Counting partitions inside a rectangle

Transcription

1 Counting partitions inside a rectangle Stephen Melczer, Greta Panova, and Robin Peantle May 2, 28 Abstract We consider the nuber of partitions of n whose Young diagras fit inside an l rectangle; equivalently, we study the coefficients of the q-binoial coefficient ) +l. We obtain sharp asyptotics q throughout the regie l = Θ) and n = Θ 2 ). Previously, sharp asyptotics were derived by Takács [Tak86] only in the regie where n l/2 = O ll + )) using a local central liit theore. Our approach is to solve a related large deviation proble: we describe the tilted easure that produces configurations whose bounding rectangle has the given aspect ratio and is filled to the given proportion. Our results are sufficiently sharp to yield the first asyptotic estiates on the consecutive differences of these nubers when n is increased by one and, l reain the sae, hence significantly refining Sylvester s uniodality theore. AMS subject classification: 5A5, 6C5, 6F5, 6F; Key words and phrases: partitions, q-binoial coefficients, large deviations, local CLT. University of Pennsylvania Departent of Matheatics 29 South 33rd Street Philadelphia, PA 94 {selczer,panova,peantle}@ath.upenn.edu Melczer partially supported by an NSERC postdoctoral fellowship and NSF grant DMS Panova partially supported by NSF grant DMS-5834 and IAS von Neuann Fellowship Peantle partially supported by NSF grant DMS

2 Introduction A partition λ of n is a sequence of weakly decreasing nonnegative integers λ = λ λ 2...) whose su λ = λ +λ 2 + is equal to n. The study of integer partitions is a classic subject with applications ranging fro nuber theory to representation theory and cobinatorics. Integer partitions with various restrictions on properties, such as part sizes or nuber of parts, occupies the field of partition theory [And76]. The generating functions for integer partitions play a role in nuber theory and the theory of odular fors. In representation theory, integer partitions index the conjugacy classes and irreducible representations of the syetric group S n ; they are the signatures of the irreducible polynoial representation of GL n. Partitions also give a basis for the ring of syetric functions. More recently, partitions have appeared in the study of interacting particle systes and other statistical echanics odels. The nuber of partitions of n, typically denoted by pn) but here unconventionally by N n, was iplicitly deterined by Euler via the generating function N n q n = q i. n= There is no exact explicit forula for the nubers N n. The asyptotic forula N n := #{λ n} ) 2n 4n 3 exp π, ) 3 obtained by Hardy and Raanujan [HR8], is considered to be the beginning of the use of coplex variable ethods for asyptotic enueration of partitions the so-called circle ethod). Our goal is to obtain asyptotic forulas siilar to ) for the nuber of partitions λ of n whose Young diagra fits inside an l rectangle, denoted i= N n l, ) := #{λ n : λ l, lengthλ) }. These nubers are also the coefficients in the expansion of the q-binoial coefficient ) l + q = l+ i= qi ) l i= qi ) i= qi ) = l n= N n l, )q n. The q-binoial coefficients are theselves central to enuerative and algebraic cobinatorics. They are the generating functions for lattice paths restricted to rectangles and taking only north and east steps under the area statistic, given by the paraeter n. They are also the nuber of l-diensional subspaces of Fq l+ and appear in any other generating functions as the q-analogue generalization of the ubiquitous binoial coefficients. Notably, the nubers N n l, ) for a syetric uniodal sequence = N l, ) N l, ) N l/2 l, ) N l l, ) =, a fact conjectured by Cayley in 856 and proven by Sylvester in 878 via the representation theory of sl 2 [Syl78]. One hundred forty years later, no previous asyptotic ethods have been able to prove this uniodality. Our ain result is an asyptotic forula for N n l, ) in the regie l/ A and n/ 2 B for any fixed A > B >. This is the regie in which a liit shape of the partitions exists: l/ A iplies the We use the notation N n to distinguish scenarios of probability with those of enueration, both of which occur in the present anuscript.

3 aspect ratio has a liit and n/ 2 B, A) iplies the portion of the l rectangle that is filled approaches a value that is neither zero nor. The existence of a liit shape was not previously verified for this regie but follows fro our ethods see Section 3 below). By asyptotic forula we ean a forula giving N n l, ) up to a factor of + o); such asyptotic equivalence is denoted with the sybol. By the obvious syetry N n l, ) = N l n l, ) it suffices to consider only the case A 2B >. To state our results, given A 2B > we define three quantities c, d and. The quantities c and d are the unique positive real solutions see Lea 7) to the siultaneous equations A = B = e c dt dt = d log where we recall the dilogarith function e c+d ) e c, 2) t e c dt dt 2 = d log e c d ) + dilog e c ) dilog e c d ) d 2, 3) dilog x) = x log t t dt = k= x) k for x <. The quantity, which will be seen to be strictly positive, is defined by = 2Bec e d ) + 2Ae c ) d 2 e d+c )e c ) k 2 A2 d 2. 4) Theore. Given, l and n, let A := l/ and B := n/ 2 and define c, d and as above. Let K be any copact subset of {x, y) : x 2y > }. As with l and n varying so that A, B) reains in K, where c and d vary in a Lipschitz anner with A, B) K. e )] [ca+2db log e c d N n l, ) 2π 2 e c ) e c d ), 5) Reark. In the special case B = A/2, the paraeters take on the eleentary values ) A + d =, c = log, and = A2 A + ) 2. A 2 In this case we understand the exponent and leading constant to be their liits as d, giving [ ] 3 A + ) A+ N A2 /2A, ) Aπ 2 A A. To explain Theore we consider the special case B = A/2, when the Young diagra fills half the area of the rectangle. Takács [Tak86] observed that for typical partitions of this type, the gaps between part sizes behave like independent geoetric rando variables with ean A. Counting the partitions is therefore equivalent as further explained below) to coputing the probability that these + geoetric rando variables will su to precisely l and that the area of the ensuing Young diagra will be precisely n. A local central liit theore iediately yields a sharp asyptotic estiate. With further work, Takács obtained bounds on the relative error that are of order + l) 3. These error bounds are eaningful for n differing fro l/2 by up to a few ultiples of log + l) standard deviations. If l = θ), this eans that B A/2 2 = Θ 3/2 log ). When B A/2 /2 log, the error is uch bigger than the ain ter of the Gaussian estiate provided by the LCLT. 2

4 We circuvent this liitation on the use of the LCLT using a technique fro the theory of large deviations. Specifically, we eploy a so-called tilted easure for which axiu likelihood occurs at any desired pair A, B). The tilted easure replaces the IID geoetric rando variables by independent but not identically distributed geoetric rando variables, where the paraeter p i for the ith variable varies in a log-linear anner. This requires extending a two-variable lattice LCLT to handle non-identically distributed suands. While this result, Lea 4, is copletely predictable, we could not find it in the literature, hence we include a proof in the Appendix. Previous results on N n ) and N n l, ) There are nuerous previous results on N n ), the nuber of partitions of n with part sizes bounded by. Erdös and Lehner [EL4] showed N n ) n! )! for = on /3 ) in 94, which was generalized by Szekeres, and others, ultiately leading to asyptotics of N n ) for all in 953 [Sze53]. Szekeres siplified his arguents a nuber of ties, ultiately giving asyptotics using only a saddle-point analysis, without needing results on odular functions. His arguent has been referred to as the Szekeres circle ethod. Mann and Whitney [MW47] showed, through the study of an equivalent statistical proble, that the size of a unifor rando partition in the l rectangle satisfies a noral distribution. As previously noted, Takács [Tak86] used a local central liit theore to show that when l + N n l, ) l ) φ n l/2 σ l, ) n l/2 < Kσ l, = O ll + ), ) σ l, 6) where σ l, = ll + + )/2, φx) = e x2 /2 / 2π, and K is a positive constant; see also [AA9]. Figure shows the predicted exponential growth of Takács copared to the actual exponential growth of partitions outside the valid asyptotic region. Figure : Exponential growth of N B 2, ) predicted by Takács forula blue, above) copared to the actual exponential growth given by Theore red, below). The results of Erdös and Lehner [EL4] iply that the expected axial part and thus also nuber of parts) in a partition of size n is typically f n logf n ), where f n := 6n π. Near this expected axiu, N n) 3

5 behaves like a doubly-exponential distribution in [Sze9]. When 6n log n 6 n 3/4 < l, < 4π π log n Szekeres [Sze9, Theore ] used saddle-point techniques to express N n l, ) in ters of N n, λ := µ := π 6n. If, in fact, ) 6n 6n log n π 4 + ε log n < l, < π πl 6n and for soe ε >, then the distributions defined by l and are independent and equal, and Szekeres forula siplifies to [ 6n N n l, ) N n exp λ + µ) e λ + e µ)]. π The Szekeres circle ethod has ore recently been applied to the study of partitions by Richond [Ric8]. Uniodality Sylvester s proof of uniodality of N n l, ) in n [Syl78], and ost subsequent proofs [Sta84, Sta85, Pro82], are algebraic, viewing N n l, ) as diensions of certain vector spaces, or their differences as ultiplicities of representations. While there are also purely cobinatorial proofs of uniodality, notably O Hara s [O H9] and the ore abstract one in [PR86], they do not give the desired syetric chain decoposition of the subposet of the partition lattice. These ethods do not give ways of estiating the asyptotic size of the coefficients or their difference. It is now known that N n l, ) is strictly uniodal [PP3], and the following lower bound on the consecutive difference was obtained in [PP7, Theore.2] using a connection between integer partitions and Kronecker coefficients: N n l, ) N n l, ).4 2 s where n l/2 and s = in{2n, l 2, 2 }. In particular, when l = we have s = 2n., 7) s9/4 Any sharp asyptotics of the difference appears to be out of reach of these algebraic ethods, however as a consequence of Theore we obtain the following estiate. Theore 2. Given, l and n, let A := l/ and B := n/ 2 and define d as above. Suppose, l and n go to infinity so that A, B) reains in a copact subset of {x, y) : x 2y > } and n l/2. Then N n+ l, ) N n l, ) d N nl, ). Reark. The condition n l/2 is equivalent to A B/2 and also to d / O ). It is autoatically satisfied whenever K is a copact subset of {x, y) : x > 2y > }. 4

6 2 A discretized analogue to Theore Large deviations heuristic Although we do not need to invoke any results fro the theory of large deviations, it ight be helpful to know the LD origins of the probability odel by which the proof of Theore is reduced to a local central liit theore. The proof of Takács result ay be suarized as follows. Let {λ j : j } denote the parts, in decreasing size, of a partition of n into at ost parts of size at ost l, padded with zeros at the end if necessary. Defining λ := l and λ + :=, the nubers x j := λ j λ j+ satisfy the following two identities see Figure 2), x j = l ; jx j = n. 8) j= j= l λ λ 2 x x x ix i λ i λ i+ x i λ Figure 2: The total area n of a partition is coposed of rectangles of area jx j By the reduced geoetric distribution with paraeter p we ean one less than a geoetric with ean /p; that is, X has this distribution if PX = k) = p q k where q := p. Let {X j : j } be a collection of independent reduced geoetric rando variables with paraeter p = /2. This distribution has the crucial property that for any set of values x,..., x, the probability PX j = x j : j =,..., ) depends only on the su l := j= x j and is equal to p + q l. Let P l) denote the su of PX = x) over all + )-vectors x with coordinate su l. If Nl) denotes the nuber of such vectors suing to l, we see iediately that P l) = p + q l Nl), hence Nl) [p + q l ]. In fact, this inequality is good because P l) is not all that sall: it is of order /2. Now let Nl, n) count those vectors satisfying both identities in 8) and P l, n) be the probability that the geoetric variables lie in this set. Takács gave a sharp asyptotic estiate of P l, n) c 2 when n = 2 /2 + O 3/2 ), thereby showing that Nl, n) c 2 [p + q l ]. Central liit theores do not provide a sharp estiate when n 2 /2 3/2. However, because the constraints on the vectors counted by Nl, n) are linear, the theory of large deviations [DZ98] iplies that P l, n) is well estiated by tilting the independent laws of the {X j } so that one is in the central liit regie of the tilted laws. Having solved for the correct tilt, we ay dispense with the LD theory and prove the estiates directly. Tilting preserves the reduced geoetric faily, altering only the paraeters; it turns out that the correct tilt akes q j := p j log-linear. 5

7 Stateent of discretized result With c, d to be specified later, let q j := e c jd/, let p j := q j and let L := log p j. j= Let P be a probability law aking the rando variables {X j : j } independent reduced geoetrics with respective paraeters p j. Define rando variables S and T by S := X i ; T := ix i, 9) i= corresponding to the unique partition λ satisfying X j = λ j λ j+. We first prove a result siilar to Theore, except that the paraeters c and d that solve integral Equations 2) and 3) are replaced by c and d satisfying the discrete suation Equations ) and ) below. These equations say that ES = l and ET =. Writing this out, using EX j = /p j = / e c dj/), gives l = j= n = i= + ) ) e c dj/ j/ + ). ) e c dj/ 2 j= Let M denote the covariance atrix for S, T ). The entries ay be coputed fro the basic identity Var X j ) = q j /p 2 j, resulting in e c dj/ Var S ) = ) e c 2 2) d j/ Cov S, T ) = Var T ) = j= e c dj/ j ) e c 2 3) d j/ j= j= j 2 e c dj/ e c d j/ ) 2. 4) Theore 3 discretized analogue). Let c and d satisfy ) ). Define α, β and γ n to be the noralized entries of the covariance atrix α := Var S ) ; β := 2 Cov S, T ) ; γ := 3 Var T ), which are O) as. Again, let A := l/ and B := n/ 2 and := α γ β. 2 Then { N n l, ) 2π 2 exp L )} + c A + d B. 5) Proof of discretized result The atoic probabilities P X = x) depend only on S and T as log P X = x) = log p j + x j log q j ) j= 6

8 = L j= = L c c + j d j= x j ) x j d jx j. j= In particular, for any x satisfying 8), log PX = x) = L c l d n. 6) These three things are equivalent: i) the vector X satisfies the identities 8); ii) the pair S, T ) is equal to l, n); iii) the partition λ = λ,..., λ ) defined by λ j λ j+ = X j for 2 j, together with λ = l X and λ = X, is a partition of n fitting inside a l rectangle. Thus, N n l, ) = P [S, T ) = l, n)] exp L + c l + d ) n [ = P [S, T ) = l, n)] exp L )] + c A + d B. 7) Coparing 5) to 7), the proof is copleted by an application of the following LCLT, for which a proof is given in the Appendix. This result is stated for an arbitrary sequence of paraeters p,..., p bounded away fro and, though we need it only for p j = e c dj/. For a 2 2 atrix M, denote by Ms, t) := [s, t] M [s, t] T the corresponding quadratic for. Lea 4 LCLT). Fix < δ < and let p,..., p be any real nubers in the interval [δ, δ]. Let {X j } be independent reduced geoetrics with respective paraeters {p j }, S := j= X j, and T := j= jx j. Let M be the covariance atrix for S, T ), written ) α β M = 2 β 2 γ 3, Q denote the inverse atrix to M, and = 4 det M = α γ β. 2 Let µ and ν denote the respective eans ES and ET. Denote p a, b) := PS, T ) = a, b)). Then sup 2 p a, b) a,b Z 2πdet M ) /2 e 2 Qa µ,b ν) 8) as, uniforly in the paraeters {p j } in the allowed range. In particular, if the sequence a, b ) satisfies Q a µ, b ν ) then PS = a, T = b ) = 2π + O 3/2)). 2 The following consequence will be used to prove Theore 2. Corollary 5 LCLT consecutive differences). Let N a, b) := e 2πdet M) /2 2 Qa µ,b ν) be the noral approxiation in Equation 8). Using the notation of Lea 4, sup pa, b + ) pa, b) N a, b + ) N a, b) ) = O 4 ). a,b Z 7

9 3 Liit shape Using the independent rando variables X i we can derive the liit shape for the partitions of n inside a rectangle; i.e., the curve which approxiates ost Young diagras of λ n. The existence of the liit j curve follows fro the easy fact that the axiu discrepancy ax j i= X i /p i ), conditional on i= X i = l, is o) in probability. We now discuss how to copute the liit shape function x Eλ x. A, B) =, /k) k = 2,..., 5 A, B) = 5/k, /k) k = 2,..., 5 Liit curve of A, B) =, /3) and rando partitions of size 2, 2 and 3. Figure 3: Liit shapes of scaled partitions as. Let λ be a partition defined by X, X,... in our setup. Then λ i = l X + X + + X i ) and hence E[λ i ] = l i j= /p j ). Using the integral approxiation, setting i := x we have that y := E [ ] x λi = A + x e c dt dt = A + x d ln e xd+c ) e c, and the liit curve of partitions scaled to the A rectangle is given by points x, y) satisfying this equation. It can be rewritten, after expressing c in ters of d, as when A > 2B, and e A+)d = e d )e da y) + e Ad )e d x) 9) = e c )e da y) + e c e dx, y = A x) when A = 2B and d = ; see Figure 3 for soe exaples of liit curves. Note that the transforation x, y) x, A y) gives the shape of the copleentary partition inside the A rectangle. 4 Existence and Uniqueness of c, d We now show that for any A 2B > there exists unique positive constants c and d satisfying Equations 2) and 3). If A = B/2 then d = and c can be deterined uniquely, so we ay assue A > 2B >. The following lea will be used to show uniqueness. 8

10 Lea 6. Let ψ denote the ap taking the pair c, d) to A, B) defined by the two integrals in Equations 2) and 3), and let K be a copact subset of {x, y) : x > 2y > }. The Jacobian atrix J := D[ψ] is negative definite for all c, d), ) 2, and all entries of ψ and J respectively ψ and J ) are Lipshitz continuous on ψ [K] respectively K). Proof. Differentiating under the integral sign shows that the partial derivatives coprising the entries of D[ψ] are given by J A,c = J A,d = J B,c = J B,d = e c+dt) e c+dt) ) 2 dt t e c+dt) e c+dt) ) 2 dt t e c+dt) e c+dt) ) 2 dt t 2 e c+dt) e c+dt) ) 2 dt ; note that each ter is negative. Let ρ denote the finite easure on [, ] with density e c+dt) / e c+dt) ) 2 and let E ρ denote expectation with respect to ρ. Then and J A,c = E ρ [ ], J A,d = J B,c = E ρ [ t], J B,d = E ρ [ t 2 ], det J = E ρ [] E ρ [t 2 ] E ρ [t]) 2 = E ρ [] 2 Var σ [t], where Var σ [t] denotes the variance of t with respect to the noralized easure σ = ρ/e ρ []. In particular, det J is positive, and bounded above and below when c and d are bounded away fro, iplying the stated results on Lipshitz continuity. As J is real and syetric, it has real eigenvalues. Since the trace of J is negative while its deterinant is positive, the eigenvalues of J have negative su and positive product, eaning both are strictly negative and J is negative definite for any c, d >. Lea 7. For any A > and B, A/2) there exist unique c, d > satisfying Equations 2) and 3). Moreover, for a fixed A, when B decreases fro A/2 to, d increases strictly fro to and c decreases strictly fro log ) A+ A to. Proof. Solving Equation 2) for c assuing d ) gives e A+)d ) c = log. e A+)d e d Substituting this into Equation 3) gives an explicit expression for B in ters of A and d, and shows that for fixed A > as d goes fro to infinity B goes fro A/2 to. By continuity, this iplies the existence of the desired c and d. It also shows that, for a fixed A, c is a decreasing function of d with the given axial and inial values as d goes fro to. To prove uniqueness, we note that for x, y R 2 Stokes theore iplies so that ψy) ψx) = x y) T ψy) ψx)) = D[ψ] tx + t)y) x y) dt [ x y)t D[ψ] tx + t)y) x y) ] dt. 9

11 When x y, negative-definiteness of D[ψ] iplies that the last integrand is strictly negative on [, ], and ψy) ψx). Thus, distinct values of c and d give distinct values of A and B. To see the onotonicity, let A be fixed and let F B d) = B be the equation obtained after substituting c = ca, d) above in equation 3), i.e. F B d) = ψ 2 ca, d), d). Then d is a decreasing function of B and vice versa since F B d) d = J B,dJ A,c J A,d J B,c J A,c = det D[ψ] J A,c <. 5 Proof of Theore fro the discretized result Here we show how c and d fro the discretized result are related to c, d defined independently of. The proof below also shows that c and d exist and are unique. The Euler-MacLaurin suation forula [db58, Section 3.6] gives an expansion L = log e c dt ) dt + log e c ) + log e c d ) 2 + O 2 ) = dilog ) dilog e e c d c ) + log ) + log e e c c d ) + O 2 ) 2) d 2 of the su L in ters of c and d. Assue that there is an asyptotic expansion c = c + u + O 2 ) 2) d = d + v + O 2 ) 22) as, where u and v are constants depending only on A and B. Under such an assuption, substitution of Equations 2) and 22) into Equation 2) iplies L = dilog e c d ) dilog e c ) ua + vb + + O 2 ) d = log e c d ua + vb ) db + + O 2 ). 23) Substituting Equations 2) 23) into Equation 5) of Theore 3 and taking the liit as then gives Theore, as e c dt ) e c dt ) 2 dt t 2 e c dt ) e c dt ) 2 dt te c dt ) 2 e c dt ) 2 dt =. It reains to show the expansions in Equations 2) and 22). For x, y >, define S x, y) := T x, y) := j= j= e x+yj/), j/ e x+yj/) 2.

12 Another application of the Euler-MacLaurin suation forula iplies with S c, d) = A + A c, d) + O 2 ), 24) T c, d) = B + B c, d) + O 2 ), 25) A = ) 2 e c + e c d and B = Let J denote the Jacobian D[ψ] of the ap ψ with respect to c and d, and let c, d ) = c, d) J A, B /2) T. A Taylor expansion around the point c, d) gives ) ) S c, d ) S c, d) T c, d = J + O )) ) T c, d) ) A / = + O 2) B /2 ) S c =, d ) + O 2), T c, d ) 2 e c d ). J A B )) + O 2 ) where Equations 24) and 25) were used to approxiate the Jacobian of ψ : x, y) S x, y), T x, y) ) with respect to x and y. The ap ψ is Lipschitz for a siilar reason as its continuous analogue. Naely, consider the partial derivatives J S,x = j= J S,y = 2 j= J T,x = 2 j= J S,y = 3 j= e x yj/ e x yj/ ) 2 j e x yj/ e x yj/ ) 2 j e x yj/ e x yj/ ) 2 j2 e x yj/ e x yj/ ). 2 Let ρ be a discrete finite easure on R := {, /, 2/,..., } with density e x yt / e x yt ) for t R and otherwise, and let E ρ be the expectation with respect to ρ. Then and J S,x = E ρ [ ], J T,x = J S,y = E ρ [ t], J T,y = E ρ [ t 2 ] det D[ψ ] = E ρ []E ρ [t 2 ] E ρ [t] 2 = E ρ [] 2 Var σ [t], where σ is the probability function ρ /E ρ []. For any fixed and x, y) in a copact neighborhood of A, B), both the variance and the expectation are finite and bounded away fro, as is the Jacobian deterinant. Moreover, the trace TrD[ψ] = E ρ [ + t 2 ] is bounded away fro and infinity, so the Jacobian is negative definite with locally bounded eigenvalues, and hence ψ is locally Lipschitz. Since the nor of the Jacobian is bounded away fro and infinity, we have that the inverse ap ψ is also locally Lipschitz in a neighborhood of ψ A, B). Moreover, siilarly to proof of existence and uniqueness of c and

13 d in Section 4, we have that there indeed are c and d as unique solutions of Equations ) and ) since the Jacobian is negative sei-definite. The trapezoid forula iplies J S,c J A,c = O ), and siilar bounds for the other differences of partial derivatives in the continuous and discrete settings. Hence, the bounds for the nors and eigenvalues of D[ψ ] are within O ) of the ones for D[ψ], and ψ and its inverse) is Lipschitz with a constant independent of. Thus, O 2 ) = ψ c, d ) ψ c, d ) C c c, d d ) for soe constant C, so that the expansions 2) and 22) hold. 6 Proof of Theore 2 We will prove Theore 2 fro Equation 7) and Corollary 5. Let p l, n) = P [S, T ) = l, n)] and let Then c and d are the solutions to L x, y) := A x, y) := B x, y) := log e x yj/ ), 26) j= + ), 27) e x yj/ j= j= j/ e + x yj/ 2 A c, d ) = l = A, B c, d ) = n/ = B.. 28) Let c, d be the solutions to A c, d ) = l and B c, d ) = n+)/, and let x = c c = O 2 ) and y = d d = O 2 ) by the Lipschitz properties proven in Section 4. Observe that L x, y) x = A x, y) and L x, y) y = B x, y). 29) Using the Taylor expansion for L c, d ) around c, d ) and the L partial derivatives fro Equation 29), L c, d ) = L c + x, d + y) = L c, d ) x A c, d ) y B c, d y ) + O 3 ), so that L c, d ) + c + x)l + d + y)n + ) = L c, d ) + c l + d n + ) + O 3 ). To lighten notation, we now write L := L c, d ) and L := L c, d ). Then [ ] [ N n+ l, ) N n l, ) = p l, n + ) exp L + c l + d n + ) p l, n) exp L + c l + d ] n [ = p l, n) exp L + c l + d ] [ n e d/ ] 3) [ + [p l, n + ) p l, n)] exp L + c l + d ] n + ) 3) 2

14 We now bound each of these suands. + p l, n + ) e L +c l+d n+)/ e L+cl+dn+)/). 32) Since d = d + O ), Equation 7) iplies that the quantity on line 3) equals ) d N n l, ) + O 2 ) as long as d / O ). This holds when A B/2 / O ) as d = when A = B/2 and the ap taking A, B) to c, d) is Lipschitz. By Corollary 5, [p l, n + ) p l, n)] N l, n + ) N l, n) + O 4 ) ) = O 2 e 2 Q,) + O 4 ) = O 4 ), where Q is the inverse of the covariance atrix of S, T ). O 4 2 N n l, )) = O 2 N n l, )). Let Thus, the quantity on line 3) is [ ψ := exp L + c l + d n + ) L + c l + d n + ) ) ] = O 3 ). As p l, n + ) = p l, n) + O 4 ), it follows that the quantity on line 32) is p l, n + ) e L+cl+dn+)/ ψ = N n l, ) ψ e d/ + O 4 e d/ e L+cl+dn/ ψ ) = O 3 N n l, )). Putting everything together, as desired. N n+ l, ) N n l, ) = N n l, ) ) d + O 2 ), Appendix: Proof of the Local Central Liit Theore Throughout this section, /2 δ > is fixed and {p j : j } are arbitrary nubers in [δ, δ]. The variables {X j } and S, T ) are as in Lea 4; we drop the index on the reaining quantities α, β, γ,, µ, ν, p a, b) and the atrices M and Q. Recall the quadratic for notation Ms, t) := [s, t] M [s, t] T. Lea 8. The constants α, β, γ and are bounded below and above by positive constants depending only on δ. [ Proof: Upper and lower bounds on α, β and γ are eleentary: α [ ] δ δ) and γ, 3 δ) 2 3δ 2. The upper bound on follows fro these. 3 δ δ), δ) 2 δ 2 ] [, β δ δ), 2 δ) 2 2δ 2 ]

15 ) For the lower bound on, let M αn β = n denote M without the factors of. We show is β n γ n bounded fro below by the positive constant 4 3)δ/6. A lower bound for the deterinant of M is λ 2 where λ is the least odulus eigenvalue of M; note that λ 2 = inf θ Mcos θ, sin θ). We copute Mcos θ, sin θ) = E cos θs + sin θt ) 2 δ k= cos θ + k sin θ ) 2 > δ cos 2 θ + cos θ sin θ + ) 3 sin2 θ. This is at least 4 3 δ for all θ, proving the lea. 6 Lea 9. Let X p denote a reduced geoetric with paraeter p. For every δ, /2) there is a K such that siultaneously for all p [δ, δ], ) log E expiλx qp q2 p) i λ 2p 2 λ2 Kλ 3. Proof: For fixed p this is Taylor s reainder theore together with the fact that the characteristic function φ p λ) of X p is thrice differentiable. The constant Kp) one obtains this way is continuous in p on the interval, ), therefore bounded on any copact sub-interval. Proof of the LCLT: The proof of Lea 4 coes fro expressing the probability as an integral of the characteristic function, via the inversion forula, and then estiating the integrand in various regions. Let φs, t) := Ee iss+tt ) denote the characteristic function of S, T ). Centering the variables at their eans, denote Ŝ := S µ, T := T ν, and φs, t) := Ee isŝ+t T ) so that φs, t) = φs, t)e isµ+itν. Then pa, b) = 2π) 2 = 2π) 2 π π π π π π π π e isa itb φs, t) ds dt e isa µ) itb µ) φs, t) ds dt. 33) Following the proof of the univariate LCLT for IID variables found in [Dur], we observe that 2πdet M) /2 e 2 Qa µ,b ν) = 2π) 2 e isa u) itb v) exp 2 ) Ms, t) ds dt. 34) Hence, coparing this to 33) and observing that e isa µ) itb ν) has unit odulus, the absolute difference between pa, b) and the left-hand side of 34) is bounded above by 2π) 2 s,t) [ π,π] 2 φs, t) e /2)Ms,t) ds dt. 35) Fix positive constants L and ε to be specified later and decopose the region R := [ π, π] 2 as the disjoint union R + R 2 + R 3, where R = [ L /2, L /2 ] [ L 3/2, L 3/2 ] R 2 = [ ε, ε] [ ε, ε ] \ R 4

16 π L//2 /2 L/ ε / ε / R 2 R L/3/2 L/3/2 π π ε ε π Figure 4: The regions R R 2 R in the proof of the LCLT. see Figure 4 for details. R 3 = R \ R R 2 ) ; As e /2)Ms,t) ds dt decays exponentially with, it suffices to obtain the following estiates R c 2 φs, t) e /2)Ms,t) ds dt = O 5/2) 36) R φs, t) e /2)Ms,t) ds dt = O 5/2 ) 37) R 2 φs, t) ds dt = o 3). 38) By independence of {X j }, log φs, t) = R 3 log Ee is+jt)xj µj). Using Lea 9 with p = p j gives log Eeis+jt)Xj qj/pj) + q j s + jt) 2 K s + jt 3. The su of q j /p 2 j )s + jt)2 is Ms, t), therefore suing the previous equation over j gives j= 2p 2 j log φs, t) + 2 Ms, t) K s + jt 3. 39) On R we have the upper bound s + jt s + t 2L /2. Thus, j= j= s + jt 3 + )8L 3 ) 3/2 = O /2). Plugging this into 39) and exponentiating shows that the left hand side of 36) is at ost R O /2 ) = O 5/2 ). To bound the integral on R 2, we define the sub-regions { ) S k := x, y) : k ax /2 x, 3/2 y } k +. 5

17 As the area of S k is 8k + 4) 2, R 2 ɛ φs, t) e /2)Ms,t) ds dt k=l S k k=l φs, t) e Ms,t)/2 dsdt ɛ 2 8k + 4) ax φs, t) e Ms,t)/2. 4) s,t) S k We break this last su into two parts, and bound each part. For s, t) R 2, we have s+jt s + t 2ε so that s + jt 3 2ε s + j t ) 2 2ε )M s, t ). j= Coparing this to 39) shows we ay choose ε sall enough to guarantee that log φs, t) + 2 Ms, t) Ms, t), 4 j= so φs, t) e /4)Ms,t). Lea 8 shows there is a positive constant c such that the iniu value of Ms, t) on R 2 is at least ck 2. Thus, for s, t) S k, φs, t) e Ms,t)/2 e Ms,t)/4 + e Ms,t)/2 2e ck 2. log If r := )/c then 8k + 4)k + ) ax φs, t) e Ms,t)/2 2 s,t) S k k=r Furtherore, for s, t) S k there exist constants C and C such that log φs, t) + Ms, t)/2 C j= k=r 8k + 4)k + )e ck = O polylog)) = O /2 ). 4) s + jt 3 C 2k + ) /2) 3 + ) = C k 3 /2. This iplies the existence of a constant K > such that for k r and s, t) S k, φs, t) e Ms,t)/2 e = Ms,t)/2 log φs,t)+ms,t)/2 e Ke ck2 k 3 /2. 2 Thus, r k=l r 8k + 4)k + ) ax φs, t) e Ms,t)/2 K /2 8k + 4)k + )k 3 e ck2 s,t) S k k=l = O /2 ). 42) Cobining 4) 42) gives 37). Finally, for 38), we clai there is a positive constant c for which φs, t) e c on R 3. To see this, observe see [Dur, p. 44]) that for each p there is an η > such that φ p λ) < η on [ π, π]\[ ε/2, ε/2]. 6

18 Again, by continuity, we ay choose one such η valid for all p [δ, δ]. It suffices to show that when either s or t is at least ε, then at least /3 of the suands log Ee is+jt)xj µj) have real part at ost η. Suppose s ε the arguent is the sae for s ε). Interpreting s + jt odulo 2π always to lie in [ π, π], the nuber of j [, ] for which s + jt [ ε/2, ε/2] is at ost twice the nuber for which s + jt [ε/2, ε], hence at ost twice the nuber for which s + jt / [ ε/2, ε/2]; thus at least /3 of the + values of s + jt lie outside [ ε/2, ε/2] and these have real part of log Ee is+jt)xj µj) η by choice of η. Lastly, if instead one assues π t ε/, then at ost half of the values of s + jt odulo 2π can fall inside any interval of length ε/2. Choosing η such that the real part of log Ee is+jt)xj µj) is at ost η outside of [ ε/4, ε/4] finishes the proof of 38) and the LCLT. Proof of Corollary 5. In order to estiate the error ters in the approxiation of pa, b) we will consider the partial differences and repeat the approxiation arguents above. Changing b to b + in Equations 33) and 34) iplies pa, b + ) pa, b) N a, b + ) N a, b) ) = e it φs, t) e /2Ms,t) dsdt. 43) [ π,π] 2 For s, t) R 3, the proof of the LCLT shows that the integral in Equation 43) decays exponentially with. As e it = 2 2 cost) t = O 3/2 ) for s, t) R, the proof of the LCLT shows that the integral in Equation 43) grows as O 3/2 5/2 ) = O 4 ). Finally, since e it t k+) 3/2 for s, t) S k following the proof of the LCLT shows that e it φs, t) e /2Ms,t) dsdt 7/2 R 2 = O 4 ). ɛ k=l 8k + 4)k + ) ax φs, t) e Ms,t)/2 s,t) S k References [AA9] Gert Alkvist and George E. Andrews, A Hardy-Raanujan forula for restricted partitions, J. Nuber Theory 38 99), no. 2, MR 367 [And76] George E. Andrews, The theory of partitions, Addison-Wesley Publishing Co., Reading, Mass.- London-Asterda, 976, Encyclopedia of Matheatics and its Applications, Vol. 2. MR 5573 [db58] N. G. de Bruijn, Asyptotic ethods in analysis, Bibliotheca Matheatica. Vol. 4, North-Holland Publishing Co., Asterda; P. Noordhoff Ltd., Groningen; Interscience Publishers Inc., New York, 958. MR [Dur] Rick Durrett, Probability: theory and exaples, fourth ed., Cabridge Series in Statistical and Probabilistic Matheatics, vol. 3, Cabridge University Press, Cabridge, 2. MR [DZ98] [EL4] [HR8] Air Debo and Ofer Zeitouni, Large deviations techniques and applications, second ed., Applications of Matheatics New York), vol. 38, Springer-Verlag, New York, 998. MR 6936 Paul Erdös and Joseph Lehner, The distribution of the nuber of suands in the partitions of a positive integer, Duke Math. J. 8 94), MR 484 G. H. Hardy and S. Raanujan, Asyptotic Forulaae in Cobinatory Analysis, Proc. London Math. Soc. 2) 7 98), MR [MW47] H. B. Mann and D. R. Whitney, On a test of whether one of two rando variables is stochastically larger than the other, Ann. Math. Statistics 8 947), 5 6. MR

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