ¯)(cid:18) (cid:19)
The claim follows from Theorem 2.8. (cid:3)
The following results are corollaries of Theorems 2.9 and 2.10.
Theorem 2.12. Suppose that k is a given nonnegative integer. Then,
1 c+k−1 2k+1 n
(2.20) 2nn! = ,
H(λ¯λ¯) 2k (k+1)! k+1
|λ¯λX¯|=2n c∈XC(λ¯λ¯)(cid:18) (cid:19) (cid:18) (cid:19)
1 n n
(2.21) 2nn! c2 =4 + .
H(λ¯λ¯) 2 1
|λ¯λX¯|=2n c∈XC(λ¯λ¯) (cid:18) (cid:19) (cid:18) (cid:19)
3. The Littlewood decomposition and corners of usual partitions
Inthissectionwerecallsomebasicdefinitionsandpropertiesforusualpartitions
(see[9],[15,p.12],[25,p.468],[12, p.75],[5]). LetW be the setofbi-infinite binary
sequences beginning with infinitely many 0’s and ending with infinitely many 1’s.
Each element w of W can be represented by (a′) = ···a′ a′ a′ a′a′a′a′ ···.
i i −3 −2 −1 0 1 2 3
However,therepresentationisnotunique,sinceforanyfixedintegerkthesequence
(a′ ) alsorepresentsw. Thecanonical representationofw isthe uniquesequence
i+k i
(a ) =···a a a a a a a ··· such that
i i −3 −2 −1 0 1 2 3
#{i≤−1,a =1}=#{i≥0,a =0}.
i i
Itwillbefurtherdenotedby···a a a .a a a a ··· withadotsymbolinserted
−3 −2 −1 0 1 2 3
between the letters a and a . There is a natural one-to-one correspondence
−1 0
between P and W (see, e.g. [25, p.468], [1] for more details). Let λ be a partition.
We encode each horizontal edge of λ by 1 and each vertical edge by 0. Reading
these (0,1)-encodings from top to bottom and from left to right yields a binary
word u. By adding infinitely many 0’s to the left and infinitely many 1’s to the
right of u we get an element w = ···000u111···∈ W. Clearly, the map λ 7→ w is
a one-to-one correspondence between P and W. For example, take λ=(6,3,3,1).
Thenu=0100110001,sothatw =(a ) =···1110100.110001000··· (seeFigure5).
i i