Introduction Difference operators Some applications on partition formulas
Difference operators for partitions and some applications
(joint work with Guo-Niu HAN)
Huan Xiong
Institut f¨ur Mathematik, Universit¨at Z¨urich
S´eminaire Lotharingien de Combinatoire(74) Ellwangen, Germany
25 March 2015
Introduction Difference operators Some applications on partition formulas
Plan of Talk
1 Introduction
2 Difference operators
3 Some applications on partition formulas
Introduction Difference operators Some applications on partition formulas
Definition
Apartitionis a finite weakly decreasing sequence of positive integers λ= (λ1, λ2, . . . , λr). The integer|λ|= P
1≤i≤r
λiis called thesizeof theλ.
A partitionλcould be identical with itsYoung diagram, which is a collection of boxes arranged in left-justified rows withλiboxes in thei-th row.
hook lengthh: the number of boxes exactly to the right, or exactly below, oritself.
hook product ofλ:H(λ) = Q
∈λ
h.
7 5 2 1 4 2 3 1 1
Figure:The Young diagram of the partition(4,2,2,1), together with the hook lengths of the corresponding boxes.
Introduction Difference operators Some applications on partition formulas
Definition
Apartitionis a finite weakly decreasing sequence of positive integers λ= (λ1, λ2, . . . , λr). The integer|λ|= P
1≤i≤r
λiis called thesizeof theλ.
A partitionλcould be identical with itsYoung diagram, which is a collection of boxes arranged in left-justified rows withλiboxes in thei-th row.
hook lengthh: the number of boxes exactly to the right, or exactly below, oritself.
hook product ofλ:H(λ) = Q
∈λ
h.
7 5 2 1 4 2 3 1 1
Figure:The Young diagram of the partition(4,2,2,1), together with the hook lengths of the corresponding boxes.
Introduction Difference operators Some applications on partition formulas
Definition
Apartitionis a finite weakly decreasing sequence of positive integers λ= (λ1, λ2, . . . , λr). The integer|λ|= P
1≤i≤r
λiis called thesizeof theλ.
A partitionλcould be identical with itsYoung diagram, which is a collection of boxes arranged in left-justified rows withλiboxes in thei-th row.
hook lengthh: the number of boxes exactly to the right, or exactly below, oritself.
hook product ofλ:H(λ) = Q
∈λ
h.
7 5 2 1 4 2 3 1 1
Figure:The Young diagram of the partition(4,2,2,1), together with the hook lengths of the corresponding boxes.
Introduction Difference operators Some applications on partition formulas
Definition
Apartitionis a finite weakly decreasing sequence of positive integers λ= (λ1, λ2, . . . , λr). The integer|λ|= P
1≤i≤r
λiis called thesizeof theλ.
A partitionλcould be identical with itsYoung diagram, which is a collection of boxes arranged in left-justified rows withλiboxes in thei-th row.
hook lengthh: the number of boxes exactly to the right, or exactly below, oritself.
hook product ofλ:H(λ) = Q
∈λ
h.
7 5 2 1 4 2 3 1 1
Figure:The Young diagram of the partition(4,2,2,1), together with the hook lengths of the corresponding boxes.
Introduction Difference operators Some applications on partition formulas
Definition
Apartitionis a finite weakly decreasing sequence of positive integers λ= (λ1, λ2, . . . , λr). The integer|λ|= P
1≤i≤r
λiis called thesizeof theλ.
A partitionλcould be identical with itsYoung diagram, which is a collection of boxes arranged in left-justified rows withλiboxes in thei-th row.
hook lengthh: the number of boxes exactly to the right, or exactly below, oritself.
hook product ofλ:H(λ) = Q
∈λ
h.
7 5 2 1 4 2 3 1 1
Figure:The Young diagram of the partition(4,2,2,1), together with the hook lengths of the corresponding boxes.
Introduction Difference operators Some applications on partition formulas
standard Young tableau (SYT): Obtained by filling in the boxes of the Young diagram with distinct entries 1 tonsuch that the entries in each row and each column are increasing.
fλ: the number of SYTs of shapeλ.
fλ/µ: the number of SYTs of skew shapeλ/µ. 1 4 5 9 2 6 3 7 8
Figure:An SYT of shape(4,2,2,1).
Introduction Difference operators Some applications on partition formulas
standard Young tableau (SYT): Obtained by filling in the boxes of the Young diagram with distinct entries 1 tonsuch that the entries in each row and each column are increasing.
fλ: the number of SYTs of shapeλ.
fλ/µ: the number of SYTs of skew shapeλ/µ.
1 4 5 9 2 6 3 7 8
Figure:An SYT of shape(4,2,2,1).
Introduction Difference operators Some applications on partition formulas
standard Young tableau (SYT): Obtained by filling in the boxes of the Young diagram with distinct entries 1 tonsuch that the entries in each row and each column are increasing.
fλ: the number of SYTs of shapeλ.
fλ/µ: the number of SYTs of skew shapeλ/µ.
1 4 5 9 2 6 3 7 8
Figure:An SYT of shape(4,2,2,1).
Introduction Difference operators Some applications on partition formulas
Theorem (Frame, Robinson and Thrall) fλ= n!
Hλ
wheren=|λ|.
RSK algorithm or representation of finite groups⇒ X
|λ|=n
fλ2=n! and therefore
1 n!
X
|λ|=n
fλ2=1.
Introduction Difference operators Some applications on partition formulas
Theorem (Frame, Robinson and Thrall) fλ= n!
Hλ
wheren=|λ|.
RSK algorithm or representation of finite groups⇒ X
|λ|=n
fλ2=n!
and therefore
1 n!
X
|λ|=n
fλ2=1.
Introduction Difference operators Some applications on partition formulas
Theorem (Nekrasov and Okounkov 2003, Han 2008)
X
n≥0
X
|λ|=n
fλ2
Y
∈λ
(t+h2)
xn n!2 =Y
i≥1
(1−xi)−1−t.
First proved by Nekrasov and Okounkov. Rediscovered and generalized by Han with a more elementary proof.
Introduction Difference operators Some applications on partition formulas
Theorem (Nekrasov and Okounkov 2003, Han 2008)
X
n≥0
X
|λ|=n
fλ2
Y
∈λ
(t+h2)
xn n!2 =Y
i≥1
(1−xi)−1−t.
First proved by Nekrasov and Okounkov. Rediscovered and generalized by Han with a more elementary proof.
Introduction Difference operators Some applications on partition formulas
1 n!
X
|λ|=n
fλ2g(λ)=??.
Han
1 n!
P
|λ|=n
fλ2 P
∈λ
h2=3n22−n.
1 n!
P
|λ|=n
fλ2 P
∈λ
h4=40n3−75n6 2+41n.
1 n!
P
|λ|=n
fλ2 P
∈λ
h6=1050n4−4060n324+5586n2−2552n.
Conjecture (Han 2008)
P(n) = 1 n!
X
|λ|=n
fλ2X
∈λ
h2k is always a polynomial ofnfor everyk∈N. Proved and generalized by Stanley.
Introduction Difference operators Some applications on partition formulas
1 n!
X
|λ|=n
fλ2g(λ)=??.
Han
1 n!
P
|λ|=n
fλ2 P
∈λ
h2=3n22−n.
1 n!
P
|λ|=n
fλ2 P
∈λ
h4=40n3−75n6 2+41n.
1 n!
P
|λ|=n
fλ2 P
∈λ
h6=1050n4−4060n324+5586n2−2552n.
Conjecture (Han 2008)
P(n) = 1 n!
X
|λ|=n
fλ2X
∈λ
h2k is always a polynomial ofnfor everyk∈N.
Proved and generalized by Stanley.
Introduction Difference operators Some applications on partition formulas
1 n!
X
|λ|=n
fλ2g(λ)=??.
Han
1 n!
P
|λ|=n
fλ2 P
∈λ
h2=3n22−n.
1 n!
P
|λ|=n
fλ2 P
∈λ
h4=40n3−75n6 2+41n.
1 n!
P
|λ|=n
fλ2 P
∈λ
h6=1050n4−4060n324+5586n2−2552n.
Conjecture (Han 2008)
P(n) = 1 n!
X
|λ|=n
fλ2X
∈λ
h2k is always a polynomial ofnfor everyk∈N.
Proved and generalized by Stanley.
Introduction Difference operators Some applications on partition formulas
Theorem (Stanley 2010)
LetF be a symmetric function. Then P(n) = 1
n!
X
|λ|=n
fλ2F(h2:∈λ) is a polynomial ofn.
Remark.Han-Stanley Theorem is a corollary of our main result.
Introduction Difference operators Some applications on partition formulas
Definition
Letλbe a partition andgbe a function defined on partitions. Difference operatorsDandD−are defined by
Dg(λ) =X
λ+
g(λ+)−g(λ) and
D−g(λ) =|λ|g(λ)−X
λ−
g(λ−),
whereλ+ranges over all partitions obtained by adding a box toλandλ− ranges over all partitions obtained by removing a box fromλ.
LetD0g=gandDk+1g=D(Dkg)fork≥0.
Introduction Difference operators Some applications on partition formulas
Definition
Letλbe a partition andgbe a function defined on partitions. Difference operatorsDandD−are defined by
Dg(λ) =X
λ+
g(λ+)−g(λ) and
D−g(λ) =|λ|g(λ)−X
λ−
g(λ−),
whereλ+ranges over all partitions obtained by adding a box toλandλ− ranges over all partitions obtained by removing a box fromλ.
LetD0g=gandDk+1g=D(Dkg)fork≥0.
Introduction Difference operators Some applications on partition formulas
Main result (Han and Xiong 2015)
Suppose thatFis a symmetric function. Then there exists somer∈Nsuch thatDr(F(h
2 :∈λ)
Hλ ) =0 for every partitionλ.
Theorem (Han and Xiong 2015)
Suppose thatgis a function defined on partitions andµis a given partition. Then we have
X
|λ/µ|=n
fλ/µg(λ) =
n
X
k=0
n k
! Dkg(µ) and
Dng(µ) =
n
X
k=0
(−1)n+k n k
! X
|λ/µ|=k
fλ/µg(λ).
Our results⇒Han-Stanley theorem, (skew) marked hook formula, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula...
Introduction Difference operators Some applications on partition formulas
Main result (Han and Xiong 2015)
Suppose thatFis a symmetric function. Then there exists somer∈Nsuch thatDr(F(h
2 :∈λ)
Hλ ) =0 for every partitionλ.
Theorem (Han and Xiong 2015)
Suppose thatgis a function defined on partitions andµis a given partition.
Then we have
X
|λ/µ|=n
fλ/µg(λ) =
n
X
k=0
n k
! Dkg(µ) and
Dng(µ) =
n
X
k=0
(−1)n+k n k
! X
|λ/µ|=k
fλ/µg(λ).
Our results⇒Han-Stanley theorem, (skew) marked hook formula, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula...
Introduction Difference operators Some applications on partition formulas
Main result (Han and Xiong 2015)
Suppose thatFis a symmetric function. Then there exists somer∈Nsuch thatDr(F(h
2 :∈λ)
Hλ ) =0 for every partitionλ.
Theorem (Han and Xiong 2015)
Suppose thatgis a function defined on partitions andµis a given partition.
Then we have
X
|λ/µ|=n
fλ/µg(λ) =
n
X
k=0
n k
! Dkg(µ) and
Dng(µ) =
n
X
k=0
(−1)n+k n k
! X
|λ/µ|=k
fλ/µg(λ).
Our results⇒Han-Stanley theorem, (skew) marked hook formula, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula...
Introduction Difference operators Some applications on partition formulas
When Dg = 0 or D
−g = 0?
Theorem
For any partitionλ, we have
D( 1 Hλ
) =0 and
D−( 1 Hλ
) =0.
(|λ|+1)fλ=X
λ+
fλ+⇒X
λ+
1 Hλ+ − 1
Hλ
=0⇒D( 1 Hλ
) =0.
fλ=X
λ−
fλ−⇒ |λ| Hλ
−X
λ−
1 Hλ−
=0⇒D−( 1 Hλ
) =0.
Introduction Difference operators Some applications on partition formulas
When Dg = 0 or D
−g = 0?
Theorem
For any partitionλ, we have
D( 1 Hλ
) =0 and
D−( 1 Hλ
) =0.
(|λ|+1)fλ=X
λ+
fλ+⇒X
λ+
1 Hλ+ − 1
Hλ
=0⇒D( 1 Hλ
) =0.
fλ=X
λ−
fλ−⇒ |λ| Hλ
−X
λ−
1 Hλ−
=0⇒D−( 1 Hλ
) =0.
Introduction Difference operators Some applications on partition formulas
When Dg = 0 or D
−g = 0?
Theorem
For any partitionλ, we have
D( 1 Hλ
) =0 and
D−( 1 Hλ
) =0.
(|λ|+1)fλ=X
λ+
fλ+⇒X
λ+
1 Hλ+ − 1
Hλ
=0⇒D( 1 Hλ
) =0.
fλ=X
λ−
fλ−⇒ |λ| Hλ
−X
λ−
1 Hλ−
=0⇒D−( 1 Hλ
) =0.
Introduction Difference operators Some applications on partition formulas
When Dg = 0 or D
−g = 0?
Theorem
D−g(λ) =0for everyλ⇒g(λ) = Ha
λ for some constant a.
Remark.WhenDg(λ) =0 for everyλ, it is not easy to determineg(λ).For example, actually we can show
D( P
∈λ
(h2 −1)−3 |λ|2 Hλ
) =0.
Introduction Difference operators Some applications on partition formulas
When Dg = 0 or D
−g = 0?
Theorem
D−g(λ) =0for everyλ⇒g(λ) = Ha
λ for some constant a.
Remark.WhenDg(λ) =0 for everyλ, it is not easy to determineg(λ).For example, actually we can show
D(
P
∈λ
(h2 −1)−3 |λ|2 Hλ
) =0.
Introduction Difference operators Some applications on partition formulas
Some properties of D and D
−Theorem
Letλbe a partition. Suppose that g1,g2are functions defined on partitions and a1,a2∈R. Then we have
D(a1g1+a2g2)(λ) =a1Dg1(λ) +a2Dg2(λ) and
D−(a1g1+a2g2)(λ) =a1D−g1(λ) +a2D−g2(λ).
Introduction Difference operators Some applications on partition formulas
Some properties of D and D
−Theorem
For any function g defined on partitions, we have D(g(λ)
Hλ
) =X
λ+
g(λ+)−g(λ) Hλ+
and
D−(g(λ) Hλ
) =X
λ−
g(λ)−g(λ−) Hλ−
.
Introduction Difference operators Some applications on partition formulas
Some properties of D and D
−For product of two functions:
Theorem
D(g1(λ)g2(λ) Hλ
) = g1(λ)D(g2(λ) Hλ
) +g2(λ)D(g1(λ) Hλ
)
+X
λ+
(g1(λ+)−g1(λ))(g2(λ+)−g2(λ)) Hλ+
and
D−(g1(λ)g2(λ) Hλ
) = g1(λ)D−(g2(λ) Hλ
) +g2(λ)D−(g1(λ) Hλ
)
−X
λ−
(g1(λ)−g1(λ−))(g2(λ)−g2(λ−)) Hλ−
.
Introduction Difference operators Some applications on partition formulas
Some properties of D and D
−For product of several functions:
Theorem
Suppose that g1,g2,· · ·,grare functions defined on partitions. Let
[r] ={1,2,· · ·,r}and∆j(λ, µ) =gj(µ)−gj(λ)for1≤j≤r . Then we have D(
Q
1≤j≤rgj(λ) Hλ
) =X
λ+
X
A∪B=[r]
A∩B=∅
A6=∅
Q
k∈A∆k(λ, λ+)Q
l∈Bgl(λ) Hλ+
and D−(
Q
1≤j≤rgj(λ) Hλ
) =−X
λ−
X
A∪B=[r]
A∩B=∅
A6=∅
Q
k∈A∆k(λ, λ−)Q
l∈Bgl(λ) Hλ−
.
Introduction Difference operators Some applications on partition formulas
Corners of partitions
For a partitionλ, theouter cornersare the boxes which can be removed to get a new partitionλ−.Let(α1, β1), . . . ,(αm, βm)be the coordinates of outer corners such thatα1> α2>· · ·αm.Letyj =βj−αjbe the contents of outer corners for 1≤j≤m.We setαm+1=β0=0 and call
(α1, β0),(α2, β1). . . ,(αm+1, βm)theinner cornersofλ. Letxi=βi−αi+1be the contents of inner corners for 0≤i≤m.
Theorem P
0≤i≤m
xi = P
1≤j≤m
yj. P
0≤i≤m
xi2− P
1≤j≤m
yj2=2|λ|.
Introduction Difference operators Some applications on partition formulas
Corners of partitions
For a partitionλ, theouter cornersare the boxes which can be removed to get a new partitionλ−.Let(α1, β1), . . . ,(αm, βm)be the coordinates of outer corners such thatα1> α2>· · ·αm.Letyj =βj−αjbe the contents of outer corners for 1≤j≤m.We setαm+1=β0=0 and call
(α1, β0),(α2, β1). . . ,(αm+1, βm)theinner cornersofλ. Letxi=βi−αi+1be the contents of inner corners for 0≤i≤m.
Theorem P
0≤i≤m
xi = P
1≤j≤m
yj.
P
0≤i≤m
xi2− P
1≤j≤m
yj2=2|λ|.
Introduction Difference operators Some applications on partition formulas
Corners of partitions
For a partitionλ, theouter cornersare the boxes which can be removed to get a new partitionλ−.Let(α1, β1), . . . ,(αm, βm)be the coordinates of outer corners such thatα1> α2>· · ·αm.Letyj =βj−αjbe the contents of outer corners for 1≤j≤m.We setαm+1=β0=0 and call
(α1, β0),(α2, β1). . . ,(αm+1, βm)theinner cornersofλ. Letxi=βi−αi+1be the contents of inner corners for 0≤i≤m.
Theorem P
0≤i≤m
xi = P
1≤j≤m
yj. P
0≤i≤m
xi2− P
1≤j≤m
yj2=2|λ|.
Introduction Difference operators Some applications on partition formulas
An example
Figure:The Young diagrams of the partition(4,2,2,1).
Outer corners:(4,1), (3,2), (1,4). {yj}={−3,−1,3}.
inner corners:(4,0),(3,1), (1,2), (0,4). {xi}={−4,−2,1,4}.
P
0≤i≤m
xi =−1= P
1≤j≤m
yj. P
0≤i≤m
xi2− P
1≤j≤m
yj2=18=2·9.
Introduction Difference operators Some applications on partition formulas
An example
Figure:The Young diagrams of the partition(4,2,2,1).
Outer corners:(4,1), (3,2), (1,4).
{yj}={−3,−1,3}.
inner corners:(4,0),(3,1), (1,2), (0,4). {xi}={−4,−2,1,4}.
P
0≤i≤m
xi =−1= P
1≤j≤m
yj. P
0≤i≤m
xi2− P
1≤j≤m
yj2=18=2·9.
Introduction Difference operators Some applications on partition formulas
An example
Figure:The Young diagrams of the partition(4,2,2,1).
Outer corners:(4,1), (3,2), (1,4).
{yj}={−3,−1,3}.
inner corners:(4,0),(3,1), (1,2), (0,4).
{xi}={−4,−2,1,4}.
P
0≤i≤m
xi =−1= P
1≤j≤m
yj. P
0≤i≤m
xi2− P
1≤j≤m
yj2=18=2·9.
Introduction Difference operators Some applications on partition formulas
An example
Figure:The Young diagrams of the partition(4,2,2,1).
Outer corners:(4,1), (3,2), (1,4).
{yj}={−3,−1,3}.
inner corners:(4,0),(3,1), (1,2), (0,4).
{xi}={−4,−2,1,4}.
P
0≤i≤m
xi =−1= P
1≤j≤m
yj.
P
0≤i≤m
xi2− P
1≤j≤m
yj2=18=2·9.
Introduction Difference operators Some applications on partition formulas
An example
Figure:The Young diagrams of the partition(4,2,2,1).
Outer corners:(4,1), (3,2), (1,4).
{yj}={−3,−1,3}.
inner corners:(4,0),(3,1), (1,2), (0,4).
{xi}={−4,−2,1,4}.
P
0≤i≤m
xi =−1= P
1≤j≤m
yj. P
0≤i≤m
xi2− P
1≤j≤m
yj2=18=2·9.
Introduction Difference operators Some applications on partition formulas
Onqγ(λ) Define
qk(λ) = X
0≤i≤m
xik
− X
1≤j≤m
yjk
and
qγ(λ) = Y
1≤l≤t
qγl(λ) for the partitionγ= (γ1, γ2, . . . , γt).
q0(λ) =1,q1(λ) =0,q2(λ) =2|λ|.
The idea to studyxi,yjandqγ(λ)comes from Kerov, Okounkov and Olshanski.
Introduction Difference operators Some applications on partition formulas
Onqγ(λ) Define
qk(λ) = X
0≤i≤m
xik
− X
1≤j≤m
yjk
and
qγ(λ) = Y
1≤l≤t
qγl(λ) for the partitionγ= (γ1, γ2, . . . , γt).
q0(λ) =1,q1(λ) =0,q2(λ) =2|λ|.
The idea to studyxi,yjandqγ(λ)comes from Kerov, Okounkov and Olshanski.
Introduction Difference operators Some applications on partition formulas
Onqγ(λ) Define
qk(λ) = X
0≤i≤m
xik
− X
1≤j≤m
yjk
and
qγ(λ) = Y
1≤l≤t
qγl(λ) for the partitionγ= (γ1, γ2, . . . , γt).
q0(λ) =1,q1(λ) =0,q2(λ) =2|λ|.
The idea to studyxi,yjandqγ(λ)comes from Kerov, Okounkov and Olshanski.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
LetS(∅,r) =0 andS(λ,r) = P
∈λ r
Q
i=1
(h2−i2)for|λ|≥1.
Step1:We want to show that there exist somebγ∈Qsuch that S(λ,r) = X
|γ|≤2r+2
bγqγ(λ) for every partitionλ.
Suppose thatfis a function defined on integers. Let F1(n) =
n
X
k=1
f(k) and
F2(n) =
n
X
k=1
F1(k).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
LetS(∅,r) =0 andS(λ,r) = P
∈λ r
Q
i=1
(h2−i2)for|λ|≥1.
Step1:We want to show that there exist somebγ∈Qsuch that S(λ,r) = X
|γ|≤2r+2
bγqγ(λ) for every partitionλ.
Suppose thatfis a function defined on integers. Let F1(n) =
n
X
k=1
f(k) and
F2(n) =
n
X
k=1
F1(k).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
LetS(∅,r) =0 andS(λ,r) = P
∈λ r
Q
i=1
(h2−i2)for|λ|≥1.
Step1:We want to show that there exist somebγ∈Qsuch that S(λ,r) = X
|γ|≤2r+2
bγqγ(λ) for every partitionλ.
Suppose thatfis a function defined on integers. Let F1(n) =
n
X
k=1
f(k) and
F2(n) =
n
X
k=1
F1(k).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
For 0≤j<i≤m, let
Bij ={(a,b)∈λ:αi+1+1≤a≤αi, βj+1≤b≤βj+1}.
Then
λ= [
0≤j<i≤m
Bij
and thus
X
∈λ
f(h) = X
0≤j<i≤m
X
∈Bij
f(h). The multiset of hook lengths ofBij are
xi−xj−1
[
a=xi−yj+1
{a,a−1,a−2, . . . ,a−(xi−yi−1)}.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
For 0≤j<i≤m, let
Bij ={(a,b)∈λ:αi+1+1≤a≤αi, βj+1≤b≤βj+1}.
Then
λ= [
0≤j<i≤m
Bij
and thus
X
∈λ
f(h) = X
0≤j<i≤m
X
∈Bij
f(h).
The multiset of hook lengths ofBij are
xi−xj−1
[
a=xi−yj+1
{a,a−1,a−2, . . . ,a−(xi−yi−1)}.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
For 0≤j<i≤m, let
Bij ={(a,b)∈λ:αi+1+1≤a≤αi, βj+1≤b≤βj+1}.
Then
λ= [
0≤j<i≤m
Bij
and thus
X
∈λ
f(h) = X
0≤j<i≤m
X
∈Bij
f(h).
The multiset of hook lengths ofBijare
xi−xj−1
[
a=xi−yj+1
{a,a−1,a−2, . . . ,a−(xi−yi−1)}.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
This means that
X
∈Bij
f(h) =
xi−xj−1
X
a=xi−yj+1 xi−yi−1
X
b=0
f(a−b)
=
xi−xj−1
X
a=xi−yj+1
(F1(a)−F1(a−xi+yi))
= F2(xi−xj−1) +F2(yi−yj+1−1)
−F2(xi−yj+1−1)−F2(yi−xj−1).
ReplaceP
∈λf(h)byS(λ,r).There exist somebk ∈Qsuch that for every partitionλ,we have
S(λ,r) = X
1≤k≤r+1
bk( X
0≤i≤j≤m
(xi−xj)2k+ X
1≤i≤j≤m
(yi−yj)2k
− X
0≤i≤m
X
1≤j≤m
(xi−yj)2k).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
This means that
X
∈Bij
f(h) =
xi−xj−1
X
a=xi−yj+1 xi−yi−1
X
b=0
f(a−b)
=
xi−xj−1
X
a=xi−yj+1
(F1(a)−F1(a−xi+yi))
= F2(xi−xj−1) +F2(yi−yj+1−1)
−F2(xi−yj+1−1)−F2(yi−xj−1).
ReplaceP
∈λf(h)byS(λ,r).There exist somebk ∈Qsuch that for every partitionλ,we have
S(λ,r) = X
1≤k≤r+1
bk( X
0≤i≤j≤m
(xi−xj)2k+ X
1≤i≤j≤m
(yi−yj)2k
− X
0≤i≤m
X
1≤j≤m
(xi−yj)2k).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Compare the coefficients ofz2k(1≤k≤r+1)on both sides of
( X
0≤i≤m
exp(xiz)− X
1≤j≤m
exp(yjz))( X
0≤i≤m
exp(−xiz)− X
1≤j≤m
exp(−yjz))
= X
0≤i≤m
X
0≤j≤m
exp((xi−xj)z) + X
1≤i≤m
X
1≤j≤m
exp((yi−yj)z)
− X
0≤i≤m
X
1≤j≤m
exp((xi−yj)z)− X
0≤i≤m
X
1≤j≤m
exp((yj−xi)z).
There exist somebγ∈Qsuch that S(λ,r) = X
|γ|≤2r+2
bγqγ(λ) for every partitionλ.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Step2:Suppose thatδis a partition. Then there exist somebγ ∈Qsuch that D(qδ(λ)
Hλ
) = X
|γ|≤|δ|−2
bγ
qγ(λ) Hλ
for every partitionλ.
For convenience, we just show the caseδ= (k)here. Letλk+=λ∪(αk+1+1, βk+1). First we have
Q
∈λk+g(h) Q
∈λg(h) =g(1) Y
0≤i≤k−1
g(xk −xi) g(xk−yi+1)
Y
k+1≤i≤m
g(xi−xk) g(yi−xk). In particular, we have
Hλk+
Hλ
= Q
0≤i≤m i6=k
(xk−xi) Q
1≤j≤m
(xk−yj).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Step2:Suppose thatδis a partition. Then there exist somebγ ∈Qsuch that D(qδ(λ)
Hλ
) = X
|γ|≤|δ|−2
bγ
qγ(λ) Hλ
for every partitionλ.
For convenience, we just show the caseδ= (k)here.
Letλk+=λ∪(αk+1+1, βk+1). First we have Q
∈λk+g(h) Q
∈λg(h) =g(1) Y
0≤i≤k−1
g(xk −xi) g(xk−yi+1)
Y
k+1≤i≤m
g(xi−xk) g(yi−xk). In particular, we have
Hλk+
Hλ
= Q
0≤i≤m i6=k
(xk−xi) Q
1≤j≤m
(xk−yj).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Step2:Suppose thatδis a partition. Then there exist somebγ ∈Qsuch that D(qδ(λ)
Hλ
) = X
|γ|≤|δ|−2
bγ
qγ(λ) Hλ
for every partitionλ.
For convenience, we just show the caseδ= (k)here.
Letλk+=λ∪(αk+1+1, βk +1). First we have Q
∈λk+g(h) Q
∈λg(h) =g(1) Y
0≤i≤k−1
g(xk −xi) g(xk−yi+1)
Y
k+1≤i≤m
g(xi−xk) g(yi−xk). In particular, we have
Hλk+
Hλ
= Q
0≤i≤m i6=k
(xk−xi) Q
1≤j≤m
(xk−yj).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Let
g1(λ) = X
0≤i≤m
g(xi)− X
1≤j≤m
g(yj).
Then
D(g1(λ) Hλ
) = X
0≤i≤m
g(xi+1) +g(xi−1)−2g(xi) Hλi+
.
In particular, letg(z) =zk andg1(λ) =qk(λ),then we obtain D(qk(λ)
Hλ
) = X
0≤i≤m
2P
1≤l≤k 2
k 2l
xik−2l
Hλi+
.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Let
g1(λ) = X
0≤i≤m
g(xi)− X
1≤j≤m
g(yj).
Then
D(g1(λ) Hλ
) = X
0≤i≤m
g(xi+1) +g(xi−1)−2g(xi) Hλi+
.
In particular, letg(z) =zk andg1(λ) =qk(λ),then we obtain D(qk(λ)
Hλ
) = X
0≤i≤m
2P
1≤l≤k 2
k 2l
xik−2l
Hλi+
.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Let
g(z) = Y
1≤j≤m
(1−yjz)− X
0≤i≤m
Hλ
Hλi+
Y
0≤l≤m l6=i
(1−xlz) be a polynomial ofzwith degreem.
Then we obtain g(1
xt
) = Y
1≤j≤m
(1−yj
xt
)− Hλ
Hλt+
Y
0≤l≤m l6=t
(1−xl
xt
)
= Y
1≤j≤m
(1−yj
xt
)− Q
1≤j≤m
(xt−yj) Q
0≤l≤m l6=t
(xt−xl)· Y
0≤l≤m l6=t
(1−xl
xt
)
= 0.
This means thatg(z)has at leastm+1 roots and thereforeg(z) =0. Now we have
X
0≤i≤m
Hλ
Hλi+
· 1 1−xiz =
Q
1≤j≤m(1−yjz) Q
0≤i≤m(1−xiz).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Let
g(z) = Y
1≤j≤m
(1−yjz)− X
0≤i≤m
Hλ
Hλi+
Y
0≤l≤m l6=i
(1−xlz) be a polynomial ofzwith degreem. Then we obtain
g(1 xt
) = Y
1≤j≤m
(1−yj
xt
)− Hλ
Hλt+
Y
0≤l≤m l6=t
(1−xl
xt
)
= Y
1≤j≤m
(1−yj
xt
)− Q
1≤j≤m
(xt−yj) Q
0≤l≤m l6=t
(xt−xl)· Y
0≤l≤m l6=t
(1−xl
xt
)
= 0.
This means thatg(z)has at leastm+1 roots and thereforeg(z) =0. Now we have
X
0≤i≤m
Hλ
Hλi+
· 1 1−xiz =
Q
1≤j≤m(1−yjz) Q
0≤i≤m(1−xiz).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Let
g(z) = Y
1≤j≤m
(1−yjz)− X
0≤i≤m
Hλ
Hλi+
Y
0≤l≤m l6=i
(1−xlz) be a polynomial ofzwith degreem. Then we obtain
g(1 xt
) = Y
1≤j≤m
(1−yj
xt
)− Hλ
Hλt+
Y
0≤l≤m l6=t
(1−xl
xt
)
= Y
1≤j≤m
(1−yj
xt
)− Q
1≤j≤m
(xt−yj) Q
0≤l≤m l6=t
(xt−xl)· Y
0≤l≤m l6=t
(1−xl
xt
)
= 0.
This means thatg(z)has at leastm+1 roots and thereforeg(z) =0.
Now we have
X
0≤i≤m
Hλ
Hλi+
· 1 1−xiz =
Q
1≤j≤m(1−yjz) Q
0≤i≤m(1−xiz).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
Let
g(z) = Y
1≤j≤m
(1−yjz)− X
0≤i≤m
Hλ
Hλi+
Y
0≤l≤m l6=i
(1−xlz) be a polynomial ofzwith degreem. Then we obtain
g(1 xt
) = Y
1≤j≤m
(1−yj
xt
)− Hλ
Hλt+
Y
0≤l≤m l6=t
(1−xl
xt
)
= Y
1≤j≤m
(1−yj
xt
)− Q
1≤j≤m
(xt−yj) Q
0≤l≤m l6=t
(xt−xl)· Y
0≤l≤m l6=t
(1−xl
xt
)
= 0.
This means thatg(z)has at leastm+1 roots and thereforeg(z) =0.
Now we have
X
0≤i≤m
Hλ
Hλi+
· 1 1−xiz =
Q
1≤j≤m(1−yjz) Q
0≤i≤m(1−xiz).
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
This means that X
0≤i≤m
Hλ
Hλi+
(X
k≥0
(xiz)k) = exp( X
1≤j≤m
ln(1−yjz)− X
0≤i≤m
ln(1−xiz)
= exp(X
k≥1
qk(λ) k zk).
By comparing the coefficients ofzk on both sides, we obtain there exist some bγ∈Qsuch that
X
0≤i≤m
Hλ
Hλi+
xik
= X
|γ|≤k
bγqγ(λ) for every partitionλ.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
This means that X
0≤i≤m
Hλ
Hλi+
(X
k≥0
(xiz)k) = exp( X
1≤j≤m
ln(1−yjz)− X
0≤i≤m
ln(1−xiz)
= exp(X
k≥1
qk(λ) k zk).
By comparing the coefficients ofzk on both sides, we obtain there exist some bγ∈Qsuch that
X
0≤i≤m
Hλ
Hλi+
xik
= X
|γ|≤k
bγqγ(λ) for every partitionλ.
Introduction Difference operators Some applications on partition formulas
Outline of proof of D
r(
F(h2 :∈λ)
Hλ
) = 0.
But we have
D(qk(λ) Hλ
) = X
0≤i≤m
2P
1≤l≤k2 k 2l
xik−2l
Hλi+
.
Thus there exist somebγ∈Qsuch that D(qk(λ)
Hλ
) = X
|γ|≤k−2
bγ
qγ(λ) Hλ
for every partitionλ.
Forqδ(λ), the proof is similar.