SOFT ASYMPTOTICS FOR GENERALIZED SPT ... AWS
SOFT ASYMPTOTICS FOR GENERALIZED SPT-FUNCTIONS ROBERT C. RHOADES Abstract. We derive a leading order asymptotic for the difference between the second moments of Garvan’s generalized rank statistics. The approach uses Dixit and Yee’s combinatorial interpretation of the difference and soft probabilistic techniques to deal with random partitions.
1. Introduction Andrews [2] introduced the study of the spt-function. For each partition λ of n let wspt (λ) be the number of occurrences of the smallest part in λ and X spt(n) = wspt (λ) λ
where the sum is over all partitions of n. Andrews proved that 1 spt(n) = (M2 (n) − N2 (n)) 2 where M2 is the second moment of the crank statistic and N2 is the second moment of the rank statistic. Later, Ji [17] gave a combinatorial proof. This yields a striking relation between the difference of the moments and proves that it is always positive. Garvan [15] later generalized the spt-function to prove that M2` (n) − N2` (n) > 0 for each n > 0 and ` > 0 where N2` and M2` are the 2`th moment of the rank and crank statistic, respectively. The study of rank and crank moments, initiated by Andrews [1], has attracted a lot of further study. See, for instance, the works of the Bringmann, Mahlburg, and the author [7] for detailed discussion of the asymptotics for the moments. The rank and crank were generalized by Garvan [13] to a family of partition statistics called the generalized ranks. For each j ∈ N Garvan defined a statistic rankj (λ), referred to as the j-rank. rank1 is the crank statistic and rank2 is the rank statistic. The moments of the j-rank statistic are defined by X rankj (λ)2k j N2k (n) := λ
where the sum is over all partitions of n. The odd moments, by symmetry, are 0. Recently Dixit and Yee [8] generalized Garvan’s higher order spt construction to prove the inequality j N2k (n)
>
j+1 N2k (n)
for all j ≥ 1, k > 0, and n ≥ 1. Date: May 21, 2012. 2000 Mathematics Subject Classification. 11F37,11F25,11F30. Key words and phrases. crank, rank, partition statistic. The author is supported by an NSF Postdoctoral Fellowship. 1
2
ROBERT C. RHOADES
Following the works of Andrews [2] and Garvan [15], Dixit and Yee [8] provide an elegant combinatorial interpretation of the value 1 (M2 (n) − j+1 N2 (n)) 2 called the generalized spt-functions. It is striking that modular techniques can be used to provide nearly exact formulas for the Sptj (n) and in fact for any j N2k (n) as n → ∞. Such results exploit the “mock” Jacobi form structure of the j-rank generating functions and circle method techniques developed by Bringmann and Mahlburg [5, 6]. This program has been carried out by Waldherr [19]. While modular techniques provide very strong theorems, they can become unwieldy for large j (see [18]) and ignore the combinatorial interpretation of the statistics being considered. The purpose of this note is to use soft combinatorial and probabilistic tools to provide leading order asymptotics for Sptj (n) for each j. This approach has the advantage that is uses the combinatorial description of this functions and exploits the typical shape of a partition. For instance, the proof gives some insight into why Sptj+1 (n) > Sptj (n) or equivalently why Sptj (n) :=
j N2 (n)
>
j+1 N2 (n)
for each n. Theorem 1.1. For each j > 0, as n → ∞ Sptj (n) =
j √
r
2π 2n
exp π
2n 3
! (1 + oj (1)) .
The argument presented here gives Sptj (n) ∼ jspt(n). Moreover, our proof shows that the contribution to Sptj (n) from each partition of n is almost always equal to j times the number of 1s in λ. Unwinding definitions, this tells us that j N2 (n) > j+1 N2 (n) for each j and large enough n because the number of 1s is almost surely larger than j. Following Dixit and Yee [8], to interpret Sptj (n) we associated to each λ a partition of n a weight Wj (λ). The weight counts the multiplicity numbers of the parts that appear in the lower Durfee squares of the partition. See Section 2.1 for details. As a result, the weight is determined by the smallest parts of the partition. In particular, if λ has at least j parts of size 1, then the weight is determined by the number of 1s. Fristedt’s results [12] or classical q-series results and Tauberian theorems give that the probability that a partition of size n has less than j parts of size 1 is asymptotic to √πj6n . Our proof is devoted to showing that the contribution from such partitions is negligible. The following result for the second generalized rank moments is straightforward from Theorem 1.1. Corollary 1.2. For each j ≥ 0, as n → ∞ r j+1 N2 (n)
= exp π
2n 3
!
1 2j + 1 √ − √ + oj (1) . 2 3 2π 2n
In particular, Dyson proved M2 (n) = 1 N2 (n) = 2np(n) where p(n) is the number of partitions of n. Therefore, Rademacher’s formula for p(n) gives an exact formula for M2 (n). Consequentially, it gives the above corollary for j = 0. It is an interesting problem to use Garvan’s combinatorial interpretation of the higher order Spt functions and techniques from random partitions to deduce asymptotics for differences of other
SOFT ASYMPTOTICS FOR GENERALIZED spt-FUNCTIONS
3
rank moments. It seems this approach would lead to identities for special values of the Riemann zeta-function. In Section 2 we give background about the combinatorial description of the Spt-functions. We also give some basic results about random partitions. In Section 3 we give the proof of the main theorem. 2. Probability and Combinatorics Background In this section we give some combinatorial background on the generalized ranks and probabilistic background on random partitions. Throughout we let Xj (λ) be the number of times j occurs as a part in the partition λ. P 2.1. Combinatorics Background. We briefly recall the definition of Sptj (n) = λ Wj (λ) where the sum is over all partitions of n. To define the weight Wj Dixit and Yee used the lower-Durfee squares. The largest square that fits inside the Ferrers diagram of λ starting from the lower left corner is the lower-Durfee square. The second lower-Durfee square is the largest square that fits inside of λ right above the first lower-Durfee square. Successive lower-Durfee squares are defined similarly. We mark each part of a partition with a ‘multiplicity number’. For example, if λ = (7, 6, 5, 5, 2, 1, 1, 1, 1) then the marked version of λ is (71 , 61 , 51 , 52 , 21 , 11 , 12 , 13 , 14 ). That is, if a positive integer occurs as a part in λ, we mark all of its occurences with positive integers in an increasing order from the left to the right. If λ is a partition, we define Wj (λ) as the the sum of the marks of each part in the first (j − 1) successive lower-Durfee squares and the mark of the part directly above the (j − 1)st lower-Durfee square. If λ has fewer than j − 1 successive lower-Durfee squares, Wj (λ) is defined to be the sum over all of the marks in the partition. For example, when λ = (7, 6, 5, 5, 2, 1, 1, 1, 1) we have W1 (λ) = 4, W2 (λ) = 4 + 3 = 7, W3 (λ) = 4 + 3 + 2 = 9, W4 (λ) = 4 + 3 + 2 + 1 = 10, and W5 (λ) = 4 + 3 + 2 + 1 + 1 = 11. Also, W6 (λ) = 4 + 3 + 2 + 1 + (1 + 2) + 1 = 14, since the first four lower-Durfee squares are of size 1 and the fifth lower-Durfee square is 2 × 2 and contains the part of size 2 and the second part of size 5. Likewise, W7 (λ) = 4 + 3 + 2 + 1 + (1 + 2) + (1 + 1 + 1) = 16 and Wj (λ) = 16 for all j > 7 because there are exactly six lower-Durfee squares. Lemma 2.1. Let Xj (λ) be the number of parts of size j in the partition λ. Fix j ≥ 1. If X1 (λ) ≥ j then Wj (λ) = jX1 (λ) − j(j−1) 2 . If X1 (λ) = j − r with r ∈ {1, · · · , j} and X2 (λ) ≥ 2r, then j(j + 1) − r(r − 1) − 2jr . 2 Proof. Assume that X1 (λ) ≥ j. Then the first j − 1 lower-Durfee squares are of size 1. They contain the parts of size one with marks X1 (λ), X1 (λ) − 1, · · · , X1 (λ) − (j − 2). Moreover, the first part after the (j − 1)st lower-Durfee square is of size 1 and has mark X1 (λ) − (j − 1). This gives the first claim. The second claim follows in a similar way, by guaranteeing that the first (j − 1) successive lower-Durfee squares are all of size 1 or 2. Additionally, we have Wj (λ) = 2rX2 (λ) +
Wj (λ) =X1 (λ) + · · · + (X1 (λ) − (j − r − 1)) + [(X2 (λ) + (X2 (λ) − 1)) + · · · ((X2 (λ) − (2r − 2)) + (X2 (λ) − (2r − 1)))] + (X2 (λ) − (2r))
4
ROBERT C. RHOADES
2.2. Probability Background. In this subsection we give some results dealing with statistics of random partitions of n for large n. Recall the generating function for the number of partitions of n is ∞ ∞ Y X 1 (2.1) P (q) = = p(n)q n . 1 − qn n=1
n=0
p π2 e 6 and Moreover, the modularity of the generating function gives P (e− ) ∼ 2π r ! 1 1 2n 1+O √ (2.2) p(n) = √ exp π . 3 n 4 3n We will make use of the following Tauberian theorem of Ingham [16]. In the version given here it can be found in [4]. P n Theorem 2.2 (Ingham). Let f (z) = ∞ n=0 a(n)z be a power series with real nonnegative coefficients and radius of convergence equal to 1. If there exists A > 0, λ, α ∈ R such that A α f (z) ∼ λ(− log(z)) exp − log(z) as z → 1− , then n X
as n → ∞.
α
1
√ λ A 2 −4 a(m) ∼ √ An α 1 exp 2 2 π n 2 +4 m=0 √
Fristedt [12] proved that for large n a typical partition of n has about proved a distributional result for the number of parts of any small size.
6n π
partitions. In fact, he
Theorem 2.3 (Fristedt). Let Pn be the uniform measure on partitions of size n. We have π Pn { √ X1 (λ) ≤ y} → 1 − e−y 6n as n → ∞ uniformly in y. We use a q-series approach to establish the following result on the number of partitions with a constrained number of 1s orP 2s. Let fa (n) be the number of partitions of n that contain < a parts n a of size 1. So that Fa (q) = ∞ n=0 fa (n)q =P(1 − q )P (q). Similarly let fa,b (n) be the number of ∞ a 2b partitions of n with < a 1s and < b 2s. Then (n)q n = (1−q Straightforward P∞ n=0 fa,b P∞)(1−q )P (q). n applications of Theorem 2.2 with (1−q) n=0 fa (n)q and (1−q) n=0 fa,b (n)q n give the following. Proposition 2.4. For any a, b ≥ 0, as n → ∞ fa,b (n) πa abπ 2 fa (n) ∼√ and ∼ p(n) p(n) n 6n We give the following result which follows easily from Fristedt’s results on random partitions. Proposition 2.5. Let F(n) be the the set of partitions of n with any fixed number of 1s or the set of partitions with any fixed number of 1s or 2s. Then, as n → ∞ ` √ X X 1 Xj (λ) = O ( n log(n))` |F(n)| λ∈F (n)
j
SOFT ASYMPTOTICS FOR GENERALIZED spt-FUNCTIONS
5
for ` = 1, 2. Moreover, if F(n) is the set of partitions of n with a fixed number of 1s and at least b 2s, then X √ 1 X2 (λ) = O n . |F(n)| λ∈F (n)
Proof. The proof of each result is the same. Fristedt’s results prove that the number of 1s or 2s in 1 a partition is independent from the number of parts of size k for all k = o(n 4 ). Therefore, the final claim follows immediately and from Fristedt’s result that as n → ∞ 2π Pn { √ X2 ≤ v} → 1 − e−v . 6n The proof of the claim for proving
P
j
Xj (λ) is completely analogous to the proof of Fristedt’s result
π X Pn { √ Xj − log 6n j
√
! 6n −v } → e−e π
or one could follow Erdös-Lehner’s [11] original proof of thisPresult. In particular, considering partitions without 1s or 2s does not effect the expected size of j Xj (λ).
3. Proof of Theorem 1.1 In this section we give the proof of Theorem 1.1. Throughout this section all sums on λ range over the partitions of n, for some fixed n. We begin with the proof in the case of j = 1. Notice that if X1 (λ) > 0 then wspt (λ) = X1 (λ). Therefore, X
wspt (λ) =
X
λ
λ
X
X1 (λ) + O
X
Xj (λ)
λ:X1 (λ)=0 j
√
6n p(n) √ =p(n) (1 + o(1)) + O √ n log(n) π n r ! 2n 1 = √ exp π (1 + o(1)). 3 2π 2n where the second equality follows Theorem 2.3 and Propositions 2.4 and 2.5 and the final equality follows from (2.2). To handle the case of Sptj (n) with j > 1 we apply Lemma 2.1 and note that
Wj (λ) ≤
i (λ) X XX
i
k=1
i=
X1 i
2
Xi (λ)(Xi (λ) − 1).
6
ROBERT C. RHOADES
Thus applying Lemma 2.1 there are constants cj,b such that X X (j − 1)j jX1 (λ) − Wj (λ) = 2 λ
λ:X1 (λ)≥j
+
X
X
b∈{1,··· ,j}
λ: X1 (λ)=j−b X2 (λ)≥2b+1
(2bX2 (λ) + cj,b ) + O
X
X
a,b λ∈Fa,b (n)
X
Xi (λ)2 .
i
where Fa,b (n) is the set of all partitions of n with exactly a 1s and b 2s and the sum on a, b is over those pairs satisfying a < j and b < 2a + 1. Using Theorem 2.3 and Propositions 2.4 and 2.5 gives √ X 6n p(n) p(n) √ 2 Wj (λ) = j p(n)(1 + o(1)) + O √ · n + O · n log (n) π n n λ
An application of (2.2) gives the result. References [1] G. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), 37–73. [2] G. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008), 133-142. [3] G. E. Andrews and F. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167–171. [4] K. Bringmann, A. Holroyd, K. Mahlburg, and M. Vlasenko, k-run overpartitions and mock theta functions. preprint. [5] K. Bringmann and K. Mahlburg, An extension of the Hardy-Ramanujan Circle Method and applications to partitions without sequences. American Journal of Math 133 (2011) 1151–1178. [6] K. Bringmann and K. Mahlburg, Coefficients of Kac-Wakimoto Series, preprint. [7] K. Bringmann, K. Mahlburg, R.C. Rhoades, Taylor Coefficients of Jacobi Forms and Partition Statistics, to appear Math. Proc. Camb. Phil. Soc. [8] A. Dixit and A. J. Yee, Generalized Higher Order spt-functions. preprint. [9] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. [10] F. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169–180. [11] P. Edös and J. Lehner, The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, (1941) 335–345. [12] B. Fristedt, The Structure of Random Partitions of Large Integers. Trans. Amer. Math. Soc., 337 2, 703–735. [13] F. Garvan, Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84 (1994), no. 3-4, 343–359. [14] F. Garvan, Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank, Int. J. Number Theory 6 (2010), 1-29. [15] F. Garvan, Higher order spt-functions. Adv. Math. 228 (2011), no. 1, 241–265. [16] A. Ingham, A Tauberian theorem for partitions. Ann. of Math. 42 (1941) 1075–1090. [17] K-Q, Ji, A combinatorial proof of Andrews’ smallest parts partition function. Electron. J. Combin. 15 (2008), no. 1, Note 12, 6 pp. [18] M. Waldherr, Arithmetic and Asymptotic Properties of Mock Theta Functions and Mock Jacobi Forms. Ph. D. thesis Universität zu Köln. 2012. [19] M. Waldherr, Asymptotics for Moments of Higher Ranks. preprint. Stanford University, Serra Mall bldg. 380, Stanford, CA 94305 E-mail address: [email protected]
1. Introduction Andrews [2] introduced the study of the spt-function. For each partition λ of n let wspt (λ) be the number of occurrences of the smallest part in λ and X spt(n) = wspt (λ) λ
where the sum is over all partitions of n. Andrews proved that 1 spt(n) = (M2 (n) − N2 (n)) 2 where M2 is the second moment of the crank statistic and N2 is the second moment of the rank statistic. Later, Ji [17] gave a combinatorial proof. This yields a striking relation between the difference of the moments and proves that it is always positive. Garvan [15] later generalized the spt-function to prove that M2` (n) − N2` (n) > 0 for each n > 0 and ` > 0 where N2` and M2` are the 2`th moment of the rank and crank statistic, respectively. The study of rank and crank moments, initiated by Andrews [1], has attracted a lot of further study. See, for instance, the works of the Bringmann, Mahlburg, and the author [7] for detailed discussion of the asymptotics for the moments. The rank and crank were generalized by Garvan [13] to a family of partition statistics called the generalized ranks. For each j ∈ N Garvan defined a statistic rankj (λ), referred to as the j-rank. rank1 is the crank statistic and rank2 is the rank statistic. The moments of the j-rank statistic are defined by X rankj (λ)2k j N2k (n) := λ
where the sum is over all partitions of n. The odd moments, by symmetry, are 0. Recently Dixit and Yee [8] generalized Garvan’s higher order spt construction to prove the inequality j N2k (n)
>
j+1 N2k (n)
for all j ≥ 1, k > 0, and n ≥ 1. Date: May 21, 2012. 2000 Mathematics Subject Classification. 11F37,11F25,11F30. Key words and phrases. crank, rank, partition statistic. The author is supported by an NSF Postdoctoral Fellowship. 1
2
ROBERT C. RHOADES
Following the works of Andrews [2] and Garvan [15], Dixit and Yee [8] provide an elegant combinatorial interpretation of the value 1 (M2 (n) − j+1 N2 (n)) 2 called the generalized spt-functions. It is striking that modular techniques can be used to provide nearly exact formulas for the Sptj (n) and in fact for any j N2k (n) as n → ∞. Such results exploit the “mock” Jacobi form structure of the j-rank generating functions and circle method techniques developed by Bringmann and Mahlburg [5, 6]. This program has been carried out by Waldherr [19]. While modular techniques provide very strong theorems, they can become unwieldy for large j (see [18]) and ignore the combinatorial interpretation of the statistics being considered. The purpose of this note is to use soft combinatorial and probabilistic tools to provide leading order asymptotics for Sptj (n) for each j. This approach has the advantage that is uses the combinatorial description of this functions and exploits the typical shape of a partition. For instance, the proof gives some insight into why Sptj+1 (n) > Sptj (n) or equivalently why Sptj (n) :=
j N2 (n)
>
j+1 N2 (n)
for each n. Theorem 1.1. For each j > 0, as n → ∞ Sptj (n) =
j √
r
2π 2n
exp π
2n 3
! (1 + oj (1)) .
The argument presented here gives Sptj (n) ∼ jspt(n). Moreover, our proof shows that the contribution to Sptj (n) from each partition of n is almost always equal to j times the number of 1s in λ. Unwinding definitions, this tells us that j N2 (n) > j+1 N2 (n) for each j and large enough n because the number of 1s is almost surely larger than j. Following Dixit and Yee [8], to interpret Sptj (n) we associated to each λ a partition of n a weight Wj (λ). The weight counts the multiplicity numbers of the parts that appear in the lower Durfee squares of the partition. See Section 2.1 for details. As a result, the weight is determined by the smallest parts of the partition. In particular, if λ has at least j parts of size 1, then the weight is determined by the number of 1s. Fristedt’s results [12] or classical q-series results and Tauberian theorems give that the probability that a partition of size n has less than j parts of size 1 is asymptotic to √πj6n . Our proof is devoted to showing that the contribution from such partitions is negligible. The following result for the second generalized rank moments is straightforward from Theorem 1.1. Corollary 1.2. For each j ≥ 0, as n → ∞ r j+1 N2 (n)
= exp π
2n 3
!
1 2j + 1 √ − √ + oj (1) . 2 3 2π 2n
In particular, Dyson proved M2 (n) = 1 N2 (n) = 2np(n) where p(n) is the number of partitions of n. Therefore, Rademacher’s formula for p(n) gives an exact formula for M2 (n). Consequentially, it gives the above corollary for j = 0. It is an interesting problem to use Garvan’s combinatorial interpretation of the higher order Spt functions and techniques from random partitions to deduce asymptotics for differences of other
SOFT ASYMPTOTICS FOR GENERALIZED spt-FUNCTIONS
3
rank moments. It seems this approach would lead to identities for special values of the Riemann zeta-function. In Section 2 we give background about the combinatorial description of the Spt-functions. We also give some basic results about random partitions. In Section 3 we give the proof of the main theorem. 2. Probability and Combinatorics Background In this section we give some combinatorial background on the generalized ranks and probabilistic background on random partitions. Throughout we let Xj (λ) be the number of times j occurs as a part in the partition λ. P 2.1. Combinatorics Background. We briefly recall the definition of Sptj (n) = λ Wj (λ) where the sum is over all partitions of n. To define the weight Wj Dixit and Yee used the lower-Durfee squares. The largest square that fits inside the Ferrers diagram of λ starting from the lower left corner is the lower-Durfee square. The second lower-Durfee square is the largest square that fits inside of λ right above the first lower-Durfee square. Successive lower-Durfee squares are defined similarly. We mark each part of a partition with a ‘multiplicity number’. For example, if λ = (7, 6, 5, 5, 2, 1, 1, 1, 1) then the marked version of λ is (71 , 61 , 51 , 52 , 21 , 11 , 12 , 13 , 14 ). That is, if a positive integer occurs as a part in λ, we mark all of its occurences with positive integers in an increasing order from the left to the right. If λ is a partition, we define Wj (λ) as the the sum of the marks of each part in the first (j − 1) successive lower-Durfee squares and the mark of the part directly above the (j − 1)st lower-Durfee square. If λ has fewer than j − 1 successive lower-Durfee squares, Wj (λ) is defined to be the sum over all of the marks in the partition. For example, when λ = (7, 6, 5, 5, 2, 1, 1, 1, 1) we have W1 (λ) = 4, W2 (λ) = 4 + 3 = 7, W3 (λ) = 4 + 3 + 2 = 9, W4 (λ) = 4 + 3 + 2 + 1 = 10, and W5 (λ) = 4 + 3 + 2 + 1 + 1 = 11. Also, W6 (λ) = 4 + 3 + 2 + 1 + (1 + 2) + 1 = 14, since the first four lower-Durfee squares are of size 1 and the fifth lower-Durfee square is 2 × 2 and contains the part of size 2 and the second part of size 5. Likewise, W7 (λ) = 4 + 3 + 2 + 1 + (1 + 2) + (1 + 1 + 1) = 16 and Wj (λ) = 16 for all j > 7 because there are exactly six lower-Durfee squares. Lemma 2.1. Let Xj (λ) be the number of parts of size j in the partition λ. Fix j ≥ 1. If X1 (λ) ≥ j then Wj (λ) = jX1 (λ) − j(j−1) 2 . If X1 (λ) = j − r with r ∈ {1, · · · , j} and X2 (λ) ≥ 2r, then j(j + 1) − r(r − 1) − 2jr . 2 Proof. Assume that X1 (λ) ≥ j. Then the first j − 1 lower-Durfee squares are of size 1. They contain the parts of size one with marks X1 (λ), X1 (λ) − 1, · · · , X1 (λ) − (j − 2). Moreover, the first part after the (j − 1)st lower-Durfee square is of size 1 and has mark X1 (λ) − (j − 1). This gives the first claim. The second claim follows in a similar way, by guaranteeing that the first (j − 1) successive lower-Durfee squares are all of size 1 or 2. Additionally, we have Wj (λ) = 2rX2 (λ) +
Wj (λ) =X1 (λ) + · · · + (X1 (λ) − (j − r − 1)) + [(X2 (λ) + (X2 (λ) − 1)) + · · · ((X2 (λ) − (2r − 2)) + (X2 (λ) − (2r − 1)))] + (X2 (λ) − (2r))
4
ROBERT C. RHOADES
2.2. Probability Background. In this subsection we give some results dealing with statistics of random partitions of n for large n. Recall the generating function for the number of partitions of n is ∞ ∞ Y X 1 (2.1) P (q) = = p(n)q n . 1 − qn n=1
n=0
p π2 e 6 and Moreover, the modularity of the generating function gives P (e− ) ∼ 2π r ! 1 1 2n 1+O √ (2.2) p(n) = √ exp π . 3 n 4 3n We will make use of the following Tauberian theorem of Ingham [16]. In the version given here it can be found in [4]. P n Theorem 2.2 (Ingham). Let f (z) = ∞ n=0 a(n)z be a power series with real nonnegative coefficients and radius of convergence equal to 1. If there exists A > 0, λ, α ∈ R such that A α f (z) ∼ λ(− log(z)) exp − log(z) as z → 1− , then n X
as n → ∞.
α
1
√ λ A 2 −4 a(m) ∼ √ An α 1 exp 2 2 π n 2 +4 m=0 √
Fristedt [12] proved that for large n a typical partition of n has about proved a distributional result for the number of parts of any small size.
6n π
partitions. In fact, he
Theorem 2.3 (Fristedt). Let Pn be the uniform measure on partitions of size n. We have π Pn { √ X1 (λ) ≤ y} → 1 − e−y 6n as n → ∞ uniformly in y. We use a q-series approach to establish the following result on the number of partitions with a constrained number of 1s orP 2s. Let fa (n) be the number of partitions of n that contain < a parts n a of size 1. So that Fa (q) = ∞ n=0 fa (n)q =P(1 − q )P (q). Similarly let fa,b (n) be the number of ∞ a 2b partitions of n with < a 1s and < b 2s. Then (n)q n = (1−q Straightforward P∞ n=0 fa,b P∞)(1−q )P (q). n applications of Theorem 2.2 with (1−q) n=0 fa (n)q and (1−q) n=0 fa,b (n)q n give the following. Proposition 2.4. For any a, b ≥ 0, as n → ∞ fa,b (n) πa abπ 2 fa (n) ∼√ and ∼ p(n) p(n) n 6n We give the following result which follows easily from Fristedt’s results on random partitions. Proposition 2.5. Let F(n) be the the set of partitions of n with any fixed number of 1s or the set of partitions with any fixed number of 1s or 2s. Then, as n → ∞ ` √ X X 1 Xj (λ) = O ( n log(n))` |F(n)| λ∈F (n)
j
SOFT ASYMPTOTICS FOR GENERALIZED spt-FUNCTIONS
5
for ` = 1, 2. Moreover, if F(n) is the set of partitions of n with a fixed number of 1s and at least b 2s, then X √ 1 X2 (λ) = O n . |F(n)| λ∈F (n)
Proof. The proof of each result is the same. Fristedt’s results prove that the number of 1s or 2s in 1 a partition is independent from the number of parts of size k for all k = o(n 4 ). Therefore, the final claim follows immediately and from Fristedt’s result that as n → ∞ 2π Pn { √ X2 ≤ v} → 1 − e−v . 6n The proof of the claim for proving
P
j
Xj (λ) is completely analogous to the proof of Fristedt’s result
π X Pn { √ Xj − log 6n j
√
! 6n −v } → e−e π
or one could follow Erdös-Lehner’s [11] original proof of thisPresult. In particular, considering partitions without 1s or 2s does not effect the expected size of j Xj (λ).
3. Proof of Theorem 1.1 In this section we give the proof of Theorem 1.1. Throughout this section all sums on λ range over the partitions of n, for some fixed n. We begin with the proof in the case of j = 1. Notice that if X1 (λ) > 0 then wspt (λ) = X1 (λ). Therefore, X
wspt (λ) =
X
λ
λ
X
X1 (λ) + O
X
Xj (λ)
λ:X1 (λ)=0 j
√
6n p(n) √ =p(n) (1 + o(1)) + O √ n log(n) π n r ! 2n 1 = √ exp π (1 + o(1)). 3 2π 2n where the second equality follows Theorem 2.3 and Propositions 2.4 and 2.5 and the final equality follows from (2.2). To handle the case of Sptj (n) with j > 1 we apply Lemma 2.1 and note that
Wj (λ) ≤
i (λ) X XX
i
k=1
i=
X1 i
2
Xi (λ)(Xi (λ) − 1).
6
ROBERT C. RHOADES
Thus applying Lemma 2.1 there are constants cj,b such that X X (j − 1)j jX1 (λ) − Wj (λ) = 2 λ
λ:X1 (λ)≥j
+
X
X
b∈{1,··· ,j}
λ: X1 (λ)=j−b X2 (λ)≥2b+1
(2bX2 (λ) + cj,b ) + O
X
X
a,b λ∈Fa,b (n)
X
Xi (λ)2 .
i
where Fa,b (n) is the set of all partitions of n with exactly a 1s and b 2s and the sum on a, b is over those pairs satisfying a < j and b < 2a + 1. Using Theorem 2.3 and Propositions 2.4 and 2.5 gives √ X 6n p(n) p(n) √ 2 Wj (λ) = j p(n)(1 + o(1)) + O √ · n + O · n log (n) π n n λ
An application of (2.2) gives the result. References [1] G. Andrews, Partitions, Durfee symbols, and the Atkin-Garvan moments of ranks, Invent. Math. 169 (2007), 37–73. [2] G. Andrews, The number of smallest parts in the partitions of n, J. Reine Angew. Math. 624 (2008), 133-142. [3] G. E. Andrews and F. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167–171. [4] K. Bringmann, A. Holroyd, K. Mahlburg, and M. Vlasenko, k-run overpartitions and mock theta functions. preprint. [5] K. Bringmann and K. Mahlburg, An extension of the Hardy-Ramanujan Circle Method and applications to partitions without sequences. American Journal of Math 133 (2011) 1151–1178. [6] K. Bringmann and K. Mahlburg, Coefficients of Kac-Wakimoto Series, preprint. [7] K. Bringmann, K. Mahlburg, R.C. Rhoades, Taylor Coefficients of Jacobi Forms and Partition Statistics, to appear Math. Proc. Camb. Phil. Soc. [8] A. Dixit and A. J. Yee, Generalized Higher Order spt-functions. preprint. [9] F. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. [10] F. Dyson, Mappings and symmetries of partitions, J. Combin. Theory Ser. A 51 (1989), 169–180. [11] P. Edös and J. Lehner, The distribution of the number of summands in the partitions of a positive integer. Duke Math. J. 8, (1941) 335–345. [12] B. Fristedt, The Structure of Random Partitions of Large Integers. Trans. Amer. Math. Soc., 337 2, 703–735. [13] F. Garvan, Generalizations of Dyson’s rank and non-Rogers-Ramanujan partitions. Manuscripta Math. 84 (1994), no. 3-4, 343–359. [14] F. Garvan, Congruences for Andrews’ smallest parts partition function and new congruences for Dyson’s rank, Int. J. Number Theory 6 (2010), 1-29. [15] F. Garvan, Higher order spt-functions. Adv. Math. 228 (2011), no. 1, 241–265. [16] A. Ingham, A Tauberian theorem for partitions. Ann. of Math. 42 (1941) 1075–1090. [17] K-Q, Ji, A combinatorial proof of Andrews’ smallest parts partition function. Electron. J. Combin. 15 (2008), no. 1, Note 12, 6 pp. [18] M. Waldherr, Arithmetic and Asymptotic Properties of Mock Theta Functions and Mock Jacobi Forms. Ph. D. thesis Universität zu Köln. 2012. [19] M. Waldherr, Asymptotics for Moments of Higher Ranks. preprint. Stanford University, Serra Mall bldg. 380, Stanford, CA 94305 E-mail address: [email protected]
