aa r X i v : . [ m a t h . N T ] J u l DISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS WILLIAM CRAIG AND ANNA PUN
Abstract.
Partitions, the partition function p ( n ), and the hook lengths of their Ferrers-Youngdiagrams are important objects in combinatorics, number theory and representation theory. Forpositive integers n and t , we study p et ( n ) (resp. p ot ( n )), the number of partitions of n with aneven (resp. odd) number of t -hooks. We study the limiting behavior of the ratio p et ( n ) /p ( n ),which also gives p ot ( n ) /p ( n ) since p et ( n ) + p ot ( n ) = p ( n ). For even t , we show thatlim n →∞ p et ( n ) p ( n ) = 12 , and for odd t we establish the non-uniform distributionlim n →∞ p et ( n ) p ( n ) =
12 + 12 ( t +1) / if 2 | n, − ( t +1) / otherwise.Using the Rademacher circle method, we find an exact formula for p et ( n ) and p ot ( n ), and thisexact formula yields these distribution properties for large n . We also show that for sufficientlylarge n , the signs of p et ( n ) − p ot ( n ) are periodic. Introduction and Statement of Results
For a positive integer n , a partition λ = ( λ ≥ λ ≥ · · · ≥ λ k ) of n is defined as a weaklydecreasing sequence of positive integers whose sum is n . For every n , the partition function p ( n ) is defined as the number of partitions of n . The study of partitions and the partitionfunction has been of historical importance both in combinatorics and number theory. Oneof the most important tools used in the combinatorial study of partitions are Ferrers-Youngdiagrams , which are a geometric representation of a partition ( λ ≥ · · · ≥ λ k ) as a grid of k rowsof left-aligned square cells, with row i containing λ i cells. For instance, the partition (5 , ,
1) of10 has the Ferrers-Young diagram .One important set of quantities associated with the Ferrers-Young diagram of a partition λ arethe hook numbers H λ ( i, j ), the number of cells ( a, b ) in the diagram of λ such that i ≤ a and j ≤ b . It is common to represent all hook numbers of λ by placing H λ ( i, j ) in the interior ofeach cell of the diagram. For instance, doing so for the partition λ = (5 , ,
1) yields7 5 4 3 15 3 2 11 .
For a partition λ of n , a Young tableau of shape λ is a labelling of each cell in the Ferrers-Young diagram of shape λ by a distinct number from 1 through n . A standard Young tableauis a Young tableau such that entries are increasing in rows (from left to right) and columns(from top to bottom). These tableau are central objects in the representation theory of S n [11, 7, 10], as the irreducible representations of S n are in one-to-one correspondence with thepartitions of n . For example, given a partition λ , the degree of the irreducible representationof S n corresponding to λ is the number of the standard Young tableaux of shape λ , denoted by f λ . The hook numbers play an important role in calculating f λ via the Frame-Thrall-Robinsonhook length formula [2]: f λ = n ! Q ( i,j ) H λ ( i, j ) . Hook numbers and related quantities arise in other ways in combinatorics and representationtheory. The multiset of hook numbers of λ , denoted H ( λ ), appears in the famous Nekrosov-Okounkov formula (formula (6.12) in [8]) X λ ∈P x | λ | Y h ∈H ( λ ) (1 − z/h ) = Y k ≥ (1 − x k ) z − , which represents an extraordinary generalization of identities of Euler and Jacobi. Restrictingto hook numbers that are multiples of a fixed t is natural and leads to the study of t -corepartitions, which are defined as partitions for which no hook number is a multiple of t . Thenumbers of t -core partitions have been the subject of much research, including the famous resultof Granville and Ono proving that t -core partitions exist for all positive integers when t ≥ t -cores is fundamental to the modular representation theory of symmetric groups.For the positive integer t ≥ λ a partition of any positive integer, we wish to study theparity of H t ( λ ), defined to be the number of hook numbers of λ which are multiples of t . Todo so, we define the partition functions p et ( n ) and p ot ( n ) to count the number of partitions λ of n for which the t -hook number H t ( λ ) is even and odd, respectively. That is, we have p et ( n ) := { λ ⊢ n : H t ( λ ) ≡ } ,p ot ( n ) := { λ ⊢ n : H t ( λ ) ≡ } . (1.1)Of particular interest here is the distribution of the parity of H t ( λ ). To study the distributionproperties of the t -hook numbers, we define the functions δ et ( n ) := p et ( n ) p ( n ) and δ ot ( n ) := p ot ( n ) p ( n ) .There have been recent developments in studying distribution properties that make use of theproperties of t -hooks. For example, Peluse has proved that the density of odd values in thecharacter table of S n goes to zero as n → ∞ [9]. We perform a similar distribution analysis forthe parity of the number of t -hooks.By definition, we have p et ( n ) + p ot ( n ) = p ( n ), and so δ et ( n ) + δ ot ( n ) = 1. Naively, one wouldexpect an even distribution of parities as n becomes large, that is, we would expect that δ et ( n ) → / δ ot ( n ) → /
2. Numerically, this initial speculation receives support for smallvalues of t like t = 2 , ,
6, and 8, as the following table suggests.
ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 3 t δ et (100) δ et (1000) δ et (10000) · · · ∞ · · · · · · · · · · · · However, numerical evidence below for the cases t = 3 , , t δ et (100) δ et (500) δ et (1000) δ et (1500) · · · ∞ · · · · · · · · · · · · t δ et (101) δ et (501) δ et (1001) δ et (1501) · · · ∞ · · · · · · · · · · · · In this paper, we prove two main theorems which explain this data, and offer exact valueson the limiting values in these distributions.
Theorem 1.1.
Assuming the notation above, the following are true.1) If t is a positive even integer, then lim n →∞ δ et ( n ) = lim n →∞ δ ot ( n ) = 12 .
2) If t is a positive odd integer, then we have lim n →∞ δ et ( n ) =
12 + 12 ( t +1) / if | n, − ( t +1) / if ∤ n, and lim n →∞ δ ot ( n ) = − ( t +1) / if | n,
12 + 12 ( t +1) / if ∤ n. We also study the sign pattern of p et ( n ) − p ot ( n ), for n → ∞ , which determines when p et ( n ) >p ot ( n ) and p ot ( n ) > p et ( n ). Theorem 1.2.
For t > a fixed positive integer, write t = 2 s ℓ with integers s, ℓ such that s ≥ and ℓ odd. Then for sufficiently large n , the signs of p et ( n ) − p ot ( n ) are periodic with period s +1 .In particular, the sign of p et ( n ) − p ot ( n ) is alternating when t is odd and for sufficiently large n . WILLIAM CRAIG AND ANNA PUN
Example.
For example, if t = 6 , we find, for sufficiently large n , that p et ( n ) − p ot ( n ) ( > if n ≡ , < if n ≡ , . This paper is organized as follows. In Section 2, we state an exact formula for the difference A t ( n ) := p et ( n ) − p ot ( n ) (see Theorem 2.1), and prove the exact formula by applying the circlemethod of Rademacher to the generating function of p et ( n ) − p ot ( n ). In Section 3, we usethis exact formula to analyze the limiting behavior of the distributions of δ et ( n ) and δ ot ( n ).Although the consequences regarding distribution properties only require an asymptotic formulafor p et ( n ) − p ot ( n ), for completeness we provide the exact formula.2. Exact Formulas
Auxiliary Partition Functions.
Since p et ( n ) + p ot ( n ) = p ( n ), A t ( n ) can serve a usefulauxiliary role in our study. The utility of the function A t ( n ) comes from the generating function G t ( x ) := X n ≥ A t ( n ) x n = Y k ≥ (1 − x tk ) t (1 − x tk ) t (1 − x tk ) t (1 − x k ) , (2.1)proven in Corollary 5.2 of [6]. Using the generating function given in (2.1), we prove thefollowing exact formula for A t ( n ), given as a Rademacher-type infinite series expansion. Theorem 2.1. If n, t are positive integers with t > , then A t ( n ) = X k ∈ Z + ( k,t )=1 k odd X ≤ h 1) exp (cid:18) πi · (cid:0) (4 t ) − (cid:1) k Hj − nhk (cid:19) · (cid:18) t − j n − (cid:19) I π k r ( t − j )(24 n − t ! + X k ∈ Z + k ( k,t ) ≡ X ≤ h Here we illustrate Theorem 2.1 using the numbers A t ( m ; n ) , which denotes the for-mula summed over k = 1 , , . . . , m . Theorem 2.1 is the conclusion that lim m →∞ A t ( m, n ) = A t ( n ) .We offer some examples in the table below. n m 10 100 1000 · · · ∞ ≈ . ≈ . ≈ . · · · ≈ . ≈ . ≈ . · · · Table 1. Values of A ( m, n )This result gives the following corollary. Corollary 2.2. For t > a fixed positive integer, write t = 2 s ℓ with integers s, ℓ such that s ≥ and ℓ is odd. Then as n → ∞ we have A t ( n ) ∼ π s + t (cid:18) · s n − (cid:19) I (cid:18) π p (1 + 3 · s )(24 n − · s +1 (cid:19) X First we note that X The Circle Method. The approach that will be utilized in the proof of Theorem 2.1 iscommonly referred to as the “circle method”. Initially developed by Hardy and Ramanujanand refined by Rademacher, the circle method has been employed with great success for thepast century in additive number theory. The crowning achievement of the circle method lies inproducing an exact formula for the partition function p ( n ), and it has been utilized to producesimilar exact formulas for variants of the partition function. A helpful and instructive sketch ofthe application of Rademacher’s circle method to the partition function p ( n ) is given in Chapter5 of [1]. Here, we will provide a summary of the circle method, in order to clarify the key stepsand the general flow of the argument.The function A t ( n ) has as its generating function G t ( x ). Our objective is to use G t ( x ) toproduce an exact formula for A t ( n ). Consider the Laurent expansion of G t ( x ) /x n +1 in thepunctured unit disk. This function has a pole at x = 0 with residue p ( n ) and no other poles.Therefore, by Cauchy’s residue theorem we have A t ( n ) = 12 πi Z C G t ( x ) x n +1 dx, (2.2)where C is any simple closed curve in the unit disk that contains the origin in its interior. Thetask of the circle method is to choose a curve C that allows us to evaluate this integral, and thisis achieved by choosing C to lie near the singularities of G t ( x ), which are the roots of unity. Forevery positive integer N and every pair of coprime non-negative integers 0 ≤ h < k ≤ N , wecan choose a special contour C in the complex upper half-plane, and divide this contour intoarcs C h,k near the roots of unity e πih/k . Then integration over C can be expressed as a finitesum of integrals over the arcs C h,k , and elementary functions ψ h,k are chosen with behaviorsimilar to G t ( x ) near the singularity e πih/k . This is done by using properties of G t ( x ) deducedfrom the functional equation of the Dedekind eta function η ( τ ) := e πiτ/ Q n ≥ (1 − e πinτ ) andthe relation between G t ( x ) and η ( τ ) given by G t ( e πiτ ) = η ( tτ ) t η (4 tτ ) t η ( τ ) η (2 tτ ) t . (2.3)The error created by replacing G t ( x ) by ψ h,k ( x ) can be estimated, and the integrals of the ψ h,k along C h,k evaluated. This procedure produces estimates that can be used to formulate aconvergent series for A t ( n ). ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 7 Transformation formula for G t ( x ) . We first recall the transformation formula for thegenerating function of p ( n ) (see, for example [5] or P.96 in [1]). Theorem 2.3. Let x = exp (cid:18) π · hi − zk (cid:19) and x ′ = exp (cid:18) π · Hi − z − k (cid:19) , where Re( z ) > , k > , ( h, k ) = 1 and hH ≡ − k ) . If F ( x ) := ∞ Q m =1 − x m , then F ( x ) = √ z · exp (cid:18) π ( z − − z )12 k + πis ( h, k ) (cid:19) F ( x ′ ) , (2.4) where s ( h, k ) is the Dedekind sum defined by s ( h, k ) = k X u =1 (cid:16)(cid:16) uk (cid:17)(cid:17)(cid:18)(cid:18) huk (cid:19)(cid:19) and (( m )) = if m ∈ Z ,m − [ m ] − otherwise . By (2.3), G t ( x ) can be expressed in terms of F ( x ): G t ( x ) = F ( x ) (cid:2) F (cid:0) x t (cid:1)(cid:3) t (cid:2) F (cid:0) x t (cid:1)(cid:3) t (cid:2) F (cid:0) x t (cid:1)(cid:3) t . We can therefore apply Theorem 2.3 to find a similar transformation formula for G t ( x ). Wemust first define some notation. Let h, k be integers with k > h, k ) = 1. Let D = ( t, k ), K = kD , T = tD (so ( K, T ) = 1). Let T ′ be an integer such that T T ′ ≡ K ). Notethat ( T h, K ) = 1 because ( T, K ) = 1 and ( h, K ) = 1. Let H be an integer so that hH ≡ − k ). In light of the definition of T ′ , it will be useful to make a similar definition for anyparameter or constant. For instance, the integer 2 ′ will be defined so that 2(2 ′ ) ≡ K )when K is understood from context. Lemma 2.4. For j = 1 , , , define x := x t , x := x t , and x := x t . Then the followingtransformation formulas for F ( x j ) hold.(a) When K is odd, for j = 1 , , we have the transformation formulas F ( x j ) = √ j − T z · exp (cid:20) π K (cid:18) j − T z − j − T z (cid:19) + πis (2 j − T h, K ) (cid:21) F ( x ′ j ) hold, where x ′ j = exp (cid:20) π (cid:18) H j K i − j − KT z (cid:19)(cid:21) and constants H j are fixed so that j − T hH j ≡ − K ) .(b) Suppose K ≡ , and define K = 4 K + 2 . Then we have the transformationformula F ( x ) = √ T z · exp " π K (cid:18) T z − T z (cid:19) + πis ( T h, K ) F ( x ′ ) WILLIAM CRAIG AND ANNA PUN and for j = 2 , we have F ( x j ) = √ j − T z · exp " π K (cid:18) j − T z − j − T z (cid:19) + πis (2 j − T h, K + 1) F ( x ′ j ) , where x ′ = exp (cid:20) π (cid:18) T ′ HK i − KT z (cid:19)(cid:21) , x ′ = exp (cid:20) π (cid:18) T ′ H K + 1 i − K + 1) T z (cid:19)(cid:21) , and x ′ =exp (cid:20) π (cid:18) ( K + 1) T ′ H K + 1 i − KT z (cid:19)(cid:21) . (c) Suppose K ≡ , and define K = 4 K . Then for j = 1 , , we have the transfor-mation formulas F ( x j ) = √ T z · exp " − j π K (cid:18) T z − T z (cid:19) + πis ( T h, − j K ) F ( x ′ j ) , where x ′ j = exp " π (cid:18) T ′ H − j K i − − j K T z (cid:19) for all j .Proof. We only prove (a), as (b) and (c) follow by making appropriate changes. By definition, x = exp (cid:18) π · T hi − T zK (cid:19) . Since ( T h, K ) = 1 and T hH ≡ − K ), from Theorem 2.3we may infer that F ( x ) = √ T z · exp " π K (cid:18) T z − T z (cid:19) + πis ( T h, K ) F ( x ′ )for x ′ = exp " π (cid:18) H K i − KT z (cid:19) . Defining the constants H j = (2 j − ) ′ T ′ H , the correspondingtransformation formulas for F ( x ) and F ( x ) follow as well. (cid:3) Now, to derive further transformation formulas, we use the changes of variables y :=exp (cid:20) π (cid:18) ′ T ′ Hk i − kT z (cid:19)(cid:21) when K is odd, y := exp (cid:20) π (cid:18) ( K + 1) T ′ Hk i − kT z (cid:19)(cid:21) when K ≡ y := exp (cid:20) π (cid:18) T ′ Hk i − kT z (cid:19)(cid:21) when K ≡ K is odd choose an integer M satisfying 4 · ′ T ′ T = M K + 1, if K ≡ M so that T T ′ = M K + 1, and if K ≡ M so that T T ′ = M K + 1. Then the definitions of the x ′ j imply the identities x ′ = y T · exp (cid:18) − πi · M HD (cid:19) = y T · exp (cid:20) − π · M ( K + 2) + 12 D Hi (cid:21) = y T exp (cid:18) − π · M HD i (cid:19) ,x ′ = y D = − y D = y D , x ′ = y D = y D = y D , x ′ = y D = y D = y D . We now proceed to use these in order to derive transformation formulas for G t ( x ). Usingthese identities for each case, the transformation laws ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 9 G t ( x ) = t/ √ z exp (cid:20) π k (cid:18) − D z − z (cid:19)(cid:21) w ( t, h, k ) F ( x ′ )[ F ( x ′ )] t [ F ( x ′ )] t [ F ( x ′ )] t if K ≡ , t/ √ z exp (cid:20) π k (cid:18) D z − z (cid:19)(cid:21) w ( t, h, k ) F ( x ′ ) (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t if K ≡ √ z exp (cid:20) π k ( z − − z ) (cid:21) w ( t, h, k ) F ( x ′ ) (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t if K ≡ w ( t, h, k ) := exp h πi (cid:0) s ( h, k )+3 ts (2 T h, K ) − ts ( T h, K ) − ts (4 T h, K ) (cid:1)i , w ( t, h, k ) :=exp h πi (cid:0) s ( h, k )+3 ts ( T h, K +1) − ts ( T h, K ) − ts (2 T h, K +1) (cid:1)i , and w ( t, h, k ) := exp (cid:2) πi (cid:0) s ( h, k )+3 ts ( T h, K ) − ts ( T h, K ) − ts ( T h, K ) (cid:1)(cid:3) . We can derive series expansions of F ( x ′ ) (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t by changing variables to each y j . These expansions are defined by ∞ X n =0 c j ( n, M j , H, D ) y nj := F ( x ′ ) (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t (cid:2) F ( x ′ ) (cid:3) t for j = 1 , , c j ( n, M j , H, D ). To make use of these transformation laws, we require exact formulasfor w j ( t, h, k ) for j = 1 , , 3. We first state a result by Hagis [4] that helps in this direction. Proposition 2.5. If u, v are nonzero integers, then exp( πis ( u, v )) = (cid:18) uv (cid:19) i v − exp (cid:20) πiq u − u ′ gv (cid:21) if v is odd, where g = (3 , v ) , (cid:18) vu (cid:19) i b ( v +1)2 exp (cid:20) πiq u − u ′ gv (cid:21) if v is even, where g = 8(3 , v ) , b ≡ u ′ (mod 8) , where uu ′ ≡ − gv ) , f q ≡ gv ) , and f g = 24 . Here (cid:18) vu (cid:19) is the Jacobi symbol. As before, let t, h, k be integers with t, k > h, k ) = 1. Let D = ( t, k ), K = kD , and T = tD . Let H, T ′ be integers such that T T ′ ≡ K ) and hH ≡ − k ). Forrelatively prime integers a, n , we also define ( a − ) n to be any multiplicative inverse of a modulo n . Lemma 2.6. Suppose K is odd. Define the constants s k := ( if ∤ k, if | k, α k := ( if ∤ k, if | k, δ K := ( if | K, if ∤ K, ˆ s k := 9 − s k , ˆ α k := 4 − α k , γ k := s k α k , and ˆ γ k := ˆ s k ˆ α k . Then w ( t, h, k ) := β (1) t,h,k exp (cid:20) πi ( γ − k ) ˆ γ k k ( h + ( h − ) ˆ γ k k ) − ˆ α k ((4 s k T h ) − ) δ K K Dt ˆ γ k k (cid:21) , where β (1) t,h,k := (cid:18) hk (cid:19)(cid:18) K (cid:19) t i k − if ∤ k and β (1) t,h,k := (cid:18) kh (cid:19)(cid:18) K (cid:19) t i b ( k +1)2 if | k , where b is chosenso that b ≡ − ( h − ) α k k (mod 8) .Proof. Recall that w ( t, h, k ) is defined for K odd by w ( t, h, k ) = exp h πi (cid:0) s ( h, k ) + 3 ts (2 T h, K ) − ts ( T h, K ) − ts (4 T h, K ) (cid:1)i . Applying Proposition 2.5 and simplifying, we have w ( t, h, k ) = exp (cid:2) πis ( h, k ) (cid:3)(cid:18) K (cid:19) t exp (cid:20) πiqgK h − t (2 T h ) ′ + 2 t ( T h ) ′ + t (4 T h ) ′ i(cid:21) , where g = (3 , K ) , qg ≡ gK ) , and ( a ) ′ is defined by a ( a ) ′ ≡ − gK ) for any a . Since ( T h ) ′ ≡ T h ) ′ ≡ T h ) ′ (mod gK ), we may suppose that ( T h ) ′ = 4(4 T h ) ′ and(2 T h ) ′ = 2(4 T h ) ′ . Therefore, w ( t, h, k ) = exp (cid:2) πis ( h, k ) (cid:3)(cid:18) K (cid:19) t exp (cid:20) πiqgK h − t · T h ) ′ + 2 t · (4 T h ) ′ + t (4 T h ) ′ i(cid:21) = (cid:18) K (cid:19) t exp (cid:2) πis ( h, k ) (cid:3) exp (cid:20) πiq · t (4 T h ) ′ gK (cid:21) . Here, we split into cases. First, suppose that 3 ∤ K . Then g = 1 and q = (24 − ) K , whichimplies 4 T h (4 T h ) ′ ≡ − K ). Noting that 4 T h · (4 − ) K T ′ H ≡ − K ), we may take(4 T h ) ′ = (4 − ) K T ′ H , and so w ( t, h, k ) = (cid:18) K (cid:19) t exp (cid:2) πis ( h, k ) (cid:3) exp (cid:20) πi · t (32 − ) K T ′ HK (cid:21) . Suppose that k is odd. If 3 ∤ k , then by Proposition 2.5 taking u = h and v = k , we have g = 1 , f = 24 , q = (24 − ) k , u ′ = H andexp (cid:2) πis ( h, k ) (cid:3) = (cid:18) hk (cid:19) i k − exp (cid:20) πi (24 − ) k h − Hk (cid:21) . Simplifying then yields the appropriate identity, all other cases are derived similarly. (cid:3) Lemma 2.7. Suppose K ≡ . Define β (2) t,h,k := (cid:18) kh (cid:19)(cid:18) K + 1 (cid:19) t ( − tK · i tb + B ( k +1)2 with b ≡ − (( T h ) − ) ,K ) K (mod 8) , B ≡ − ( h − ) α k k , K = 4 K + 2 . Then if ∤ K we have w ( t, h, k ) = β (2) t,h,k exp (cid:20) πi (3 − ) k h + ( h − ) k k (cid:21) · exp (cid:20) πi (cid:18) − (3 − ) K (cid:2) t ( K + 1) T ′ H + t (( T h ) − ) K (cid:3) K (cid:19)(cid:21) , ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 11 and otherwise w ( t, h, k ) = β (2) t,h,k exp (cid:20) πi h + ( h − ) k k (cid:21) · exp " πi K + 1) m h tT h + 5 t (cid:0) (2 T h ) − (cid:1) K +1) i + 5 t (cid:0) (2 T h ) − (cid:1) K +1) − t (( T h ) − ) K K ! . Proof. Recall that w ( t, h, k ) is defined for K ≡ w ( t, h, k ) = exp h πi (cid:0) s ( h, k ) + 3 ts ( T h, K + 1) − ts ( T h, K ) − ts (2 T h, K + 1) (cid:1)i . As in Lemma 2.6, we apply Proposition 2.5 on each of s ( T h, K +1) , s ( T h, K ) and s (2 T h, K +1), and after simplifying we have w ( t, h, k ) = (cid:18) K + 1 (cid:19) t ( − tK · i tb exp (cid:2) πis ( h, k ) (cid:3) exp (cid:20) πiq · tT h − t ( T h ) ′ + t (2 T h ) ′ g (2 K + 1) (cid:21) · exp (cid:20) πiq · − tT h − t (( T h ) − ) g K g K (cid:21) , where g = (3 , K + 1) , g = 8(3 , K ) , b ≡ − (( T h ) − ) g K (mod 8). Furthermore, T h ( T h ) ′ ≡ T h (2 T h ) ′ ≡ − g (2 K + 1)) which means we can set ( T h ) ′ = 2(2 T H ) ′ . Then we have w ( t, h, k ) = (cid:18) K + 1 (cid:19) t ( − tK · i tb exp (cid:2) πis ( h, k ) (cid:3) · exp (cid:20) πi (cid:18) q (cid:0) tT h − t (2 T h ) ′ (cid:1) g (2 K + 1) − q (cid:0) tT h + 2 t (( T h ) − ) g K (cid:1) g K (cid:19)(cid:21) . Suppose 3 ∤ K . Then g = 1 , q = (24 − ) K +1 , g = 8 , q = (3 − ) K , b ≡ − (( T h ) − ) K (mod 8)all hold. Note that 2 T h · ( K +1) T ′ H ≡ − K +1), we can choose (2 T h ) ′ = ( K +1) T ′ H .Simplifying, we obtain w ( t, h, k ) = (cid:18) K + 1 (cid:19) t ( − tK · i tb exp (cid:2) πis ( h, k ) (cid:3) · exp (cid:20) πi (cid:18) (3 − ) K +1 (cid:2) tT h − t ( K + 1) T ′ H (cid:3) − (3 − ) K (cid:0) tT h + t (( T h ) − ) K (cid:1) K (cid:19)(cid:21) . Since 3 · (3 − ) K ≡ K + 1), we can set (3 − ) K +1 = (3 − ) K , and so w ( t, h, k ) = (cid:18) K + 1 (cid:19) t ( − tK i tb exp (cid:2) πis ( h, k ) (cid:3) · exp (cid:20) πi − (3 − ) K (cid:2) t ( K + 1) T ′ H + t (( T h ) − ) K (cid:3) K (cid:21) . The proof when 3 | K is similar. (cid:3) Lemma 2.8. Let | K , and set K = 4 K . Further define b ≡ ( h − ) k (3 ,k ) (mod 8) , b ≡ (cid:0) ( T h ) − (cid:1) K (3 ,K ) (mod 8) , b ≡ (cid:0) ( T h ) − (cid:1) K (3 ,K ) (mod 8) . Define β (3) t,h,k := (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+ tb − K t (1 − K if ∤ K , ∤ K , (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K if ∤ K , | K , (cid:18) kh (cid:19)(cid:18) T h (cid:19) t i b k +1) − tb K if | K . Then there is an integer m ∈ Z for which w ( t, h, k ) = β (3) t,h,k exp (cid:20) πi (cid:18) ( α − k )( h + ( h − ) ˆ γ k k ) − km (cid:2) T h + (( T h ) − ) K (cid:3) ˆ γ k k (cid:19)(cid:21) if ∤ K , | K ,β (3) t,h,k exp (cid:20) πi ( α − k )( h + ( h − ) ˆ γ k k )ˆ γ k k (cid:21) otherwise . Proof. Recall that w ( t, h, k ) is defined for K ≡ w ( t, h, k ) = exp h πi (cid:0) s ( h, k ) + 3 ts ( T h, K ) − ts ( T h, K ) − ts ( T h, K ) (cid:1)i . As in Lemma 2.6, we apply Proposition 2.5 on s ( h, k ) , s ( T h, K ) , s ( T h, K ) and simplifying weobtain w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t i b k +1)+3 tb K − tb K +1)2 exp (cid:20) πiq h + ( h − ) g k g k (cid:21) · exp " πiq t h T h + (cid:0) ( T h ) − (cid:1) g K i g K exp (cid:20) πiq − t h T h + (cid:0) ( T h ) − (cid:1) g K i g K (cid:21) exp (cid:2) − ts ( T h, K ) (cid:3) , where g = 8(3 , k ) , g = 8(3 , K ) = 8(2 , K ) , g = 8(3 , K ) = 8(3 , K ) = g , q g ≡ g k ) , q g ≡ g K ) , q g ≡ g K ) , b ≡ ( h − ) g k (mod 8) , b ≡ (cid:0) ( T h ) − (cid:1) g K (mod 8), and b ≡ (cid:0) ( T h ) − (cid:1) g K (mod 8) . We now split into cases based on the parity of K . First, suppose K is odd. Immediately, w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K exp (cid:20) πiq h + ( h − ) g k g k (cid:21) · exp " πi q t h T h + (cid:0) ( T h ) − (cid:1) g K i g K − q t h T h + (cid:0) ( T h ) − (cid:1) g K i g K − q t h T h + (cid:0) ( T h ) − (cid:1) g K i g K ! , where g = (3 , K ) , q g ≡ g K ). If we additionally suppose 3 ∤ K , then 3 ∤ K and 3 ∤ K , so g = g = 8 , g = 1 , q = (3 − ) K = (3 − ) K , q = (3 − ) K , and q = (24 − ) K . Notingthat 3 q ≡ K ), we set q = (3 − ) K = q . Similarly, T h · (cid:0) ( T h ) − (cid:1) K ≡ K ), ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 13 so we set (cid:0) ( T h ) − (cid:1) K and (cid:0) ( T h ) − (cid:1) K as (cid:0) ( T h ) − (cid:1) K , which implies b = b ≡ (cid:0) ( T h ) − (cid:1) K (mod 8). Then simplifying, we obtain w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+ tb − K t (1 − K exp (cid:20) πi q (cid:0) h + ( h − ) g k (cid:1) g k (cid:21) · exp " πi (cid:0) · (3 − ) K − − ) K (cid:1) t h T h + (cid:0) ( T h ) − (cid:1) K i K ! . Since 3 · (3 − ) K ≡ K ), we can set (3 − ) K = (3 − ) K . Therefore, w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+ tb − K t (1 − K exp (cid:20) πi q (cid:0) h + ( h − ) g k (cid:1) g k (cid:21) , where g = 8 , q = (3 − ) k if 3 ∤ k and g = 24 , q = 1 if 3 | k . Similarly, if 3 | K , then g = g = g = 24 , g = 3 , q = q = q = 1 , q = (8 − ) K . This implies w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K exp (cid:20) πi h + ( h − ) k k (cid:21) · exp " πi t h T h + (cid:0) ( T h ) − (cid:1) K i − t h T h + (cid:0) ( T h ) − (cid:1) K i K − t (8 − ) K h T h + (cid:0) ( T h ) − (cid:1) K i K ! . Since T h · (cid:0) ( T h ) − (cid:1) K ≡ K ), we may fix (cid:0) ( T h ) − (cid:1) K = (cid:0) ( T h ) − (cid:1) K , so w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K exp (cid:20) πi h + ( h − ) k k (cid:21) · exp " πi t h T h + (cid:0) ( T h ) − (cid:1) K i K − t (8 − ) K h T h + (cid:0) ( T h ) − (cid:1) K i K ! = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K exp (cid:20) πi h + ( h − ) k k (cid:21) · exp " πi t h T h + (cid:0) ( T h ) − (cid:1) K i K − t (3 K m + 1) h T h + (cid:0) ( T h ) − (cid:1) K i K ! for some integer m . Since T h · (cid:0) ( T h ) − (cid:1) K ≡ K ), we set (cid:0) ( T h ) − (cid:1) K = (cid:0) ( T h ) − (cid:1) K without loss of generality. So, w ( t, h, k ) = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K exp (cid:20) πi h + ( h − ) k k (cid:21) · exp " πi − K m h T h + (cid:0) ( T h ) − (cid:1) K i K ! = (cid:18) kh (cid:19)(cid:18) K T h (cid:19) t (cid:18) T hK (cid:19) t i b k +1)+3 tb K − tb K +1)+ t (1 − K · exp " πi h + ( h − ) k − km h T h + (cid:0) ( T h ) − (cid:1) K i k ! . This completes the proof of all formulae for K odd, and the result follows mutatis mutandis for K even. (cid:3) Using these exact formulas for w j , we compute the magnitudes of each w j . Proposition 2.9. Let t, h, k be integers such that t, k > and ( h, k ) = 1 . Then for r = 1 , , and integers M, n, j where n > and j ≥ , we may write w r ( t, h, k ) = C r ( t, h, k ) exp (cid:20) πi q [ U h + V ( h − ) gk ] gk (cid:21) , where | C r ( t, h, k ) | = 1 , and U, V, g, q ∈ Z satisfy g ∈ { , , , } and g q ≡ gk ) .Furthermore, C r ( t, h, k ) depends on k and t only if we fix integers a, d with d odd such that h ≡ a (mod k ) and h ≡ d (mod 8) .Proof. By Lemma 2.6, w ( t, h, k ) is given by w ( t, h, k ) = β (1) t,h,k exp (cid:20) πi ( γ − k ) ˆ γ k k ( h + ( h − ) ˆ γ k k ) − ˆ α k ((4 s k T h ) − ) δ K K Dt ˆ γ k k (cid:21) , where all parameters defined as in Lemma 2.6. An elementary calculation shows that we may set( x − ) u = ( x − ) gk if u | gk . Using this as a factorization trick on w ( t, h, k ), we see that w ( t, h, k )is of the form w ( t, h, k ) = β (1) t,h,k exp (cid:20) πi q [ U h + V ( h − ) gk ] gk (cid:21) . We then set C ( t, h, k ) = β (1) t,h,k .Since (cid:18) · k (cid:19) is periodic modulo k , C ( t, h, k ) depends on t, k only if a is fixed and k is odd. If k is even, let k = 2 α k ′ where k ′ is odd and α is a positive integer, then (cid:18) kh (cid:19) = (cid:18) h (cid:19) α (cid:18) k ′ h (cid:19) =( − α ( h − · ( − ( h − k − (cid:18) hk ′ (cid:19) and i b ( k +1)2 depend on k if h ≡ a (mod k ) and h ≡ d (mod 8)for some fixed integers a and d . As a result, C ( t, h, k ) depends on k and t only if we fix h ≡ a (mod k ) and h ≡ d (mod 8) for some fixed integers a and d with d odd. The proofs for r = 2 , (cid:3) ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 15 Lemma 2.10. Using the notation of Proposition 2.9 for w r ( t, h, k ) , then for any positive integer n , ( qU − gn, gk ) ≤ n − .Proof. Consider the case when K is odd. If k is odd and 3 ∤ k , then ( qU − gn, gk ) = (1 − n, k ) ≤ n − 1. If k is odd and 3 | k , then ( qU − gn, gk ) = ((8 − ) k − n, k ) = (8 · (8 − ) k − n, k ) =(3 kA + 1 − n, k ) = (1 − n, k ) ≤ n − A is any integer. If k is even and 3 ∤ k ,then ( qU − gn, gk ) = ((3 − ) k − n, k ) = (3 · (3 − ) k − n, k ) = (8 kA + 1 − n, k ) =(1 − n, k ) ≤ n − A is any integer. If k is even and 3 | k , then ( qU − gn, gk ) =(1 − n, k ) ≤ n − 1. The proof is analogous when K is 0 or 2 modulo 4. (cid:3) By replacing t + 1 and the condition that h ≡ a (mod D ) in Theorem 2 of [5] by t ≡ h ≡ a (mod k ), and replacing the last paragraph of the proof of Theorem 2 of [5] by Lemma 2.10, weobtain the following theorem. Theorem 2.11. Let n, t, h, H, k, M be integers such that t, k > , t ≤ n, ( h, k ) = 1 and hH ≡− k ) where s ≤ H < s (mod k ) for some integers s , s such that ≤ s < s ≤ k .Then for any fixed a such that ( a, k ) = 1 , the sum Y := X h mod k w r ( t, h, k ) exp (cid:20) πi − hn + H ′ Mk (cid:21) where the summation runs through all h such that ≤ h < k and h ≡ a (mod k ) , is subject tothe estimate O (cid:0) n / k / ǫ (cid:1) where the multiplicative constant implied by the O -symbol dependson t only. A convergent series for G t ( x ) . We follow closely to the notations and proofs in Chapter5 of [1].Let t, h, k be integers with t, k > h, k ) = 1. Let D = ( t, k ), K = kD , T = tD . Let H, T ′ be an integer such that T T ′ ≡ K ) and hH ≡ − k ). Recall that G t ( x ) := X λ ∈P x | λ | ( − H t ( λ ) = X n ≥ X λ ⊢ n ( − H t ( λ ) x n = X n ≥ A t ( n ) x n . By Cauchy’s residue theorem, we have A t ( n ) = 12 πi Z C G t ( x ) x n +1 dx, where C is any positively oriented simple closed curve in a unit disk that contains the origin inits interior. Using the transformations x = e πiτ and z = − ik (cid:18) τ − hk (cid:19) , A t ( n ) = N X k =1 " ik X ≤ h 1, it follows that X tail | S j,t,h,k,n ( z ) | ≤ | z | exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) X tail exp (cid:20) − π T (cid:18) j − T (4 − D )24 (cid:19)(cid:21)(cid:12)(cid:12) c ( j, M, H, D ) (cid:12)(cid:12) . Note that X tail exp (cid:20) − π T (cid:18) j − T (4 − D )24 (cid:19)(cid:21)(cid:12)(cid:12) c ( j, M, H, D ) (cid:12)(cid:12) is a convergent sum becausewe know exp (cid:20) − π T (cid:18) j − T (4 − D )24 (cid:19)(cid:21) < j > T (4 − D )24 . Furthermore, | z | ≤ max {| z | , | z |} < √ kN , and therefore X tail | S j,t,h,k,n ( z ) | ≤ C | z | exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) < C k N − exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) , where C = X tail exp (cid:20) − π T (cid:18) j − T (4 − D )24 (cid:19)(cid:21)(cid:12)(cid:12) c ( j, M, H, D ) (cid:12)(cid:12) . Note that the path of integra-tion can be moved so that we integrate along the chord joining z ( h, K ) and z ( h, k ), and thelength of chord | z ( h, k ) z ( h, k ) | ≤ √ kN − , so with C ′ = 2 C , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z z ( h,k ) z ( h,k ) X tail S j,t,h,k,n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z z ( h,k ) z ( h,k ) X tail | S j,t,h,k,n ( z ) | dz< C k N − exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) = C ′ k N − exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) . Now consider the partial sum not included within P tail , that is with index values 0 ≤ j ≤ T (4 − D )24 . Note that this sum is empty unless D = ( t, k ) = 1 in which case K = k, T = t, T ′ = ( t − ) k and T (4 − D )24 = t 24 . In this case, (cid:4) T (4 − D (cid:5)X j =0 | S j,t,h,k,n ( z ) | = | z | (cid:4) t (cid:5)X j =0 e π Re( z ) k (cid:0) n − (cid:1) e π Re( 1 z )2 t (cid:0) t − j (cid:1)(cid:12)(cid:12) c ( j, M, H, (cid:12)(cid:12) ≤ k N − exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) · C , where C = (cid:4) t (cid:5)P j =0 exp (cid:20) π t (cid:18) t − j (cid:19)(cid:21)(cid:12)(cid:12) c ( j, M, H, (cid:12)(cid:12) . Noting that the length of the arc joining 0to z ( h, k ) is less than π | z ( h, k ) | < √ kπN , we therefore have by simple inequalities that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z z ( h,k )0 (cid:4) T (4 − D (cid:5)X j =0 S j,t,h,k,n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z z ( h,k )0 (cid:4) t (cid:5)X j =0 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S j,t,h,k,n ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dz ≤ C ′ k N − exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) , where C ′ = C π . Similarly, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z z ( h,k ) (cid:4) T (4 − D (cid:5)X j =0 S j,t,h,k,n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C k N − exp (cid:20) πk (cid:18) n − (cid:19)(cid:21) for a constant C . Assuming D = ( k, t ) = 1 and 0 ≤ j ≤ t 24 , we now consider the requiredintegral around K ( − ). We adopt the notation I ∗ j,t,k,n ( C ) = Z C √ z exp (cid:20) − π tz (cid:18) j − t (cid:19) + 2 πzk (cid:18) n − (cid:19)(cid:21) dz for an integral of this form over any curve C . Letting w = 1 z and r = πw t (cid:18) t − j (cid:19) , I ∗ j,t,k,n ( K ( − )) = Z i ∞ − i ∞ − w − exp (cid:20) − πw t (cid:18) j − t (cid:19) + 2 πk w (cid:18) n − (cid:19)(cid:21) dw = 2 πi · k / / t / (cid:18) t − j n − (cid:19) I π k r ( t − j )(24 n − t ! , where c = π t (cid:18) t − j (cid:19) ≥ I is a modified Bessel function of the first kind. Then wehave I ( t, h, k, n ) = ( √ t w ( t, h, k ) √ k (cid:4) T (4 − D (cid:5)X j =0 + X tail ! Z z ( h,k ) z ( h,k ) S j,t,h,k,n ( z ) dz ! . Since Z z ( h,k ) z ( h,k ) = Z K ( − ) − Z z ( h,k )0 − Z z ( h,k ) , we can evaluate A t ( n ) using the previous work. Toaid in notation, define the integrals J = I ∗ j,t,k,n ([ z ( h, k ) , z ( h, k )]), J := I ∗ j,t,k,n ([0 , z ( h, k )]), J := I ∗ j,t,h,k ([ z ( h, k ) , J := I ∗ j,t,h,k ( K ( − )). Also, write c := c ( j, M, H, 1) and w := ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 19 w ( t, h, k ). Then we have N X k =1 ik S , ( t, h, k, n ) = X ≤ k ≤ N ( k,t )=1 k odd X ≤ h Preliminaries. We start by proving that the Kloosterman sum X ≤ h Let S k ( n ) be the Kloosterman sum defined by S k ( n ) := 12 r k X x (mod 24 k ) x ≡− n +1 (mod 24 k ) χ ( x ) e (cid:18) x k (cid:19) , (3.1) where χ ( x ) = (cid:18) x (cid:19) is the Kronecker symbol and e ( x ) := e πix . If k is a power of , then S k ( n ) = 0 for all positive integers n .Proof. Let n ≥ 1, and let k = 2 s for an integer s ≥ 0. To show that S k ( n ) = 0, we need onlyshow that the summation given in (3.1) is nonzero. To evaluate this sum, consider the conditionon x that x ≡ − n +1 (mod 24 k ). Since − n +1 ≡ x ≡ − n +1 (mod 2 s +3 ) hasexactly 4 incongruent solutions, and so the congruence x ≡ − n + 1 (mod 24 k ) has exactly8 incongruent solutions. For any given solution x , we can see that all of 12 k − x , 12 k + x , and24 k − x are also solutions and are pairwise distinct.Now, let x, y (mod 24 k ) be solutions to x ≡ − n +1 (mod 24 k ) such that y is not congruentto any of x , 12 k − x , 12 k + x , or 24 k − x , so that the summation in (3.1) runs over the setof eight values {± x, ± (12 k + x ) } ∪ {± y, ± (12 k + y ) } . Taking real parts in the summation in(3.1) yields the value 4 a + 4 b , where a = χ ( x ) cos ( πx/ k ) and b = χ ( y ) cos ( πy/ k ). Theequivalences known about x and y imply that χ ( x ) , χ ( y ) = 0, and so the proof reduces todemonstrating that | a | 6 = | b | . If | a | = | b | , then x ≡ y (mod 6 k ) must hold, so we may fix y = 6 k − x . Since x is odd, y = x − kx + 36 k ≡ − n + 1 + 12 k + 36 k (mod 24 k ) , andthe equivalence modulo 6 k of x and y implies 12 k + 36 k ≡ k (1 + 3 k ) ≡ k ). This ISTRIBUTION PROPERTIES FOR t -HOOKS IN PARTITIONS 21 requires that 1 + 3 k be even, which is a contradiction since k = 2 s . Therefore, | a | 6 = | b | , and itthen follows that S k ( n ) = 0 for all n . (cid:3) Lemma 3.2. For t > a fixed positive integer, write t = 2 s ℓ with integers s, ℓ such that s ≥ and ℓ is odd. If k = 2 s +1 , then X We are now ready to prove the main theorems. Proposition 3.3. Let n be positive integers. Then for t fixed, as n → ∞ we have A t ( n ) p ( n ) ∼ ( ( − n / ( t − / if ∤ t, if | t. Furthermore, A t ( n ) p ( n ) ∼ as n, t → ∞ .Proof. Recall that p ( n ) satisfies p ( n ) ∼ π (24 n − / I (cid:18) π √ n − (cid:19) as n → ∞ . Then byCorollary 2.2, as n → ∞ we have A t ( n ) p ( n ) ∼ (1 + 3 · s ) / s +1+ t · I (cid:18) π s(cid:18) s +1 + 34 (cid:19) (24 n − (cid:19) I (cid:18) π √ n − (cid:19) X By Corollary 2.2, we have A t ( n ) ∼ π s + t (cid:18) · s n − (cid:19) I (cid:18) π p (1 + 3 · s )(24 n − · s +1 (cid:19) X We would like to thank Ken Ono for suggesting this problem and his guidance. References [1] Tom M. Apostol. Modular functions and Dirichlet series in number theory , volume 41 of Graduate Textsin Mathematics . Springer-Verlag, New York, second edition, 1990.[2] J. S. Frame, G. de B. Robinson, and R. M. Thrall. The hook graphs of the symmetric groups. Canad. J.Math. , 6:316–324, 1954.[3] Andrew Granville and Ken Ono. Defect zero p -blocks for finite simple groups. Trans. Amer. Math. Soc. ,348(1):331–347, 1996.[4] Peter Hagis, Jr. A root of unity occurring in partition theory. Proc. Amer. Math. Soc. , 26:579–582, 1970.[5] Peter Hagis, Jr. Partitions with a restriction on the multiplicity of the summands. Trans. Amer. Math.Soc. , 155:375–384, 1971.[6] Guo-Niu Han. The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension andapplications. Ann. Inst. Fourier (Grenoble) , 60(1):1–29, 2010.[7] Gordon James and Adalbert Kerber. The representation theory of the symmetric group , volume 16 of Encyclopedia of Mathematics and its Applications . Addison-Wesley Publishing Co., Reading, Mass., 1981.With a foreword by P. M. Cohn, With an introduction by Gilbert de B. Robinson.[8] Nikita A. Nekrasov and Andrei Okounkov. Seiberg-Witten theory and random partitions. In The unity ofmathematics , volume 244 of Progr. Math. , pages 525–596. Birkh¨auser Boston, Boston, MA, 2006.[9] Sarah Peluse. On even values in the character table of the symmetric group. arXiv preprint .[10] G. de B. Robinson. Representation theory of the symmetric group . Mathematical Expositions, No. 12.University of Toronto Press, Toronto, 1961.[11] A. Young. On Quantitative Substitutional Analysis (Second Paper). Proc. Lond. Math. Soc. , 34:361–397,1902. Department of Math, University of Virginia, Charlottesville, VA 22904 E-mail address : [email protected] Department of Math, University of Virginia, Charlottesville, VA 22904 E-mail address ::