The asymptotic normality of (s,s+1) -cores with distinct parts
aa r X i v : . [ m a t h . C O ] S e p The asymptotic normality of ( s, s + 1)-cores with distinct parts
J´anos Koml´os ∗ Emily Sergel † G´abor Tusn´ady ‡ Abstract
Simultaneous core partitions are important objects in algebraic combinatorics. Recentlythere has been interest in studying the distribution of sizes among all ( s, t )-cores for coprime s and t . Zaleski (2017) gave strong evidence that when we restrict our attention to ( s, s + 1)-cores with distinct parts, the resulting distribution is approximately normal. We prove hisconjecture by applying the Combinatorial Central Limit Theorem and mixing the resultingnormal distributions. A partition of n is a weakly decreasing sequence λ = ( λ ≥ λ ≥ · · · ≥ λ k >
0) whose parts sum to n , i.e., λ + λ + · · · + λ k = n . We say that n is the size of λ and k is its length . For example, thepartition (4 , , , ,
2) has size 15 and length 5.To each partition, we associate a diagram, known as a Ferrers diagram. The (french) Ferrersdiagram of a partition λ consists of boxes which are left-justified and whose i th row from thebottom contains λ i boxes. For example, see Figure 1.Figure 1: The Ferrers diagram of the partition (4 , , , , hook length . The hooklength of a cell c is the number of boxes strictly right of c (known as the arm of the cell) plus thenumber of boxes strictly above c (the leg ) plus one. For example, the cell c indicated in Figure 2has hook length 4. The cell marked a is the only one in the arm and the two cells marked ℓ formthe leg. ∗ Department of Mathematics, Rutgers University † [email protected]. Department of Mathematics, University of Pennsylvania. Partially supported by NSF grantDMS-1603681. ‡ MTA R´enyi Alfr´ed Matematikai Kutat´o Int´ezet ℓc a Figure 2: The arm and leg of a cell of a Ferrers diagram.For convenience, we will sometimes write the hook length of each cell into the Ferrers diagram. Wesay that a partition is an s -core if none of its cells have hook-length s . A partition is an ( s, t )-coreif it is simultaneously an s -core and a t -core. See Figure 3. The number of ( s, t )-cores is finite ifand only if gcd( s, t ) = 1. Jaclyn Anderson [And02] gives a beautiful bijection between ( s, t )-coresand certain lattice paths from (0 ,
0) to ( s, t ) which proves this result and much more. ∅ , t = ds ± d ∈ N , they arenaturally in bijection with certain regions of the d -Shi arrangement in type A. Drew Armstrong,Christopher Hanusa, and Brant Jones [AHJ14] extended this work to type C and related simul-taneous cores to rational Catalan combinatorics. Purely enumerative questions have yielded deepconnections as well. For instance, Armstrong [AHJ14] initially conjectured a simple formula forthe average size of an ( s, t )-core in 2011. Paul Johnson [Joh15] gave the first proof of Armstrong’sconjecture by relating cores to polytopes.As Shalosh B. Ekhad and Doron Zeilberger [EZ15] note “the average is just the first question onecan ask about a probability distribution”. They determine the distribution obtained by fixing t − s , taking the size of a random ( s, t )-core, normalizing, and letting s → ∞ . Surprisingly thesedistributions are not normal and are not known to be associated with any other combinatorialproblems. However, Anthony Zaleski [Zal17] gave strong experimental evidence that if t = s + 1and only cores with distinct parts are considered, then the resulting limit distribution is indeednormal. We prove this in the following form.For a positive integer s , let X s be the random variable given by the size of an ( s, s + 1)-core withdistinct parts which is chosen uniformly at random. Let µ and σ be the mean and variance of X s .Let Φ denote the standard normal distribution function. Theorem 1.
For all positive integers s , sup x ∈ R (cid:12)(cid:12) P ( X s ≤ µ + xσ ) − Φ( x ) (cid:12)(cid:12) = O (1 / √ s ) . (1)Here, and throughout the paper, the implied constants in error bounds O ( . ) are universal constantsnot depending on any of our parameters. That is, Theorem 1 says: There is a universal constant2 such that, for all s and x , (cid:12)(cid:12) P ( X s ≤ µ + xσ ) − Φ( x ) (cid:12)(cid:12) ≤ C / √ s. To prove this we introduce a new tool to this discussion: the Combinatorial Central Limit Theorem(CCLT). The original form of the CCLT is due to Wassily Hoeffding [Hoe51], but we will use thetail bounds given by Erwin Bolthausen [Bol84].Our main tools are two classical results: Proposition 1 on page 4 (CCLT) and Proposition 2 onpage 6 (about generating functions with only real roots). (These two existing tools are namedPropositions and they are numbered separately. All other statements (theorem, corollary, lemma)are labeled in one single sequence.)The rest of the paper is organized as follows. In Section 2, we review the Combinatorial CentralLimit Theorem. In Section 3, we prove that the distribution of size among ( s, s + 1)-cores withdistinct parts is already approximately normal when the number of parts is fixed.
In Section 4, werecall that the weights needed to mix these distributions together are also approximately normal.In Section 5 we mix these distributions together to prove Theorem 1. Section 6 contains the proofsof some technical lemmas used in Section 5.
Let A = ( a ij ) be an m × m matrix of real numbers. We are interested in the random sum S A = X i a iπ ( i ) where π ∈ S m is a random permutation of { , , . . . , m } chosen uniformly from among all m !permutations. Following [Bol84] we write a i · = 1 m X j a ij , a · j = 1 m X i a ij , and a ·· = 1 m X i,j a ij and set ˙ a ij = a ij − a i · − a · j + a ·· to normalize the row- and column-sums of our matrix to 0. Furthermore, we write µ A = ma ·· and σ A = 1 m − X i,j ˙ a ij for the mean and variance of S A , and consider the normalized sum T A = S A − µ A σ A = X i b a iπ ( i ) where b a ij = ˙ a ij /σ A A is of rank 1, this gives a tail bound for the classical resultof Abraham Wald and Jacob Wolfowitz [WW44]. Proposition 1.
There is an absolute constant K such that for all A with σ A > , sup t | P ( T A ≤ t ) − Φ( t ) | ≤ K X i,j | b a ij | /m . Armin Straub [Str16] gave the following beautiful characterization of our chosen objects:
A partition λ into distinct parts is an ( s, s + 1) -core if and only if it has perimeter ℓ ( λ ) + λ − ≤ s − . Let k and s be fixed non-negative integers. By the above characterization, a partition λ consistingof k distinct parts is an ( s, s + 1)-core if and only if the largest part λ is at most s − k . We naturallyassociate to each such partition a vector of length s − k by recording a 1 at position λ i for 1 ≤ i ≤ k and 0 elsewhere. For example, the vector (0 , , , , ,
0) corresponds to the (9 , , , s, s + 1)-cores with k distinct parts is just (cid:0) s − kk (cid:1) . Summingshows that the total number of ( s, s + 1)-cores with any number of distinct parts is the Fibonaccinumber F ib s +1 . This fact was originally conjectured by Tewodros Amdeberhan [Amd16] and provedby Straub [Str16].We can also see that the size of the initial core is just the sum of the positions of 1’s in the resultingvector, i.e., the inner product of this vector and (1 , , , . . . , s − k ). With this rephrasing we areable to apply the CCLT: simply take the matrix A to be the outer product of the vector (1 k , s − k )and the vector (1 , , , . . . , s − k ).In general, suppose A = ( a ij ) is an m × m rank 1 matrix, i.e., a ij = α i x j for some vectors α , x .Thus, writing ¯ α = ( P α i ) /m and ¯ x = ( P x j ) /m , we have˙ a ij = ( α i − ¯ α )( x j − ¯ x ) , µ A = m ¯ α ¯ xσ A = 1 m − X i,j ˙ a ij = m m − m X i ( α i − ¯ α ) ! m X j ( x j − ¯ x ) (2)Let α = · · · = α k = 1, α k +1 = · · · = α m = 0. Note that now S A is the sum of the elements in arandom k -subset of the list x , . . . , x m . We are interested in the special case x i = i for i = 1 , . . . , m . Theorem 2.
For this choice of parameters the following explicit bound holds: sup x ∈ R | P ( T A ≤ x ) − Φ( x ) | ≤ (cid:18) m k ( m − k ) (cid:19) / · K √ m (3) which goes to 0 when both km − / → ∞ and ( m − k ) m − / → ∞ .Proof. It is easy to see that¯ α = k/m, ¯ x = ( m + 1) / , µ A = m + 12 · k , σ A = m + 112 · k ( m − k ) . (4)4sing | ˙ a ij | = | α i − ¯ α | · | x j − ¯ x | ≤ · m = m , the right-hand side in Proposition 1 is K X i,j | b a ij | /m ≤ Km σ A < (cid:18) m k ( m − k ) (cid:19) / · K √ m which goes to 0 if km − / → ∞ and ( m − k ) m − / → ∞ .Plugging m = s − k in to (3) gives the following corollary. Corollary 3.
Let X s,k be the random variable given by the size of an ( s, s + 1) -core with k distinctparts chosen uniformly at random. Let µ k and σ k denote the mean and variance of X s,k , respectively.Then for any < k < s/ , the normalized variable ( X s,k − µ k ) /σ k satisfies the following. sup x ∈ R (cid:12)(cid:12)(cid:12)(cid:12) P (cid:18) X s,k − µ k σ k ≤ x (cid:19) − Φ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ / K ( s − k ) / ( k ( s − k )) / Hence the distribution of ( X s,k − µ k ) /σ k tends to the standard normal distribution if s → ∞ andboth ks − / → ∞ and ( s − k ) s − / → ∞ . We will use Corollary 3 only when s/ ≤ k ≤ s/
3, in which case we obtain the boundsup x ∈ R | P ( X s,k ≤ µ k + xσ k ) − Φ( x ) | < K √ s . (5) Remark.
Zaleski [Zal17] already noted that the generating function for ( s, s + 1) -cores with k dis-tinct parts is none other than the shifted q -binomial coefficient q ( k +12 ) (cid:0) s − kk (cid:1) q . It was this observationthat lead us to study the distribution when k is fixed. By taking s = n + m and k = m , Corollary3 shows that the partial sums of coefficients in the q -binomial coefficient (cid:0) nm (cid:1) q are approximatelynormally distributed. It would be interesting to see that the distribution is also locally approximatelynormal. Ultimately we will mix together the distributions of X s,k for all k with s fixed. Each distributionis weighted according to how many cores are being enumerated, namely X s,k gets weight p k = P ( W = k ) = (cid:18) s − kk (cid:19) /F ib s +1 . Here the random variable W is the number of parts in a random ( s, s + 1)-core with distinct parts.The sequence (cid:0) s − kk (cid:1) appears often in combinatorics. Its generating function is g s ( z ) = X ≤ k ≤ s (cid:18) s − kk (cid:19) z k = 1 √ z (cid:18) √ z (cid:19) s +1 − (cid:18) − √ z (cid:19) s +1 ! µ ( W ) := X k k p k = 5 − √ · s + O (1)and σ ( W ) := √ · s + O (1) . For convenience we write c = (5 − √ /
10 = 0 . .. and k = ⌊ c s ⌋ .There is a long history of normal approximations for finite non-negative real sequences whosegenerating functions have only real roots. The first appearance in combinatorics of a global normallaw similar to (6) is a result of Lawrence Harper [Har67] studying Stirling numbers. Harper’sbrilliant idea was further developed and generalized in the classical paper of Ed Bender [Ben73].Some important early results can be found in the paper [Sch55] of Isaac Schoenberg.The following proposition is from Pitman [Pit97]. It says that if a polynomial f with non-negativecoefficients has only real zeros, then its coefficients are approximately normally distributed, bothglobally and locally. For completeness, we cite both the global and the local versions. Proposition 2.
Let p , p , . . . , p n be a sequence of non-negative real numbers summing to 1 withmean µ and variance σ . Let f ( x ) = P k p k x k be its generating function. Write S k = P ki =0 p i forthe partial sums. Assume all roots of the polynomial f are real. Then, max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12) S k − Φ (cid:18) k − µσ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < . σ (6) and there exists a universal constant C such that max ≤ k ≤ n (cid:12)(cid:12)(cid:12)(cid:12) σp k − ϕ (cid:18) k − µσ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < Cσ . (7) Remark.
It is obvious that if f has only real roots, then the non-negativity of the coefficients p , . . . , p n is equivalent to all roots of f being non-positive – another traditional way of stating theresult. Our generating function g s ( x ) has only real roots, since only real numbers z ≤ − / (cid:12)(cid:12) √ z (cid:12)(cid:12) = (cid:12)(cid:12) − √ z (cid:12)(cid:12) . Hence Proposition 2 applies to our sequence of weights p k = (cid:0) s − kk (cid:1) /F ib s +1 with n = ⌊ s/ ⌋ , µ = µ ( W ), and σ = σ ( W ).The same paper [Pit97] (Formula (11) on page 284) contains exponential tail bounds for our weightdistribution (phrased in the more general setup of so-called PF-distributions). Plugging in ourspecific parameter µ ( W ) = c s + O (1), we get the following bound: for every ε > δ > C ( ε ) > X k< ( c − ε ) s p k + X k> ( c + ε ) s p k < C ( ε ) e − δs (8)We will use this tail probability estimate later with ε = min { / − c , c − / } = 0 . .. Proof of Theorem 1
Fix a positive integer s . Recall that X s is the random variable given by the size of an ( s, s + 1)-corewith distinct parts which is chosen uniformly at random. Zaleski [Zal17] shows that the mean andvariance of X s are: µ = µ ( X s ) = 110 s + O ( s ) , σ = σ ( X s ) = 2 √ s + O ( s ) . (9)Recall also that if 0 ≤ k ≤ s/
2, then X s,k is the random variable given by the size of an ( s, s + 1)-core with k distinct parts which is chosen uniformly at random. Hence the distribution of X s isthe mixture of the distributions of the ⌊ s/ ⌋ + 1 individual X s,k .Setting m = s − k in (4) gives µ k = 12 k ( s + 1 − k ) , σ k = 112 k ( s + 1 − k )( s − k ) . (10) Remark.
Zaleski’s formulas (9) could be obtained by a lengthy computation involving the generatingfunction g s ( z ) , (10) , and the Pythagorean Theorem of Probability Theory (a.k.a. the Law of TotalVariance): V ar (cid:2) ξ (cid:3) = EV ar (cid:2) ξ | η (cid:3) + V ar (cid:2) E ( ξ | η ) (cid:3) . Fix x ∈ R . Let F ( x ) := P ( X s ≤ µ + xσ ) = EP ( X s,k ≤ µ + xσ ) . (11)Here the expected value E denotes the weighted sum EP ( X s,k ≤ µ + xσ ) = X ≤ k ≤ s/ P ( X s,k ≤ µ + xσ ) p k . (12)For 0 < k < s/ P ( X s,k ≤ µ + xσ ) = P ( X s,k ≤ µ k + y k σ k ) =: F k ( y k ) , (13)where y k = 1 σ k (cid:0) ( µ − µ k ) + xσ (cid:1) . (14)For k = 0 and k = s/ s is even) we have σ k = 0, so y k is undefined. These at most twoterms of the right-hand side of (12) have weight 1 /F ib s +1 (each), so we will only work with integers k with 0 < k < s/ F ( x ) is approximately Φ( x ) with an error bound O (1 / √ s )uniformly for x ∈ R . We will accomplish this with a sequence of approximations Q , . . . , Q andseveral lemmas. Each subsequent Q introduces an error of only O (1 / √ s ). The proofs of theselemmas will be put off to Section 6. 7et Q := X 5, and t k = 5 / ( k − k ) / √ s . The next lemma says that y k is well approximated by the arithmetic progression y ∗ k = ax + bt k in the relevant range of k . Wealso write dt k = t k − t k − = 5 / / √ s . The quantity dt k (which is independent of k ) will be used asa mesh size in approximating integrals. We will also see (41) that σ k is roughly constant when k isclose to k . Lemma 4. For all integers k with < k < s/ , | y k − y ∗ k | = 1 √ s · O (1 + | xt k | + t k ) . (18)We will also show in the last section that Lemma 4 implies the following statement. Corollary 5. For all integers k with < k < s/ we have | Φ( y k ) − Φ( y ∗ k ) | = O (cid:18) √ s (cid:0) t k (cid:1)(cid:19) (19) uniformly for x ∈ R . Hence, Q = X Let m ≤ n be integers. Suppose ( U k : m ≤ k ≤ n + 1) and ( V k : m − ≤ k ≤ n ) aretwo (finite) real sequences. Then, n X k = m U k ( V k − V k − ) = n X k = m ( U k − U k +1 ) V k + (cid:2) U n +1 V n − U m V m − (cid:3) . (24)(Lemma 7 can be verified easily by comparing the two sides term by term.)Write u k = U k − U k +1 ( m ≤ k ≤ n ) and v k = V k − V k − ( m ≤ k ≤ n ).Thus (24) becomes n X k = m U k v k = n X k = m u k V k + (cid:2) U n +1 V n − U m V m − (cid:3) . (25)Note also: for all m ≤ k ≤ n , U k = U n +1 + X k ≤ i ≤ n u i and V k = V m − + X m ≤ i ≤ k v i . Corollary 8. Let m ≤ n be integers. Suppose ( U k : m ≤ k ≤ n + 1) , ( U ′ k : m ≤ k ≤ n + 1) , ( V k : m − ≤ k ≤ n ) , and ( V ′ k : m − ≤ k ≤ n ) are real sequences. Define u k , u ′ k , v k , v ′ k as inLemma 7. Write δ U = sup m ≤ k ≤ n | U k − U ′ k | , δ V = sup m ≤ k ≤ n | V k − V ′ k | . (26) Then, (cid:12)(cid:12)(cid:12) n X k = m U k v k − n X k = m U ′ k v ′ k (cid:12)(cid:12)(cid:12) ≤ δ U X | v ′ k | + δ V X | u k | + | U n +1 V n − U m V m − | + (cid:12)(cid:12) U n +1 V ′ n − U m V ′ m − (cid:12)(cid:12) . (27)9his simple corollary of Lemma 7 will be proved in the last section.Define F ∗ k = k ≤ y ∗ k ) = Φ( ax + bt k ) if 0 < k < s/ k ≥ s/ 2. (28)Then, Q = X For all integers k ∈ Z , Φ( t k ) − Φ( t k − ) = ϕ ( t k ) dt k + 1 √ s O (cid:0) | ϕ ′ ( t k ) | dt k (cid:1) + O (1 /s / ) . (36)Applying Lemma 9, we get Q = X ≤ k ≤ s/ Φ( ax + bt k ) [Φ( t k ) − Φ( t k − )]= X ≤ k ≤ s/ Φ( ax + bt k ) ϕ ( t k ) dt k + 1 √ s · O X ≤ k ≤ s/ | ϕ ′ ( t k ) | dt k + O (1 / √ s )= X ≤ k ≤ s/ Φ( ax + bt k ) ϕ ( t k ) dt k + O (1 / √ s ) . (37)For the last line we used the fact that the O ( P ... ) term is a (partial) Riemann-sum for theconvergent integral R ∞−∞ | ϕ ′ ( t ) | dt . The bounded non-negative function | ϕ ′ ( t ) | is made up of fourmonotone pieces, and our mesh size is dt k = O (1 / √ s ).The sum in the last line of (37) can be extended for all integers k with an error of only O (1 / √ s ).This is because X k< Φ( ax + bt k ) ϕ ( t k ) dt k < X k< ϕ ( t k ) dt k and the right-hand side is a Riemann sum for the function ϕ ( t ) integrated from −∞ to − / k / √ s .This integral is exponentially small in s . Since on this domain ϕ ( t ) is monotone increasing andis between 0 and 1 / √ π , the Riemann sum approximation itself only introduces an error at most dt k / √ π = O (1 / √ s ). The same applies to the sum P k>s/ Φ( ax + bt k ) ϕ ( t k ) dt k .Thus, Q = X k ∈ Z Φ( ax + bt k ) ϕ ( t k ) dt k + O (1 / √ s ) . (38)11et Q := X k ∈ Z Φ( ax + bt k ) ϕ ( t k ) dt k . Then, Q = Q + O (1 / √ s ) . (39)Define Q := Z ∞−∞ Φ( ax + bt ) ϕ ( t ) dt. (40) Lemma 10. Let h : R → R be a differentiable function. Assume V h = Z ∞−∞ | h ′ t ) | dt < ∞ . Let I j = [ ℓ j , r j ] ( j ∈ Z ) be a partition of R into intervals of lengths not exceeding δ > , and let ξ j ∈ I j be arbitrary points. Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ Z h ( ξ j ) | I j | − Z ∞−∞ h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ V h δ . We apply this lemma to the function h ( t ) = Φ( ax + bt ) ϕ ( t ) with δ = dt k = 5 / / √ s .Thus, h ′ ( t ) = ϕ ( t ) · [ b ϕ ( ax + bt ) − t Φ( ax + bt )], whence | h ′ ( t ) | ≤ ϕ ( t ) · ( | b | + | t | ).Since Z ∞−∞ | h ′ ( t ) | dt < ∞ uniformly for x ∈ R , by Lemma 10 we get Q = Q + O (1 / √ s ) . Lemma 11. Let a and b be real numbers. Then for all x ∈ R , Z ∞−∞ Φ( ax + bt ) ϕ ( t ) dt = Φ (cid:18) ax √ b (cid:19) . We apply Lemma 11 with a = p / b = − p / Q = Φ( x ) . This completes the proof of Theorem 1. Namely, we have shown that F ( x ) = Φ( x ) + O (1 / √ s )uniformly in x ∈ R . 12 Computational Proofs of the Lemmas Lemma 4. For all integers k with < k < s/ , | y k − y ∗ k | = 1 √ s · O (1 + | xt | + t k ) . Proof. Recall that µ k = k ( s +1 − k )2 , σ k = k ( s +1 − k )( s − k )12 , k = ⌊ −√ s ⌋ . Let D k = k − k . Then σ k σ k = ( k + D k )( s + 1 − k − D k )( s − k − D k ) k ( s + 1 − k )( s − k ) = 1 + O (cid:18) D k s (cid:19) . (41)Therefore y k = 1 σ k (cid:0) ( µ − µ k ) + xσ (cid:1) = (cid:20) O (cid:18) D k s (cid:19)(cid:21) · σ k (cid:0) ( µ − µ k ) + xσ (cid:1) . Let q = σ/σ k . Then y k = (cid:20) O (cid:18) D k s (cid:19)(cid:21) · q · (cid:18) µ − µ k σ + x (cid:19) . Now note that µ k = 12 − √ s ! s + 1 − − √ s ! + O ( s )= 12 − √ ! − − √ ! s + O ( s )= s 10 + O ( s ) = µ + O ( s ) . So µ − µ k = µ k − µ k + O ( s )= 12 ( k ( s + 1 − k ) − k ( s + 1 − k )) + O ( s )= 12 ( k − k ) (cid:16) − s − k + k ) (cid:17) + O ( s )= 12 ( k − k ) (cid:16) − s − k − k ) + 2 k (cid:17) + O ( s )= 12 D k (cid:16) − s − D k + 5 − √ s (cid:17) + O ( s )= − √ · sD k + O ( D k ) + O ( s ) . (Above and below we use the obvious inequality: 2 D k ≤ D k + 1.)13herefore µ − µ k σ = − √ · sD k + O ( D k ) + O ( s ) q √ s / (cid:2) O (cid:0) s (cid:1)(cid:3) = s √ − √ · D k √ s + O (cid:18) D k s / (cid:19) + O (cid:18) √ s (cid:19)! (cid:20) O (cid:18) s (cid:19)(cid:21) = − / / / · D k √ s + O (cid:18) D k s / (cid:19) + O (cid:18) √ s (cid:19) . Finally, setting t k = 5 / D k / √ s and using | D k | ≤ s gives y k = (cid:20) O (cid:18) D k s (cid:19)(cid:21) · q · (cid:18) µ − µ k σ + x (cid:19) = (cid:20) O (cid:18) t k √ s (cid:19)(cid:21) · q · x − r t k + O (cid:18) t k √ s (cid:19) + O (cid:18) √ s (cid:19)! = q · x − r t k ! + 1 √ s · O (cid:0) | xt k | + t k (cid:1) . But q is essentially a constant. That is, q = σ σ k = √ s + O ( s ) k ( s + 1 − k )( s − k ) + O ( s )= √ s + O ( s ) c (1 − c )(1 − c ) s (cid:2) O (cid:0) s (cid:1)(cid:3) = 85 + O (cid:18) s (cid:19) . So q = p / O (1 /s ). Therefore y k = r x − r t k ! + 1 √ s · O (cid:0) | xt | + t k (cid:1) = y ∗ k + 1 √ s · O (cid:0) | xt | + t k (cid:1) . Corollary 5. For all integers k with < k < s/ we have | Φ( y k ) − Φ( y ∗ k ) | = O (cid:18) √ s (cid:0) t k (cid:1)(cid:19) uniformly for x ∈ R . roof. Let K be the implied constant in (18). Let ε = p / x = 16 K /a , and s = (8 K /a ) . Special case I: | t k | ≥ s / .Then 1 + t k > s / , so √ s (1 + t k ) > 1. Hence (19) is automatically true (independent of the valueof x ). Special case II: | t k | ≥ ε | x | .Then 1 + | xt k | + t k ≤ ε + 1) t k < t k ). Special case III: | x | ≤ x .Then 1 + | xt k | + t k ≤ x | t k | + t k ≤ (1 + x / t k ) = O (1 + t k ).For the rest of this proof we will assume k is an integer with 0 < k < s/ x > x , | t k | < ε | x | , and | t k | < s / . We will first show that both y k and y ∗ k are between ax and ax . This will allow us to apply theMean Value Theorem to prove the corollary.Recall that a = p / b = − p / 5, and t k = 5 / ( k − k ) / √ s . Thus, | bt k | = p / | t k | < p / ε | x | = 12 | ax | . Consequently, y ∗ k = ax + bt k is between 12 ax and 32 ax, whence | y ∗ k | > a | x | . Now we estimate y k : | y ∗ k − y k | ≤ K √ s · (1 + | xt k | + t k ) = K √ s · (1 + t k ) + K √ s · | xt k | . The first term on the right-hand side is estimated as K √ s (1 + t k ) < K √ s (1 + s / ) = K (1 + s − / ) ≤ K ≤ a | x | for x ≥ x .For the second term we have K √ s · | xt k | < K √ s · | x | s / = K s / · | x | ≤ a | x | for s ≥ s .Consequently, | y ∗ k − y k | < a | x | , and thus y k is between 14 ax and 74 ax as desired.15y the Mean Value Theorem, there is a ξ between y k and y ∗ k such that Φ( y ∗ k ) − Φ( y k ) = ϕ ( ξ ) ( y ∗ k − y k ).As we showed above, ξ is between ax and ax , and hence | ξ | > a | x | > a ε | t k | . Consequently, since ϕ is monotone, ϕ ( ξ ) = ϕ ( | ξ | ) < ϕ (cid:18) a | x | (cid:19) and ϕ ( ξ ) < ϕ (cid:18) a ε | t k | (cid:19) . We obtain: | Φ( y ∗ k ) − Φ( y k ) | = ϕ ( ξ ) | y ∗ k − y k | ≤ ϕ ( ξ ) K √ s · (1 + | xt k | + t k ) < K √ s · (cid:20) (1 + t k ) ϕ (cid:18) a ε | t k | (cid:19) + ε x ϕ (cid:18) a | x | (cid:19)(cid:21) . Since the quantity in square brackets is bounded uniformly in k ∈ Z and x ∈ R , Corollary 5 isproved. Lemma 6. There exists a universal constant K such that for all s ∈ N , X ≤ k ≤ s/ (1 + t k ) p k ≤ K . Proof. By the definition of t k , we have t k = 5 / s ( k − k ) ≤ s · (cid:2) ( k − µ ( W )) + ( µ ( W ) − k ) (cid:3) = 25 s ( k − µ ( W )) + O (1 /s ) . Here we used ( α − γ ) ≤ α − β ) + ( β − γ ) ]. Hence, X ≤ k ≤ s/ t k p k ≤ s X ≤ k ≤ s/ ( k − µ ( W )) p k + O (1) = 25 · σ ( W ) s + O (1) = O (1)(where, as always, O (1) is independent of s ). Corollary 8. For sequences U, U ′ , V, V ′ as before, (cid:12)(cid:12)(cid:12) n X k = m U k v k − n X k = m U ′ k v ′ k (cid:12)(cid:12)(cid:12) ≤ δ U X | v ′ k | + δ V X | u k | + | U n +1 V n − U m V m − | + (cid:12)(cid:12) U n +1 V ′ n − U m V ′ m − (cid:12)(cid:12) . roof. We start with the following four identities, the non-trivial two of which follow from applyingLemma 7 twice. n X k = m U k v k − n X k = m u k V k = [ U n +1 V n − U m V m − ] . n X k = m u k V k − n X k = m u k V ′ k = n X k = m u k ( V k − V ′ k ) . n X k = m u k V ′ k − n X k = m U k v ′ k = − (cid:2) U n +1 V ′ n − U m V ′ m − (cid:3) . n X k = m U k v ′ k − n X k = m U ′ k v ′ k = n X k = m ( U k − U ′ k ) v ′ k . Adding up these four identities we get n X k = m U k v k − n X k = m U ′ k v ′ k = n X k = m ( U k − U ′ k ) v ′ k + n X k = m u k ( V k − V ′ k ) + [ U n +1 V n − U m V m − ] − (cid:2) U n +1 V ′ n − U m V ′ m − (cid:3) , from which Corollary 8 follows. Lemma 9. For all integers k ∈ Z , Φ( t k ) − Φ( t k − ) = ϕ ( t k ) dt k + 1 √ s O ( | ϕ ′ ( t k ) | dt k ) + O (1 /s / ) where dt k = 5 / / √ s .Proof. Let k ∈ Z . There exists a ξ k with t k − < ξ k < t k such thatΦ( t k ) − Φ( t k − ) = ϕ ( t k )( t k − t k − ) − ϕ ′ ( t k )( t k − t k − ) + 16 ϕ ′′ ( ξ k )( t k − t k − ) = ϕ ( t k ) dt k − ϕ ′ ( t k )( dt k ) + 16 ϕ ′′ ( ξ k )( dt k ) = ϕ ( t k ) dt k + 1 √ s O ( | ϕ ′ ( t k ) | dt k ) + O (1 /s / ) . emma 10. Let h : R → R be a differentiable function. Assume V h = R ∞−∞ | h ′ t ) | dt < ∞ . Let I j = [ ℓ j , r j ] ( j ∈ Z ) be a partition of R into intervals of lengths not exceeding δ > , and let ξ j ∈ I j be arbitrary points. Then, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ Z h ( ξ j ) | I j | − Z ∞−∞ h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ V h δ . Proof. While the statement is known in the context of total variations of functions, we give, forcompleteness, a simple direct proof by applying the bounded version below on each individualinterval I j . Observation. Let h be a differentiable function on a closed interval I = [ a, b ] ( a < b ) . Then, | h ( b ) − h ( a ) | ≤ Z ba | h ′ ( t ) | dt. Indeed, by the Fundamental Theorem of Calculus, (cid:12)(cid:12) h ( b ) − h ( a ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z ba h ′ ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z ba | h ′ ( t ) | dt. Bounded version. Let h be differentiable on a closed bounded interval I = [ a, b ] ( a < b ) . Let ξ ∈ I be arbitrary. Then, D := (cid:12)(cid:12)(cid:12)(cid:12) h ( ξ ) · ( b − a ) − Z ba h ( t ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ ( b − a ) Z ba | h ′ ( t ) | dt. Indeed, since h is continuous on I , there exists an η ∈ I such that Z ba h ( t ) dt = h ( η ) · ( b − a ) . Assume (WLOG) that η ≤ ξ . Then, by the Observation above, D = ( b − a ) · (cid:12)(cid:12) h ( ξ ) − h ( η ) (cid:12)(cid:12) ≤ ( b − a ) Z ξη | h ′ ( t ) | dt ≤ ( b − a ) Z ba | h ′ ( t ) | dt. Lemma 11. Let a and b be real numbers. Then for all x ∈ R , Z ∞−∞ Φ( ax + bt ) ϕ ( t ) dt = Φ (cid:18) ax √ b (cid:19) . roof. One could compute the two-dimensional integral corresponding to the left hand side. Wepresent instead a simple probabilistic proof. We write E for expected value.Let Z and Z be independent standard normal variables. Define Z = Z − bZ . Then Z is anormal random variable with 0 expectation and variance 1 + b . We then have Z ∞−∞ Φ( ax + bt ) ϕ ( t ) dt = E Φ( ax + bZ ) = EP ( Z ≤ ax + bZ )= EP ( Z ≤ ax ) = Φ (cid:18) ax √ b (cid:19) . References [AHJ14] Drew Armstrong, Christopher R. H. Hanusa, and Brant C. Jones. Results and conjectureson simultaneaous core partitions. European Journal of Combinatorics , 41:205–220, 2014. http://arxiv.org/abs/1308.0572 .[Amd16] Tewodros Amdeberhan. 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