A t (see [Ma72℄). Aruial stepof that proof

PAUL-OLIVIERDEHAYE ANDGUO-NIUHAN

Abstrat. A multisethooklengthformulaforintegerpartitions

is established by using ombinatorial manipulation. As speial

ases, we rederive three hook length formulas, two of them ob-

tainedbyNekrasov-Okounkov,thethirdonebyIqbal,Nazir,Raza

andSaleem,whohavemadeuseoftheylisymmetryofthetopo-

logialvertex. Amultisethook-ontentformulaisalsoproved.

1. Introdution

Reently,anelementaryproofoftheNekrasov-Okounkovhooklength

formula[NO06℄ was given by the seond author in [Han10℄, using the

Madonald identities for

A t

^{(see [Ma72℄). Aruial stepof that proof}

is the onstrution of a bijetion between

t

^{-ores and integer vetors}

satisfying some additional properties. Several further papers related

to the Nekrasov-Okounkov formula have been published. See, e.g.,

[Wes06, CLPS08, CW09, CKW09,IKS10, Sta10, Ols10, INRS10℄.

In the present paper we again take up the study of the Nekrasov-

Okounkovformulaandobtainseveralresultsinthefollowingdiretions:

(1)Thebijetionbetween

t

^{-oresandintegervetors isonstrutedfor}

any positive integer

t

^{, while in [Han10℄,}

t

^{had to be an odd positive}

integer; (2) That bijetion is shown to satisfy a multiset hook length

formula (Theorem 1) with a funtional parameter

τ

^{by using a geo-}

metri model, alled exploded tableau. The result in [Han10℄ orre-

spondstothespeialase

τ(x) = x

^{. (3)Amultisethooklengthformula}

provides another speial ase when taking

τ = sin

^{, namely Theorem}

2. (4) Three hook length formulas are derived (Corollaries 7 and 8,

Theorem 5), the rst two previously obtained by Nekrasov-Okounkov

[NO06℄, the third one by Iqbal, Nazir, Raza and Saleem[INRS10℄. (5)

Theorem 2provides aunied formulafor the Nekrasov-Okounkov for-

mula and the lassial Jaobi triple produt identity ([And98, p.21℄,

[Knu73, p.20℄). This formula solves Problem 6.4 in [Han09℄. (6) A

multisethook-ontent hook lengthformulaisalsogiven in Setion6.

2010 MathematisSubjetClassiation. 11P83,05A17,05A15,05A19.

Keywordsandphrases. integerpartitions,hooklength,

q

^{-series,ongruenere-}

lations,

t

^-ores.

The authors wish to thank the Frenh National Researh Ageny (ANR

grantBLAN07-2183619)and theForshungsintitutfürMathematik, ETHZürih,

Switzerlandformakingthis ollaborationpossible.

The basi notions needed here an be found in[Ma95, p.1℄, [Sta99,

p.287℄, [Las03, p.1℄, [Knu73, p.59℄, and [And98, p.1℄). A partition

λ

^of

size

n

^andoflength

ℓ

^{isasequene ofpositiveintegers}

λ = (λ 1 , · · · , λ ℓ )

suh that

λ 1 ≥ λ 2 ≥ · · · ≥ λ ℓ > 0

^and

n = λ 1 + λ 2 + · · · + λ ℓ

^{. We}

write

n = |λ|

^,

ℓ(λ) = ℓ

^and

λ i = 0

^for

i ≥ ℓ + 1

^{. The set of all}

partitions of size

n

^{is denoted by}

P (n)

^{. The set of all partitions is}

denoted by

P

^{, so that}

P = S

n≥0 P (n)

^{. The hook length multiset of}

λ

^,

denoted by

H(λ)

^{, is the multiset of all hook lengths of}

λ

^{. Let}

t

^{be a}

positive integer. We write

H t (λ) = {h | h ∈ H(λ), h ≡ 0 (mod t)}

^{. A}

partition

λ

^{is a}

t

^{-ore if}

H _t (λ) = ∅

^{(see [Knu73, p.69, p.612℄, [Sta99,}

p.468℄). For example,

λ = (6, 3, 3, 2)

^{is a partition of size 14 and}

of length 4. We have

H(λ) = {2, 1, 4, 3, 1, 5, 4, 2, 9, 8, 6, 3, 2, 1}

^and

H ₂ (λ) = {2, 4, 4, 2, 8, 6, 2}

^{(see also[Han10℄).}

Let

t

^{be a positive integer and}

t 0 = 0

^(resp.

t 0 = 1/2

^{) if}

t

^{is odd}

(resp. even). Consider the set of (half-) integers

Z ^′ = t ₀ + Z

. Eah vetorof(half-)integers

V ~ = (v 0 , v 1 , . . . , v t−1 ) ∈ Z ^′t

isalleda

V t

^-oding

if the following onditions hold: (i)

{v _i − i mod t : i = 0, · · · , t − 1}

is equal to

t 0 + {0, 1, · · · , t − 1}

^{, (ii)}

v 0 + v 1 + · · · + v t−1 = 0

^{, (iii)}

v ₀ > v ₁ > · · · > v _t−1

^.

Theorem 1. Let

t

^{be a positive integer and}

τ : Z → F

^{be any weight}

funtion from

Z

to a eld

F

^{. Then, there is a bijetion}

φ t : λ 7→ V ~ = (v 0 , v 1 , . . . , v t−1 )

^from

t

^{-ores onto}

V t

^{-odings suh that}

|λ| = 1

2t (v ₀ ² + v ₁ ² + · · · + v ² _t−1 ) − t ² − 1 24

(1)

and

Y

h∈H(λ)

τ (h − t)τ(h + t)

τ (h) ² =

t−1

Y

i=1

τ(−i) ^{β i (λ)} τ (i) ^{β i (λ)+t−i}

Y

0≤i<j≤t−1

τ(v i − v j ),

(2)

where

β _i (λ) = #{ ∈ λ : h( ) = t − i}

^.

The proof of Theorem 1 is given in Setion 3. With the weight

funtion

τ = sin

^{, an odd funtion, we get the} speialization stated in the next theorem. Its proof is given inSetion5.

Theorem 2. For any positiveinteger

r

^{and any omplex numbers}

z, t

^,

we have

(3)

X

λ

q ^|λ| Y

h∈H r (λ)

1 − sin ² (tz ) sin ² (hz)

= exp

∞

X

k=1

q ^k

k(1 − q ^k ) − rq ^rk k(1 − q ^rk )

sin ² (tkz) sin ² (rkz)

.

Some speializations of Equation (3) are given in Setion4.

λ (8, 4, 3, 2, 2, 1)

W (λ) {10, 5, 3, 1, 0, −2, −4, −5, −6, −7, −8, −9, −10, −11, · · · }

V (λ) {10, 3, 1, −6, −8}

W ^† (λ) {5, 0, −2, −4, −5, −7, −9, −10, −11, · · · }

M (λ) 10

m(λ) −4

C(λ) {9, 8, 7, 6, 4, 2, −1, −3}

λ ^∗ (6, 5, 3, 2, 1, 1, 1, 1)

W 2 (λ) {8, 6, 3, 1, −1, −2, −3, −4, −6, −7, −8, −9, −10, −11, −12, · · · }

V ₂ (λ) {8, 6, −1, −3, −10}

W ₂ ^† (λ) {3, 1, −2, −4, −6, −7, −8, −9, −11, −12, · · · }

M 2 (λ) 8

m ₂ (λ) −6

C ₂ (λ) {7, 5, 4, 2, 0, −5}

Table 1. The example

λ = (8, 4, 3, 2, 2, 1)

^with

t = 5

^.

Notethat this alsogives

W 1 (λ), V 1 (λ),

^etsine

W (λ) = W 1 (λ),

^et.

2. Exploded tableau

With eah partition

λ = (λ 1 , λ 2 , · · · , λ ℓ )

^{and eah positive integer}

t

we assoiate several sets of (half-)integers. All these onepts will be

illustrated for the ase

λ = (8, 4, 3, 2, 2, 1)

^and

t = 5

^{(See Table 1).}

Note that this ase is speial, as

λ

^{is itself a}

t

^{-ore, but this property}

will be assumed most of the time.

The

W

^{-set of}

λ

^isatranslationoftheshiftedparts,dened tobethe set ofallintegersoftheform

λ i − i+(t+1)/2

^for

i ∈ N \ 0

^{(thepartition}

λ

^{isviewed asaninnite}non-inreasingsequene trailingwithzeroes).

We denote this set by

W (λ)

^{. It is immediate that}

W (λ) ⊂ Z ^′

. It is also lear that there exists a smallest (half-)integral

M = M(λ)

^{and a}

largest (half-)integral

m = m(λ)

^{suh that}

{m, m − 1, · · · } ⊆ W (λ) ⊆ {M, M − 1, · · · }

^.

We say that anelement

x

^{ina set}

X

^is

t

^{-maximalif it is the largest}

in its ongruene lass modulo

t

^{. If}

t

^{is even, we have}

W (λ) ⊂ ^Z ₂

^{. By}

ongruene lasses mod

t

^{,we then mean the ongruene lasses mod}

t

^of

1/2, 3/2, · · · , t − 1/2

^{. The set of}

t

^{-maximal elements is denoted}

by

t −max(X)

^{. In the ases further onsidered, ongruene lasses will}

always ontain anelement, sono maximum willever be taken over an

empty set. It is then lear that

|t − max(X)| = t

^.

Wedenethe

V

^-set

V (λ)

^of

λ

^by

V (λ) := t − max(W (λ))

^{. It iseasily}

seen fromthe denitionof

m(λ)

^{thatnoongruenelass modulo}

t

^an

be empty. We also set

W ^† (λ) = W (λ) \ V (λ)

^{. If}

V (λ)

^{is sorted by}

dereasing order, we get a

V t

^{-oding (asproved inEquation (8)), that}

will be denoted by

V ~ (λ) = φ t (λ)

^{. Thus, the bijetion}

φ t

^{required in}

Theorem 1is onstruted.

We also dene the omplementary set

C(λ) := {M, M − 1, · · · } \ W (λ)

^{, so that the disjoint union}

W ^† (λ) ∪ V (λ) ∪ C(λ)

^{is equal to}

{M, M − 1, · · · }

^{. Note that}

m(λ) = min C(λ) − 1

^.

Theinvariantspreviouslydened,suhas

V (λ), W (λ), . . .

^willalsobe

given the subsript 1,as in

V 1 (λ), W 1 (λ), . . .

^{The invariants attahed}

to the onjugate partition

λ ^∗

^{, suh as}

V (λ ^∗ ), W (λ ^∗ ), . . .

^{will then be}

written

V 2 (λ), W 2 (λ), . . .

The exploded diagram of a partition

λ

^{, whih we now dene, is a}

basi tool in the onstrution. The reader is referred to Figure 1 for

an examplewhen

t

^{isodd, and Figure3 when}

t

^{is even. We startwith}

a two-dimensional lattie

Z ^′ × Z ^′ ⊂ R ²

, and add a

1 × 1

^{box in eah}

position

(λ i − i + ^t+1 ₂ , λ ^∗ _j − j + ^t+1 ₂ )

^{of the lattie for every}

i, j ∈ N _>0

. This meansthatthereisoneboxineahelementof

W 1 (λ) × W 2 (λ)

^{. In}

ontrast tothe lassialFerrersdiagram,the exploded diagramis thus

innite. The entry of eah box in the exploded diagramis dened to

be the the sum of the two oordinates of the box. When the entry is

expliitly writtenoneah box, we shall speak of anexploded tableau.

Boxesofonstantentrylineuponanti-diagonals. Weusethisfatto

group boxes into dierent sets. Let

∆

^(resp.

Γ ⁺

^{, resp.}

Γ ⁻

^{) be the set}

of allboxes withentries intherange

(t, ∞)

^(resp.

(0, t)

^,resp.

(−t, 0)

^).

Theset

∆

^{orrespondstotheboxesof}

λ

^{inthelassialFerrersdiagram}

(whih are shaded inFig. 1). In addition, if

(x, y) ∈ ∆

^{orresponds to}

∈ λ

^{,its entry}

x + y

^{inthe exploded tableau is equalto}

h + t

^{. The}

entries lowerthan

t

^{orrespondtooutsidehooks,andthere arethusno}

box with entry exatly

t

^.

Givenaset

X

^,wewrite

−X

^{forthesetofoppositesofelementsof}

X

^.

In the speial ase of a

t

^{-ore, many of the invariants we just dened}

are niely related.

Lemma 3. If

λ

^{is a}

t

^{-ore, then}

W (λ) = [

a∈V (λ)

(a − t N )

(4)

V ₁ (λ) = W ₁ (λ) ∩ −W ₂ (λ)

(5)

V 2 (λ) = −V ₁ (λ)

(6)

W ₁ ^† (λ) = −C ₂ (λ) ∪ {−M ₂ (λ) − 1, −M ₂ (λ) − 2, · · · }

(7)

X

v∈V (λ)

v = 0.

(8)

Proof. WerstproveEquation(4). Theset

W (λ)

^{onsistsbydenition}

of the

w i = λ i − i + (t + i)/2

^{. If}

j ≥ i > 0

^{, then all hook lengths in}

the interval

[w i − w j+1 − 1, w i − w j + 1]

^{appear inrow}

i

^of

λ

^{. Hene if}

w i − w j < t

^and

w i − w j+1 > t

^{, then}

t

^{must appear as a hook length}

18 16 13 11 9 8 7 6

13 11 8 6

11 9 6

9 7

8 6

6

4 3 2 1 0

4 3 2 1

4 2 1 0

4 2 0

3 1

4 1

4 2

3 1

2 0

1 0

-1 -2 -3 -4 -1 -2 -3 -4

-1 -3 -4

-1 -2 -3 -1 -2 -3 -4

-1 -3 -4

-1 -3

-2 -4

-3

-1 -4

-2

-1 -3

-2 -4

-3 -4

-5 -5 -6 -7 -8 -9 -10 -5 -6 -7 -8 -9 -10 -11 -12 -5 -6 -7 -8 -9 -10 -11 -12 -13 -14 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15 -5 -6 -8 -9 -10 -11 -12 -13 -14 -15 -16 -17 -5 -6 -7 -8 -10 -11 -12 -13 -14 -15 -16 -17 -18 -19 -6 -7 -8 -9 -11 -12 -13 -14 -15 -16 -17 -18 -19 -20 -5 -7 -8 -9 -10 -12 -13 -14 -15 -16 -17 -18 -19 -20 -21 -6 -8 -9 -10 -11 -13 -14 -15 -16 -17 -18 -19 -20 -21 -22 -5 -7 -9 -10 -11 -12 -14 -15 -16 -17 -18 -19 -20 -21 -22 -23 -6 -8 -10 -11 -12 -13 -15 -16 -17 -18 -19 -20 -21 -22 -23 -24 -7 -9 -11 -12 -13 -14 -16 -17 -18 -19 -20 -21 -22 -23 -24 -25 -5 -8 -10 -12 -13 -14 -15 -17 -18 -19 -20 -21 -22 -23 -24 -25 -26 -6 -9 -11 -13 -14 -15 -16 -18 -19 -20 -21 -22 -23 -24 -25 -26 -27 -5 -7 -10 -12 -14 -15 -16 -17 -19 -20 -21 -22 -23 -24 -25 -26 -27 -28

X 1

X 2

10 5 3 1 0 -2 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13

8 6 3 1 -1 -2 -3 -4 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15

∆ Γ ⁺ Γ ⁻

Figure 1. The exploded tableau of the partition

λ = (8, 4, 3, 2, 2, 1)

^,with

t = 5

^{. The partition(shaded boxes)}

appears in an orientation similar to the Frenh orienta-

tion for Ferrers diagrams. Therefore, axes are reversed

and swithed. The

W

^{-sets of}

λ

^and

λ ^∗

^{serve as oor-}

dinates and the

V

^{-sets are} underlined. For instane, onsider the third box of the seond row of the las-

sial Ferrers diagram of

λ

^{, i.e. the} seond-to-last box on that row. This box now ends up at oordinates

(4 − 2 + ⁵⁺¹ ₂ , 3 − 3 + ⁵⁺¹ ₂ ) = (5, 3)

^{, arrying the entry}

5 + 3 = 8

^{. Three diagonallines separate}

∆, Γ ⁺

^and

Γ ⁻

^.

in row

i

^and

λ

^{would not be a}

t

^{-ore. So for any}

i

^{, there must always}

exist a

j > i

^with

w _i − w _j = t

^.

It isa lassiallemmainombinatoris thatfor any partition

λ

^,the

twosets

{λ i − i + 1/2 : i ≥ 1}

^and

− {λ ^∗ _i − i + 1/2 : i ≥ 1}

^{are disjoint}

and that their union is

Z + 1/2

^{. The sets}

W 1 (λ)

^and

−W 2 (λ)

^are

merely translates of these two lassial sets. In light of Equation (4)

the sets

W 1

^and

−W 2

^{interset in just one point for eah ongruene}

lassmod

t

^{. Itiseasytoshowthatthesetofallthosepointsisatually}

V 1 (λ)

^or

−V ₂ (λ)

^{(this is Equations (5)and (6)).}

Equation(7)isaquikonsequeneofthe previousthree. Aproofof

identity (8) byusing the Durfeesquare ofthe partition

λ

^{an befound}

in [GKS90℄.

3. Proof of Theorem 1

Throughout the proofwewilluse the example of

λ = (8, 4, 3, 2, 2, 1)

and

t = 5

^{, as} illustrated in Fig. 2 (for

t

^{odd) and Fig. 3 (for}

t

^even).

When

t

^{iseven,both oordinatesare} half-integers. Theentries arestill integraland the argumentarriesthrough identially. Let

T _(a,b) : Z ^′ → Z ^′

denote the translation dened by

T (a,b) (x, y) = (x + a, y + b)

^{. We}

now need the followingeasy results:

1 W 1 ×W 2 ^† + 1

W 1 ^† ×W 2 = 1 _W

1 ×W 2 \V 1 ×V 2 + 1

W 1 ^† ×W 2 ^†

(9)

T (0,−t) (∆) = (∆ ∪ Γ ⁺ ) ∩ (W 1 × W ₂ ^† ),

(10)

T _(−t,0) (∆) = (∆ ∪ Γ ⁺ ) ∩ (W ₁ ^† × W 2 ),

(11)

T _(−t,−t) (∆) = (∆ ∪ Γ ⁺ ∪ Γ ⁻ ) ∩ (W ₁ ^† × W ₂ ^† ).

(12)

The rst one, where

1

is the indiator funtion, is ompletely trivial.

For the seond, assume

= (x, y) ∈ λ

^{, so that}

x + y = h + t

^and

x ∈ W 1

^,

y ∈ W 2

^{. Then, its}

(0, −t)

^{translate, equal to}

(x, y − t)

^{, has}

entry

x + y − t

^whihisnonnegative. Also,

y − t

^isin

W ₂ ^†

^{, sine}

y − t

^is

not

t

^{-maximal. Finally,}

y − t ∈ W ₂ ^†

^{isequivalentto}

y ∈ W 2

^{. The other}

two identities followsimilarly.

Proof of Theorem 1. For any set

B

^{of boxes in the exploded tableau}

let

||B || := Y

(x,y)∈B

τ(x + y).

The left-hand side of Equation (2)an bewritten

LHS

= ||∆|| · ||T _(−t,−t) (∆) ||

||T _(−t,0)) (∆) || · ||T _(0,−t) (∆) || .

(13)

Using relations(9-12) we an rewriteExpression (13) as

LHS

= ||∆ ∩ (V 1 × V 2 )|| · ||Γ ⁻ ∩ (W ₁ ^† × W ₂ ^† )||

||Γ ⁺ \ (V 1 × V 2 )|| .

(14)

This informationissummarizedgraphiallyinFigure2forourrunning

example (the numerator is the produt of the entries in squares on-

taining a value, while the denominatoris the produt of the entries in

irles). At this point the readerisenouraged toonsider Figure2 to

antiipate the next step: we aim to fold the boxes in the region

Γ ⁺

and interleave them withboxes inthe region

Γ ⁻

^.

18

18 16 16 13 11 9 9 8 77 6

13 11 8 6

11

11 9 9 6

99 7 7

8 6

6

4

4 33 2 2 11 0 4

4 33 2 2 11 4

4 2 11 0

4

4 22 0

3

3 11

4

4 11

44 2 2 33 1 1

2 0

11 0

-1 -2 -3 -4 -1

-1 -2 -2 -3 -3 -4 -4

-1 -3 -4

-1 -2 -3 -1 -2 -2 -3 -4 -4 -1

-1 -3 -4 -4 -1

-1 -3 -3 -2 -2 -4 -4 -3 -1 -4 -4 -2

-1 -3

-2 -4

-3 -4

-1 -2 -3 -4 -1 -2 -3 -4

-1 -3 -4

-1 -2 -3 -1 -2 -3 -4

-1 -3 -4

-1 -3

-2 -4

-3

-1 -4

-2

-1 -3

-2 -4

-3 -4

X 1

X 2

10 5 3 1 0 -2 -4 -5 -6 -7 -8 -9 -10 -11 -12 -13

8 6 3 1 -1 -2 -3 -4 -6 -7 -8 -9 -10 -11 -12 -13 -14 -15

∆ Γ ⁺ Γ ⁻

Figure 2. The produt given in Equation (14) marked

inagraphialway. First,

B

^{isthesetofallboxes(shaded}

ornot) loated in

∆

^{withentries inthe range}

[6, 9]

^{. The}

set

T (−t,−t) (B)

^ismaterializedby allthe squares appear- ing in

Γ ⁻

^{. The set}

T (−t,0) (B) ∪ T (0,−t) (B)

^{onsists of all}

the irles appearing in

Γ ⁺

^{. The middle diagonal indi-}

ates the diagonal used to fold the boxes of

Γ ⁻

^{, while}

the dashedsquaresshowtheloationswherethoseboxes

end up. One exampleis indiated with the arrow.

Consider the map

F : Z ^′ × Z ^′ → Z ^′ × Z ^′

,sending

(x, y )

^to

(−y, −x)

^.

By Equation(7), this map isabijetion between

Γ ⁻ ∩ (W ₁ ^† × W ₂ ^† )

^and

Γ ⁺ ∩ (C 1 × C 2 )

^{. It alsomerely hanges the sign of}

x + y

^{. Hene,}

||Γ ⁻ ∩ (W ₁ ^† × W ₂ ^† )|| =

t−1

Y

i=1

τ(−i) τ (i)

β i (λ)

||Γ ⁺ ∩ (C ₁ × C ₂ )||.

(15)

We nowlaim that

||Γ ⁺ ∩ (W 1 × (M 2 − N ))||

||Γ ⁺ ∩ ( Z ^′ × C 2 )|| =

t−1

Y

i=1

τ(i) ^t−i .

(16)

Theproofofthislaimworksbyobservingthatinthedenominator,the

produtoverallboxesinagivenolumnisalwaysoftheform

Q t−1 i=1 τ (i)

^.

For the numerator, the produt of all boxes in a given row is also of

the form

Q t−1

i=1 τ(i)

^{, exept for the highest}

t − 1

^{rows, whih end up}

produing the right-hand side. We are left to ount the multipliities

of the full produt

Q t−1

i=1 τ(i)

^{in numerator and} denominator, and get

|W 1 ∩ [−M 2 , ∞)| − t = |V 1 ∪ −C 2 | − t = |C 2 |

^{by Equation (7). This}

proves Equation(16). Hene

(17)

||Γ ⁺ ∩ (W 1 × W 2 )||

||Γ ⁺ ∩ (C 1 × C 2 )|| = ||Γ ⁺ ∩ (W 1 × (W 2 ∪ C 2 ))||

||Γ ⁺ ∩ ((W 1 ∪ C 1 ) × C 2 )|| =

t−1

Y

i=1

τ (i) ^t−i .

By Equations (14), (15), (17) we derive

LHS

=

t−1

Y

i=1

τ(−i) τ(i)

β i (λ)

||∆ ∩ V ₁ × V ₂ || · ||Γ ⁺ ∩ V ₁ × V ₂ ||

Q t−1

i=1 τ(i) ^t−i

=

t−1

Y

i=1

τ(−i) ^{β i (λ)} τ(i) ^{β i (λ)+t−i}

Y

(x,y)∈V 1 ×V 2

x+y≥1

τ (x + y)

=

t−1

Y

i=1

τ(−i) ^{β i (λ)} τ(i) ^{β i (λ)+t−i}

Y

0≤i<j≤t−1

τ (v i − v j ).

The last equality follows fromEquation (6). This lastterm equals the

right-hand side of Equation (2).

We still have to prove Equation (1). For this, we onveniently rely

on Equation (2) with the speial weight funtion

τ (k) = 1 + zk ²

^{. By}

onsidering the oeientof

z

^{on both sides, we get}

(18)

2|λ|t ² = X

h∈H(λ)

(h − t) ² + (h + t) ² − 2h ²

^Eq.(2)

=

−

t−1

X

k=1

k ² (t − k)

!

+ X

0≤i<j≤t−1

(v i − v j ) ² =

− 1

12 t ² (t ² − 1)

+



 t

t−1

X

i=0

v ² _i +

t−1

X

i=0

v i

! 2 

 ,

whih impliesthe result,thanks to Equation (8).

Whentheweightfuntion

τ

^{iseithereven orodd,theright-handside}

ofEquation(2)anbesimplied. InthenextCorollaryweassumethat

u = (u 0 , u 1 , . . . , u t−1 )

^{is the vetor obtained by sorting the}

V t

^-oding

V ~

^{aording to the ongruene lasses, i.e.,}

u i ≡ i + t 0 (mod t)

^for

0 ≤ i ≤ t − 1

^.

Corollary 4. We have

Y

h∈H(λ)

τ(h − t)τ(h + t)

τ (h) ² = C

Q t−1

k=1 τ(k) ^t−k

Y

0≤i<j≤t−1

τ(u i − u j ),

(19)

with

C =

−1

^if

t ≡ 3 mod 4

^while

τ

^{is odd,}

1

^otherwise.

Proof. We need to split the boxes in

Γ ⁻

^{aording to the ongruene}

lasses forthe oordinates ofthe boxes. Weknowthat

W (λ) ⊂ t 0 + Z

. It an be shown that for all

i, j ∈ {0, 1, 2 · · · , t − 1}

^,

(x, y) ∈ Γ ⁻ : x ≡ i + t ₀ mod t, y ≡ j + t ₀ mod t

= max

0,

u i − u j

t

= max(0, k i − k j − δ i<j )

if

u = (u i ) = (i + t 0 + t · k i ) ^t−1 _i=0

^{. Therefore,}

(20)

Γ ⁻ = |{ ∈ λ : h < t}| = X

u i i,j >u j

((k i − k j ) − δ i<j ) ≡ X

u i i,j >u j

((k i + k j ) + δ i<j ) mod 2

≡ (t − 1)

t−1

X

i=0

k i

!

+ t(t − 1)

2 + sgn Y

i<j

(u i − u j ) mod 2.

We know (by Equation (8)) that

0 = P t−1

i=0 u i = P t−1

i=0 (t 0 + i + k i t) ,

whih gives

− P

k i = ^t−1 ₂ + t 0

^{. Together with Equation (20), this}

easily gives

C = (−1) ^{(t 0 −1/2)(t−1)}

^{, whih is merely a} restatement of

Equation (19).

4. Speializations

We derive some speializations of the multiset hook length formula

(Theorem 1). The simplest non-trivial example is the ase where the

weight funtion

τ(x) = x

^{. Theorem 1 is then equivalent to Theorem}

1.1 in [Han10℄, whihprovides a ombinatorialproof of a hook length

formula due to Nekrasov and Okounkov [NO06, formula (6.12)℄ (see

also Equation (22)) by using the Madonald identities for

A t

^{[Ma72 ℄.}

When we take

τ = sin

^{, whih is an odd funtion, thanks to some}

properties ofthe funtion

sin

^{(see Lemma12), we deriveTheorem 2in}

Setion 5.

If

r

^{equals 1, we obtainthe following hook lengthformula.}

Theorem 5 (

r = 1

^{). For any omplex numbers}

z

^and

t

^{, we have}

(21)

X

λ

q ^|λ| Y

h∈H(λ)

1− sin ² (tz) sin ² (hz)

= exp

∞

X

k=1

q ^k k(1 − q ^k )

1− sin ² (tkz) sin ² (kz)

.

λ

^{(8,5, 4,1,1,1)}

W (λ) ₂₁

2 , ¹³ ₂ , ⁹ ₂ , ¹ ₂ , − ¹ ₂ , − ³ ₂ , − ⁷ ₂ , − ⁹ ₂ , − ¹¹ ₂ , − ¹³ ₂ , − ¹⁵ ₂ , − ¹⁷ ₂ , − ¹⁹ ₂ , − ²¹ ₂ , · · ·

V (λ) ₂₁

2 , ¹³ ₂ , − ¹ ₂ , − ⁷ ₂ , − ⁹ ₂ , − ¹⁷ ₂

W ^† (λ) ₉

2 , ¹ ₂ , − ³ ₂ , − ¹¹ ₂ , − ¹³ ₂ , − ¹⁵ ₂ , − ¹⁹ ₂ , − ²¹ ₂ , · · ·

M (λ) ²¹ ₂

m(λ) − ⁷ ₂

C(λ) { ¹⁹ ₂ , ¹⁷ ₂ , ¹⁵ ₂ , ¹¹ ₂ , ⁷ ₂ , ⁵ ₂ , ³ ₂ , − ⁵ ₂ }

λ ^∗ (6, 3, 3, 3, 2, 1, 1, 1)

W ₂ (λ) ₁₇

2 , ⁹ ₂ , ⁷ ₂ , ⁵ ₂ , ¹ ₂ , − ³ ₂ , − ⁵ ₂ , − ⁷ ₂ , − ¹¹ ₂ , − ¹³ ₂ , − ¹⁵ ₂ , − ¹⁷ ₂ , − ¹⁹ ₂ , − ²¹ ₂ , − ²³ ₂ , · · ·

V ₂ (λ) ₁₇

2 , ⁹ ₂ , ⁷ ₂ , ¹ ₂ , − ¹³ ₂ , − ²¹ ₂

W ₂ ^† (λ) ₅

2 , − ³ ₂ , − ⁵ ₂ , − ⁷ ₂ , − ¹¹ ₂ , − ¹⁵ ₂ , − ¹⁷ ₂ , − ¹⁹ ₂ , − ²³ ₂ , · · ·

M ₂ (λ) ¹⁷ ₂

m 2 (λ) − ¹¹ ₂

C ₂ (λ) { ¹⁵ ₂ , ¹³ ₂ , ¹¹ ₂ , ³ ₂ , − ¹ ₂ , − ⁹ ₂ }

Table 2. The example

λ = (8, 5, 4, 1, 1, 1)

^with

t = 6

^.

It an be shown that formula(21) is equivalent to the ombination

of thetwoidentities(2.4)and (2.7)inthe paperwrittenby Iqbaletal.

[INRS10℄. Thoseauthorshave madeuse ofthe yli symmetry of the

topologial vertex [AKMV05, ORV06℄. When

t = 0

^{inTheorem 5, we}

obtain the lassial generating funtionfor partitions.

Corollary 6 (

r = 1, t = 0

^{). We have}

X

λ

q ^|λ| = exp

∞

X

k=1

q ^k k(1 − q ^k )

!

=

∞

Y

m=1

1 1 − q ^m .

Sine

sin(az)

sin(bz) = a − a ³ z ² /6 + · · · b − b ³ z ² /6 + · · · ,

Equation (21) beomes

X

λ

q ^|λ| Y

h∈H(λ)

1 − t ²

h ²

= exp X

k

q ^k

k(1 − q ^k ) (1 − t ² ) .

when

z = 0

^{. WealsoobtainthefollowinghookformuladuetoNekrasov}

and Okounkov [NO06, Equation (6.12)℄ (see also[Han10℄):

Corollary 7 (

r = 1, z = 0

^{). For any omplex number}

β

^{we have}

(22)

X

λ

q ^|λ| Y

h∈H(λ)

1 − β h ²

= Y

m

(1 − q ^m ) ^β−1 .

19

19 15 15 14 14 13 11 11 9 8 7 15

15 11 11 10 10 9 7 7

13 9 8 7

9 88 7

55 4 33 2 2 1 1 0 5

5 44 3 3 11 0 5

5 33 22 1 1 55 4 4 33 1 1

4 3 22 0

3 3 2 2 11

5 1 0

4 0

3 3 22 11 0

-1 -2 -3 -4 -5 -1 -2 -3 -4 -5

-1

-1 -2 -3 -3 -4 -4 -5 -5 -1

-1 -2 -2 -3 -3 -5 -5 -2 -3 -4 -1 -3 -3 -4 -4 -5 -5

-1 -3 -5

-1 -2 -4 -1 -2 -3 -3 -5 -2 -3 -4 -4 -3 -4 -5 -5 -4 -5

-1 -5

-2 -3 -4 -5

-1 -2 -3 -4 -5 -1 -2 -3 -4 -5

-1 -2 -3 -4 -5 -1 -2 -3 -5

-2 -3 -4 -1 -3 -4 -5

-1 -3 -5

-1 -2 -4 -1 -2 -3 -5 -2 -3 -4 -3 -4 -5 -4 -5

-1 -5

-2 -3 -4 -5

X 1

X 2

21/2 13/2 9/2 1/2

−1/2

−3/2

−7/2

−9/2

−11/2

−13/2

−15/2

−17/2

−19/2

−21/2

−23/2

−25/2

−27/2

−29/2

17

2 9

2 7

2 5

2 1

2 - ³ ₂ - ⁵ ₂ - ⁷ ₂ - ¹¹ ₂ - ¹³ ₂ - ¹⁵ ₂ - ¹⁷ ₂ - ¹⁹ ₂ - ²¹ ₂ - ²³ ₂ - ²⁵ ₂ - ²⁷ ₂ - ²⁹ ₂ - ³¹ ₂ - ³³ ₂

Figure3. Analogof Figure2,butforthe partition

λ = (8, 5, 4, 1, 1, 1)

^,with

t = 6

^.

Let

e ^2itz = s

^and

q = qs

^{inTheorem 5. Equation (21) beomes}

(23)

X

λ

q ^|λ| Y

h∈H(λ)

s + s ² − 2s + 1 4 sin ² (hz)

= exp X

k

q ^k

k(1 − s ^k q ^k ) s ^k + s ^2k − 2s ^k + 1 4 sin ² (kz)

.

Letting

s = 0

^yields

Corollary 8 (

r = 1, e ^2itz = 0

^{). We have}

(24)

X

λ

q ^|λ| Y

h∈H(λ)

1

4 sin ² (hz) = exp X

k

q ^k 4k sin ² (kz)

.

Note that Equation (24) has the followingequivalent form:

X

λ

q ^|λ| Y

h∈H(λ)

1

2 − 2 cos(hz) = exp X

k

q ^k

2k(1 − cos(kz))

,

or, sine

sinh(x) = −i sin(ix)

^,

(25)

X

λ

q ^|λ| Y

h∈H(λ)

− 1 4 sinh ² (hz)

= exp X

k

q ^k

k − 1

4 sinh ² (kz)

.

Equation (25) and Equation (7.25) in [NO06℄ are the same. Minor

typos are to be orretedin the latter paper.

Let

s = −1

^{in Equation (23). We} immediatelyhave (26)

X

λ

q ^|λ| Y

h∈H(λ)

−1 + 1 sin ² (hz)

= exp X

k

q ^k

k(1 − (−1) ^k q ^k ) (−1) ^k + 2 − 2(−1) ^k 4 sin ² (kz)

.

Corollary 9 (

r = 1, e ^2itz = −1

^{). We have}

X

λ

q ^|λ| Y

h∈H(λ)

cot ² (zh)

= exp X

k≥1

q ^2k−1 cot ² ((2k − 1)z)

(2k − 1)(1 + q ^2k−1 ) + q ^2k 2k(1 − q ^2k )

.

When

r = 1

^and

t = 2

^{,Equation3beomestheJaobitripleprodut}

identity.

Corollary 10 (

r = 1, t = 2

^{). We have}

(27)

Y

n≥0

(1 + ax ⁿ⁺¹ )(1 + x ⁿ /a)(1 − x ⁿ⁺¹ ) =

+∞

X

n=−∞

a ⁿ x ^n(n+1)/2 .

5. Proof of Theorem 2

Weuse aMadonaldidentityfortheproofofourtheorem. Let

t

^bea

positiveinteger. Milne[Mil85℄andLebenzon[Le91℄providethefollow-

ing version of Madonald's identity [Ma72℄ for the type

A _t

^{: let}

M _t = { a = (a ₁ , a ₂ , . . . , a _t ) ∈ Z ^t | a ₁ + a ₂ + · · · a _t = 1 + 2 + · · · + t}

^{. For}

a ∈ Z

denote the residue of

a

^modulo

t

^{by res}

_t a ∈ Z /t Z

. For eah sequene

(b 1 , b 2 , . . . , b t )

^{ofresidues modulo}

t

^{denethenumber}

ǫ(b 1 , b 2 , . . . , b t )

^to

be equal to

0

^or

±1

^{aording to the following rules: if}

b i

^{is dierent}

from

b j

^whenever

i

^and

j

^{are distint,i.e.}

(b 1 , b 2 , . . . , b t )

^{is apermuta-}

tion of the sequene

(

^res

t 1,

^res

t 2, . . . ,

^res

t t)

^{, then}

ǫ(b 1 , b 2 , . . . , b t )

^{is the}

sign of the permutation; otherwise, let

ǫ(b 1 , b 2 , . . . , b t ) = 0

^{. For eah}

a = (a 1 , a 2 , . . . , a t ) ∈ Z ^t

let

ǫ( a ) = ǫ(

^res

t a 1 ,

^res

t a 2 , . . . ,

^res

t a t )

^and

Ω( a ) = 1

2t a ² ₁ + a ² ₂ + · · · + a ² _t − 1 ² − 2 ² − · · · − t ² .

The Madonald identity isthen rewritten in the followingform.

Theorem 11. For every

t ≥ 2

^{the identity}

(28)

Y

m≥1

(1 − q ^m ) ^t−1 Y

1≤j<i≤t

1 − x i

x _j q ^m−1 1 − x j

x _i q ^m

= X

a ∈ M _t

ǫ( a )q ^{Ω( a )} x ^1−a ₁ ¹ · · · x ^t−a _t ^t

holds in the ring of formal power series in

q

^{with oeients from the}

ring of Laurent polynomials in

x 1 , x 2 , . . . , x t

^.

Let

t = 2t ^′ + 1

^{be an odd integer. The right-hand side of Equa-}

tion (28) reads

A(q) = X

a ∈ M _t a i ≡i−1 mod t

X

σ∈S n

ǫ(σ)q ^{Ω( a )} x ^1−σ(a ₁ ^{1 )} · · · x ^t−σ(a _t ^{t )}

= x ¹ ₁ x ² ₂ · · · x ^t _t X

a ∈ M _t a i ≡i−1 mod t

q ^{Ω( a )} det x ^−a _i ^j

.

Let

u = (u ₀ , u ₁ , . . . , u _t−1 )

^{be a sequene dened by}

u i = a i+t ^′ +2 − t ^′ − 1,

where

a _t+j = a _j

^{. Then}

u _i ≡ a _i+t ^′ ₊₂ − t ^′ − 1 ≡ i + t ^′ + 2 − 1 − t ^′ − 1 (mod t)

^and

P t−1

i=0 u i = P t

i=1 −t(t ^′ + 1) = t(t − 1)/2 − t(t ^′ + 1) = 0

^{, so}

that the sorted vetor

V ~

^of

u

by dereasing order is a

V _t

^{-oding. Let}

λ = φ ⁻¹ _t ( V ~ )

^where

φ t

^{isdened inSetion 2. Then}

Ω( a ) = 1 2t

t

X

i=1

(a ² _i − i ² )

= 1 2t

t

X

i=1

((u i−1 + t ^′ + 1) ² − i ² )

= 1 2t

t−1

X

i=0

u i − t ² − 1 24

Eq.(1)

= |λ|.

(29)

Hene (rememberthat

t

^isodd)

A(q) = x ¹ ₁ x ² ₂ · · · x ^t _t X

u

q ^|λ| det

x ^−(u _i ^{j− 1 +t ′ +1)}

= x ¹ ₁ x ² ₂ · · · x ^t _t (x ₁ x ₂ · · · x _t ) ^{−t ′ −1} X

u

q ^|λ| det

x ^−u _i ^{j− 1} .

Let

B(q) = Y

m≥1

(1 − q ^m ) ^t−1 Y

1≤j<i≤t

Y

m≥1

1 − x i

x j

q ^m 1 − x j

x i

q ^m

.

Then

(30)

B(q) = A(q)/A(0) = P

u q ^|λ| det

x ^−u _i ^j−1 det

x ^−w _i ^{j− 1} ,

where

w = (w 0 , w 1 , . . . , w t−1 ) = (0, 1, 2, . . . , t ^′ , −t ^′ , . . . , −2, −1)

^{. Let}

x _i = e ^{−2Iz(i−1)}

^,with

I ² = −1

^{. We have}

det

x ^−u _i ^{j− 1}

= det e ^{2Iz(i−1)u j−1}

= Y

0≤i<j≤t−1

(e ^{2Izu i} − e ^{2Izu j} )

= (2I) ^n(n−1)/2 Y

0≤i<j≤t−1

sin(u i z − u j z)

(31)

and

det

x ^−u _i ^j−1

= (2I) ^n(n−1)/2 Y

0≤i<j≤t−1

sin(u i z − u j z)

= (2I) ^n(n−1)/2 (−1) ^{t ′}

t−1

Y

k=1

sin ^t−k (kz).

(32)

By the last threeequations, we have

Y

m≥1

(1 − q ^m ) ^t−1 Y

1≤j<i≤t

1 − e ^{−2Iz(i−1)} e ^{−2Iz(j−1)} q ^m

1 − e ^{−2Iz(j−1)} e ^{−2Iz(i−1)} q ^m

= Y

m≥1

(1 − q ^m ) ^t−1 Y

1≤i≤t−1

(1 − e ^{−2Iz(t−i)} q ^m ) ⁱ (1 − e ^{−2Iz(−t+i)} q ^m ) ⁱ

= (−1) ^{t ′} Q t−1

k=1 sin ^t−k (kz) X

u

q ^|λ| Y

0≤i<j≤t−1

sin(u i z − u j z).

Wenowmakeuseofthefollowingeasypropertiesofthe

sin

^funtion.

Lemma 12. Let

x, y, u 1 , u 2 , . . . , u n

^{be omplexnumberssuh that}

u 1 + u 2 + · · · + u n = 0

^{. Then}

sin(x − y) sin(x + y) = sin ² (x) − sin ² (y);

(33)

Y

1≤i<j≤n

sin(u i − u j ) = Y

1≤i<j≤n

e ^{2u i I} − e ^{2u j I}

2I .

(34)

Taking

τ (k) = sin(kz)

^{in Equation (19) and using Lemma 12 we}

obtain the followingresult.

Lemma 13. Forany omplex number

p

^{andany oddpositive integer}

t

^,

we have

(35)

X

λ∈T (t)

q ^|λ| Y

h∈H(λ)

1 − sin ² (tz) sin ² (hz)

= Y

m≥1

(1 − q ^m ) ^t−1 Y

1≤i≤t−1

(1 − e ^{−2Iz(t−i)} q ^m ) ⁱ (1 − e ^2Iz(t−i) q ^m ) ⁱ ,

where

T (t)

^{is the set of all}

t

^-ore partitions.

We an work with the logarithm of the right-hand side of Equa-

tion (35) toget

X

k

−1 k

X

m≥1

(t − 1)q ^mk +

t−1

X

i=1

ie ^{−2Iz(t−i)k} q ^mk + ie ^2Iz(t−i)k q ^mk ))

= X

k

−q ^k

k(1 − q ^k ) (t − 1) +

t−1

X

i=1

(ie ^{−2Iz(t−i)k} + ie ^2Iz(t−i)k )

= X

k

q ^k k(1 − q ^k )

1 − e ^−2Iztk + e ^2Iztk − 2 e ^−2Izk + e ^2Izk − 2

.

Lemma 13 beomesthe following lemma.

Lemma14. Forany omplexnumbers

z

^{andany oddpositiveinteger}

t

^,

we have

(36)

X

λ∈T (t)

q ^|λ| Y

h∈H(λ)

1 − sin ² (tz) sin ² (hz)

= exp

∞

X

k=1

q ^k k(1 − q ^k )

1 − sin ² (tkz) sin ² (kz)

! .

Proof of Theorem 5. It is enough to prove that Equation (23) is true

for any omplex numbers

z

^and

s

^{. Let}

n

^{be a positive integer. The}

oeient

C n (s)

^(resp.

D n (s)

^)of

q ⁿ

^{onthe left-hand side (resp. right-}

hand side) of Equation (23) is apolynomial in

s

^{of degree}

2n

^{. Forthe}

proof of

C n (s) = D n (s)

^{, it sues to nd}

2n + 1

^{expliit numerial}

values

s 0 , s 1 , . . . , s 2n

^suhthat

C n (s i ) = D n (s i )

^for

0 ≤ i ≤ 2n

^byusing

the Lagrangeinterpolationformula. The basi fatis that

Y

h∈H(λ)

s + s ² − 2s + 1 4 sin ² (hz)

= 0

for every partition

λ

^{whih is not a}

t

^{-ore (remember that}

s = e ^2tz

^).

By omparing Theorem 2 and Lemma 14 we see that Equation 23 is

true when

s = e ^2tz

^{for every odd integer}

t

^{, i.e.}

C n (e ^2tz ) = D n (e ^2tz )

^.

This guarantees

C n (s) = D n (s)

^{for every omplexnumber}

s

^.

Reall the following result obtained in[HJ11℄.

Theorem 15 (Multipliation Theorem). If the series

f α (q)

^{and the}

funtion

ρ(h)

^{satisfy the relation}

(37)

X

λ∈P

q ^|λ| Y

h∈H(λ)

ρ(αh) = f α (q),

then, for any positive integer

r

^{, the followingidentity holds:}

(38)

X

λ∈P

q ^|λ| x ^{#H r (λ)} Y

h∈H r (λ)

ρ(h) = (f r (xq ^r )) ^r Y

k≥1

(1 − q ^rk ) ^r (1 − q ^k ) .

This lastresult an be used asa transitionfrom Theorem 5toThe-

orem 2.

Proof of Theorem 2. Let

ρ(h) = 1 − sin ² (tz)/ sin ² (hz)

^{inTheorem 15.}

We get

f α (q) = X

λ∈P

q ^|λ| Y

h∈H(λ)

ρ(αh)

= X

λ∈P

q ^|λ| Y

h∈H(λ)

1 − sin ² (tz) sin ² (αhz)

Thm5

= exp X ^∞

k=1

q ^k

k(1 − q ^k ) 1 − sin ² (tkz) sin ² (αkz)

.

Hene,

X

λ

q ^|λ| Y

h∈H r (λ)

1 − sin ² (tz) sin ² (hz)

= exp r

∞

X

k=1

q ^rk

k(1 − q ^rk ) 1 − sin ² (tkz) sin ² (rkz)

Y

k≥1

(1 − q ^rk ) ^r (1 − q ^k )

= exp

∞

X

k=1

q ^k

k(1 − q ^k ) − rq ^rk k(1 − q ^rk )

sin ² (tkz) sin ² (rkz)

.

6. Multiset hook-ontent formula

In this setionwe establisha multiset hook-ontentformula. Let

s λ

be the Shur funtion orresponding to the partition

λ

^{(see [Ma95,}

p.40℄, [Sta99, p.308℄,[Las03, p.8℄). Reall the followinglassialhook-

ontent formula([Sta99, p.374℄,[Rob58℄).

Theorem 16. For any partition

λ

^{and positive integer}

n

^{we have}

(39)

s λ (1, p, p ² , . . . , p ⁿ⁻¹ ) = p ^b(λ) Y

∈λ

1 − p ^n+c

1 − p ^h ,

where

b(λ) = _i (i − 1)λ _i

^and

c = j − i

^if

∈ λ

^{ours on the}

i

^row

and

j

^{th olumn of the diagram of}

λ

^.

We now state a Theorem that provides an alternative approah to

the left-handside of Equation(2).

Theorem 17. Let

t

^{be a positive integer. There is a bijetion}

ψ _t : λ 7→ µ

^{whih maps}

t

^{-ores onto the set of all partitions}

µ

^{of length at}

most

t − 1

^{suh that}

{µ _i − i mod t : i = 1, . . . , t} = {0, 1, . . . , t − 1}

^.

Moreover, given any

τ

^from

Z

to a eld

F

^{, we have}

|λ| = −|µ| |µ| + t + t ² 2t ² +

t

X

i=1

µ ² _i + 2(t + 1 − i)µ i

2t

(40)

and

Y

∈λ

τ (h − t)τ (h + t)

τ(h ) ² =

t−1

Y

i=1

τ (−i) τ (i)

β i (λ) Y

∈µ

τ (t + c ) τ(h ) .

(41)

In fat, Theorem 2 an also be proved by using the multiset hook-

ontentformula(Theorem17)andthehook-ontentformula(Theorem

16). Conversely,the hook-ontentformula(39)anbederivedbyusing

themultisethook-ontentformula(41)andTheorem2. Thisderivation

justies the name of this setion.

Proof of Theorem 17. Wegiveanexpliitdesriptionof

µ = ψ t (λ)

^{. Let}

a = M 2 (λ) = − min V (λ)

^and

V ~ = φ t (λ)

^bethe

V t

^-odingof

λ

^{. Wealso}

set

µ = ~µ := ( V ~ i − t + i + a : i ∈ {1, · · · , t})

^{tobe the (ordered) parts}

of a partition, trailing with at least one zero. The temporary arrow

notation for vetors is meant to emphasize that it is now sorted by

dereasing order. The partition

µ

^{may be rewritten as}

~µ = V ~ + a~ 1 − ~b

where

~b = (t − 1, · · · , 0)

^and

~ 1 = (1, · · · , 1)

^{. Wealso know that}

1

2t V ~ · V ~ = |λ| +

t ² − 1 24

(byEquation(1)),andthat

V ~ · ~ 1 = 0

^{(byEquation(8)). Thestatement}

to be proved is then

|λ| = −(~µ · ~ 1) ~µ · ~ 1 + t + t ²

2t ² + ~µ · ~µ 2t + 1

t

~ 1 + ~b

· ~µ.

But this follows readily from

~b · ~b = t(t − 1)(2t − 1)/6

^and

~b · ~ 1 = t(t − 1)/2

^.

We now move on to Equation (41). Dene

τ!(i) = Q i

j=1 τ (j)

^{. On}

the other hand,we have

Y

∈µ

τ(c + t) = Q t

i=1 τ !(µ i + t − i) Q t−1

i=1 τ(i) ^t−i .

(42)

The partition

µ

^{an be viewed in the exploded tableau by the set}

{(x, y ) ∈ V 1 × ([M 2 , −M 1 ] \ −V 1 ) | x + y > 0}

^{. Hene}

(43)

Y

∈µ

τ (h ) = Y

(x,y)∈V 1 ×([M 2 ,−M 1 ]\−V 1 ) x+y>0

τ (x + y)

= Y

x∈V 1

Y

y∈([M 2 ,−M 1 ]\−V 1 ) x+y>0

τ (x + y) = Q

x∈V ¹ τ !(M 2 + x) Q

0≤i,j≤t−1 τ(v i − v j ) .

Equations (42) and (43), together with Theorem 1 sue to estab-

lish (41).

Referenes

[AKMV05℄ MinaAganagi,AlbrehtKlemm, MarosMariño, andCumrun Vafa.

Thetopologialvertex.Comm. Math.Phys.,254(2):425478,2005.

[And98℄ GeorgeE.Andrews.Thetheoryofpartitions.CambridgeMathematial

Library.CambridgeUniversityPress,Cambridge,1998.Reprintofthe

1976original.

[CKW09℄ Emily Clader, Yvonne Kemper, and Matt Wage. Launarity of er-

tainpartition-theoretigeneratingfuntions. Pro. Amer. Math. So.,

137(9):29592968,2009.

[CLPS08℄ K. Carde, J. Loubert, A. Potehin, and A. Sanborn. Proof of Han's

HookExpansionConjeture.ArXive-prints, August2008,0808.0928.

[CW09℄ Dan Collins and SallyWolfe. Congruenesfor Han's generating fun-

tion.Involve,2(2):225236,2009.

[GKS90℄ FrankGarvan,DongsuKim, andDennisStanton.Cranksand

t

^-ores.

Invent.Math.,101(1):117,1990.

[Han09℄ Guo-NiuHan.Someonjeturesandopen problemsonpartitionhook

lengths.Experiment.Math.,18(1):97106,2009.

[Han10℄ Guo-Niu Han. The Nekrasov-Okounkov hook length formula: rene-

ment,elementaryproof,extensionandappliations.Ann.Inst.Fourier

(Grenoble),60(1):129,2010.

[HJ11℄ Guo-Niu Han and Kathy Q. Ji. Combining hook length formulas

andBG-ranksforpartitions viatheLittlewood deomposition.Trans.

Amer.Math.So.,363(2):10411060,2011.

[IKS10℄ Amer Iqbal, Can Kozçaz, and Khurram Shabbir. Rened topologial

vertex,ylindri partitionsand

U (1)

^{adjointtheory. NulearPhys. B,}

838(3):422457,2010.

[INRS10℄ A.Iqbal,S.Nazir,Z.Raza,andZ.Saleem.GeneralizationsofNekrasov-

OkounkovIdentity.ArXive-prints,November2010,1011.3745.

[Knu73℄ Donald E. Knuth. The art of omputer programming. Volume 3.

Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills,

Ont.,1973.Sortingandsearhing,Addison-WesleySeriesinComputer

SieneandInformationProessing.

[Las03℄ Alain Lasoux. Symmetri funtions and ombinatorial operators on

polynomials,volume99ofCBMSRegionalConfereneSeriesinMath-

ematis.PublishedfortheConfereneBoardoftheMathematialSi-

enes,Washington,DC,2003.

[Le91℄ Z. L. Lebenzon. A simple proof of the Madonald identities for the

series

A

^{. F}unktsional. Anal. iPrilozhen.,25(3):1923,95, 1991.trans- lationin Funt. Anal. Appl.25(1991),no.3,180183(1992).

[Ma72℄ I.G.Madonald.AnerootsystemsandDedekind's

η

^{-funtion.Invent.}

Math.,15:91143,1972.

[Ma95℄ I. G. Madonald. Symmetri funtions and Hall polynomials. Oxford

Mathematial Monographs. The Clarendon Press Oxford University

Press, New York, seond edition, 1995. With ontributions by A.

Zelevinsky,OxfordSienePubliations.

[Mil85℄ S.C.Milne.AnelementaryproofoftheMadonaldidentitiesfor

A ⁽¹⁾ l

^.

Adv. inMath.,57(1):3470,1985.

[NO06℄ NikitaA.NekrasovandAndrei Okounkov.Seiberg-Wittentheoryand

randompartitions.InTheunity ofmathematis,volume244ofProgr.

Math.,pages525596.BirkhäuserBoston,Boston,MA,2006.

[Ols10℄ GrigoriOlshanski.Planherelaverages: remarksonapaperbyStanley.

Eletron.J. Combin.,17(1):ResearhPaper43,16,2010.

[ORV06℄ Andrei Okounkov, Nikolai Reshetikhin, and Cumrun Vafa. Quantum

Calabi-Yauandlassialrystals.InTheunityofmathematis,volume

244ofProgr. Math., pages597618.Birkhäuser Boston,Boston,MA,

2006.

[Rob58℄ G.deB.Robinson.AremarkbyPhilipHall.Canad. Math.Bull.,1:21

23,1958.

[Sta99℄ Rihard P. Stanley. Enumerative ombinatoris. Vol. 2, volume 62of

Cambridge Studies in Advaned Mathematis. Cambridge University

Press,Cambridge,1999.With aforewordbyGian-CarloRotaandap-

pendix1bySergeyFomin.

[Sta10℄ RihardP.Stanley.Someombinatorialpropertiesofhooklengths,on-

tents,andpartsofpartitions. Ramanujan J.,23(1-3):91105,2010.

[Wes06℄ Brue W. Westbury. Universalharatersfrom theMadonald identi-

ties.Adv.Math., 202(1):5063,2006.

Department of Mathematis, ETH Zürih, Rämistrasse 101, 8092

Zürih,Switzerland

E-mailaddress: pdehayemath.ethz.h

Institut deReherhe MathématiqueAvanée,UniversitédeStras-

bourg etCNRS, 7 rue René-Desartes,67084 Strasbourg, Frane

E-mailaddress: guoniu.hanunistra.fr