Distinct parts partitions without sequences

0.18 MB·English

by  Kathrin Bringmann

DISTINCT PARTS PARTITIONS WITHOUT SEQUENCES
5 KATHRIN BRINGMANN, KARL MAHLBURG, AND KARTHIK NATARAJ
1
0
2 Abstract. Partitionswithoutsequencesofconsecutiveintegersaspartshavebeen
studiedrecentlybymanyauthors,includingAndrews,Holroyd,Liggett,andRomik,
n
among others. Their results include a description of combinatorial properties, hy-
a
J pergeometricrepresentationsforthe generatingfunctions, andasymptoticformulas

0 for the enumeration functions. We complete a similar investigation of partitions
1 into distinct parts without sequences, which are of particular interest due to their

relationshipwiththeRogers-Ramanujanidentities. Ourmainresultsincludeadou-
]
T ble series representationfor the generating function, an asymptotic formula for the
N enumeration function, and several combinatorialinequalities.
.
h
t
a
m
[

1. Introduction and statement of results
1
v
For k 2, a k-sequence in an integer partition is any k consecutive integers that
5
≥
0 all occur as parts (a standard general reference for integer partitions is [2]). Note
3
that the case k = 1 is excluded because any part in a nonempty partition trivially
2
0 forms a “1-sequence”. The study of partitions without sequences was introduced by
1. MacMahon in Chapter IV of [20]. Let pk(n) be the number of partitions of n with no
0 k-sequences, andletp (m,n)bethenumberofsuchpartitionswithmparts. Sincethe
k
5
presence of a k-sequence in a partition also implies the presence of a (k 1)-sequence,
1
−
: MacMahon’s results on page 53 of [20] can be stated as the following generating
v
i
X
r
a Date: January 13, 2015.
Key words and phrases. Rogers-Ramanujan;integer partitions; distinct parts.
The research of the first author was supported by the Alfried Krupp Prize for Young University
Teachers of the Krupp Foundation and the research leading to these results has received funding
from the European Research Council under the European Union’s Seventh Framework Programme
(FP/2007-2013)/ERCGrantagreementn. 335220-AQSER.The secondauthorwassupportedby
NSF Grant DMS-1201435.
1
2 KATHRINBRINGMANN,KARL MAHLBURG,ANDKARTHIKNATARAJ
function for p (m,n),
2
znqn(q6;q6)
G (z;q) := p (m,n)zmqn = 1+ n−1 , (1.1)
2 2 (1 qn)(q2;q2) (q3;q3)
n,m≥0 n≥1 − n−1 n−1
X X
where the q-Pochhammer symbol is defined by (a) = (a;q) := n−1(1 aqj).
n n j=0 −
Partitions without k-sequences for arbitrary k 2 arose more recently in the
≥ Q
work of Holroyd, Liggett, and Romik on probabilistic bootstrap percolation models
[14]. These partitions were also studied by Andrews [3], who found a (double) q-
hypergeometric series expansion for the generating function,
G (z;q) := p (m,n)zmqn (1.2)
k k
n,m≥0
X
(k+1)k(r+s)2 + (k+1)(s+1)s
1 ( 1)rzkr+(k+1)sq 2 2
= − .
(zq;q) (qk;qk) (qk+1;qk+1)
∞ r,s≥0 r s
X
Andrews’ proof of this expression followed from the theory of q-difference equations.
The first two authors and Lovejoy provided an alternative bijective proof [9], as
well as some additional combinatorial insight into Andrews’ q-difference equations.
It should also be noted that Andrews gave another separate treatment of the case
k = 2 in [3, Theorem 4], where he transformed MacMahon’s expression (1.1) in order
to write G (1;q) in terms of one of Ramanujan’s famous mock theta functions [25]
2
(see [5,10,19] for a sampling of other recent results on the role of mock modular forms
in hypergeometric q-series).
In addition to the combinatorial results described above, it is also of great interest
to determine the asymptotic behavior of partitions; such study dates back to Hardy
and Ramanujan’s famous formula ((1.41) in [13]), which states that as n ,
→ ∞
1
p(n) eπ√23n. (1.3)
∼ 4√3n
In fact, such formulas for partitions without k-sequences were particularly important
in [14], as the metastability threshold of the k-cross bootstrap percolation model is
intimately related to asymptotic estimates of log(p (n)). These approximations were
k
subsequently refined in [3], [7], and [11], with the most recent progress due to Kane
DISTINCT PARTS PARTITIONS WITHOUT SEQUENCES 3
and Rhoades [17, Theorem 1.8], who proved the asymptotic formula
1
1 1 2 4 1 2 2
p (n) 1 exp π 1 n . (1.4)
k ∼ 2k (cid:18)6 (cid:18) − k(k +1)(cid:19)(cid:19) n34 s3 (cid:18) − k(k +1)(cid:19) !
Remark. The exponent in this formula was first determined by Holroyd, Liggett, and
Romik [14], who showed that
log(p (n)) 2 (λ λ )n, (1.5)
k 1 k
∼ −
where λk := π2/(3k(k+1)). Note that as kpbecomes large this expression approaches
2√λ = π√2n/3, which is the same exponent for log(p(n)) seen in (1.3). However,
1
theconvergence regime ismore intricate forthe full enumeration functions, asit is not
true that (1.4) approaches (1.3) as k , even though p (n) = p(n) for sufficiently
k
→ ∞
large k.
We note further that the value of λ was derived in [14] by way of the very inter-
k
esting auxiliary function f : [0,1] [0,1], which is defined as the unique decreasing,
k
→
positive solution to the functional equation fk fk+1 = xk xk+1. Theorem 1 of [14]
− −
gives the evaluation
1 dx
λ = log(f (x)) .
k k
− x
Z0
An alternative proof of the above evaluation is given in [4], which proceeds by rewrit-
ing λ as a double integral and then making a change of variables that essentially
k
gives the integral representation of the dilogarithm function [26].
In this paper we consider a natural variant of MacMahon’s partitions by restricting
to those partitions with no k-sequences that only have distinct parts. Following the
spirit of the results mentioned above, we provide expressions for generating functions,
describe their combinatorial properties, and determine asymptotic formulas. Let
Q (n) be the number of partitions of n with no k-sequences or repeated parts, and
k
define the refined enumeration function Q (m,n) to be the number of such partitions
k
with m parts. Furthermore, denote the generating function by
(z;q) := Q (m,n)zmqn.
k k
C
m,n≥0
X
If the parts are not counted, i.e. z = 1, then we also write (q) := (1;q).
k k
C C
We begin by considering the combinatorics of the case k = 2, which corresponds
to those partitions into distinct parts with no sequences. This case is analogous to
4 KATHRINBRINGMANN,KARL MAHLBURG,ANDKARTHIKNATARAJ
MacMahon’s original study of partitions with no sequences, and a similar argument
(using partition conjugation) leads to a generating function much like (1.1). In fact,
theseries for distinct partspartitions is even simpler, as Q (n) counts those partitions
2
of n in which each part differs by at least 2 – we immediately see that these are the
same partitions famously studied by Rogers and Ramanujan [23]! We therefore have
znqn2
(z;q) = ,
2
C (q;q)
n
n≥0
X
which specializes to the corresponding products (equations (10) and (11) of [23])
1
(1;q) = ,
C2 (q,q4;q5)
∞
1
(q;q) = .
C2 (q2,q3;q5)
∞
Remark. The Rogers-Ramanujan identities have inspired an incredible amount of
work across divergent areas of mathematics ever since their introduction more than
a century ago. For a small (and by no means exhaustive!) collection of recent work,
refer to [6,10,12,16].
Our first result gives double hypergeometric q-series expressions for our new parti-
tion functions that are analogous to (1.2).
Theorem 1.1. For k 2, we have
≥
( 1)jzkj+rq(r+kj)(r+kj+1)+kj(j−1)
2 2
(z;q) = − .
Ck (qk;qk) (q;q)
j,r≥0 j r
X
We give two proofs of this theorem; the first uses q-difference equations as in [3]
and [9], while the second follows the bijective arguments of [9].
Remark. In fact, the statement of Theorem 1.1 and equation (1.2) also hold for the
trivial case k = 1; here the q-series identities are true with G (q) = (q) = 1.
1 1
C
Wenext turntotheasymptotic study ofpartitionswithout k-sequences orrepeated
parts. As in [8], we use the Constant Term Method and a Saddle Point analysis in
order to determine the asymptotic behavior of (q) near q = 1, and then apply
k
C
Ingham’s Tauberian Theorem to obtain an asymptotic formula for the coefficients.
DISTINCT PARTS PARTITIONS WITHOUT SEQUENCES 5
Before stating our results, we introduce two auxiliary functions (see Section 2 for the
definition of the dilogarithm), namely
1
g (u) := 2π2u2 +Li e2πiu Li e2πiku ,
k 2 2
− − k
h (x) := xk+1 2x+1.(cid:0) (cid:1) (cid:0) (cid:1)
k
−
We show in Proposition 2.1 that h has a unique root w (0,1), and we let v
k k k
∈
be the point on the positive imaginary axis such that w = e2πivk. In other words,
k
v := ilog(w−1)/(2π).
k k
Theorem 1.2. Using the notation above, as n , we have
→ ∞
1
√πg (v )
Q (n) k k 4 e2√gk(vk)n.
k ∼ g′′(v )n3
− k k 4
Remark. The exponent for this resuplt can be written in a form similar to (1.5). In
particular,
log(Q (n)) 2 (γ γ )n,
k 1 k
∼ −
1 p
2
where γ := log(f (x))dx/(x(1 x)).
k k
− −
Z0
We do not present the proof of this alternative expression for the exponent, as it
follows directly from the arguments in Section 3 of [14] (with probability (1+qj)−1
for the analogous event C ). Furthermore, the values of γ do not simplify as cleanly
j k
as the λ , as the integral does not reduce to a dilogarithm evaluation. However, it is
k
true that γ decreases monotonically to 0 as k increases, since the f are decreasing in
k k
k. Additionally, a short calculation shows that γ = π2/12, which is again compatible
1
with the exponent of Hardy and Ramanujan’s asymptotic formula for partitions into
distinct parts. The corresponding enumeration function was denoted by q(n) in [13],
where they showed that
1
q(n) eπ√n3.
∼ 4 31n3
4 4
·
Remark. In the case k = 2 we find that w = φ−1, where φ := (1 + √5)/2 is the
2
golden ratio. Furthermore, the first and third special values on page 7 of [26] give the
6 KATHRINBRINGMANN,KARL MAHLBURG,ANDKARTHIKNATARAJ
evaluation
1 1
g (v ) = (logφ)2 +Li φ−1 Li φ−2
2 2 2 2
2 − 2
1 π2 (cid:0) (cid:1) π(cid:0)2 1(cid:1) π2
= (logφ)2 + (logφ)2 + (logφ)2 = .
2 10 − − 30 2 15
Plugging in to the theorem statement, this gives
√φ
c2(n) ∼ 2 31√5n3e2π√1n5,
4 4
·
which was previously proven by Lehner in his study of the Rogers-Ramanujan prod-
ucts in [18].
Theremainder ofthepaperisstructuredasfollows. InSection2wegivemanybasic
identities for hypergeometric q-series and determine thecritical points of the auxiliary
functions g and h . Section 3 contains analytic and combinatorial proofs of the
k k
doubleseriesrepresentationfromTheorem1.1,andalsopresentsseveralcombinatorial
observations. We conclude with Section 4, where we use the Constant Term Method
and a Saddle Point analysis to prove the asymptotic formula from Theorem 1.2.
2. Hypergeometric series and auxiliary functions
In this section we recall several standard facts from the theory of hypergeometric
q-series, including useful identities for special functions and modular transformations.
2.1. Definitions and identities for q-series. The dilogarithm function [26, p. 5]
is defined for complex x < 1 by
| |
xn
Li (x) := .
2 n2
n≥0
X
This function has a natural q-deformation that is known as the quantum dilogarithm
[26, p. 28], which is given by ( x , q < 1)
| | | |
xn
Li (x;q) := log(x;q) = .
2 − ∞ n(1 qn)
n≥1 −
X
Moreover, an easy calculation shows that its Laurent expansion begins with the terms
1
Li x;e−ε = Li (x) 1 log(1 x)+O(ε), (2.1)
2 ε 2 − 2 −
(cid:0) (cid:1)
DISTINCT PARTS PARTITIONS WITHOUT SEQUENCES 7
where the series converges uniformly in x as ε 0+.
→
Next, we recall two identities due to Euler, which state that [2, equations (2.2.5)
and (2.2.6)]
1 xn
= , (2.2)
(x;q) (q;q)
∞ n
n≥0
X
( 1)nxnqn(n−1)
2
(x;q) = − . (2.3)
∞
(q;q)
n
n≥0
X
Finally, Jacobi’s theta function is defined by
θ(q;x) := qn2xn. (2.4)
n∈Z
X
In order to determine the asymptotic behavior near q = 1, we use for ε > 0 the
modular inversion formula (cf. [24, p. 290]),
θ e−ε;e2πiu = π e−π2(nε+u)2. (2.5)
ε
r n∈Z
(cid:0) (cid:1) X
2.2. Auxiliary functions. We now prove several useful facts about the auxiliary
functions h and g .
k k
Proposition 2.1. Adopt the above notation.
(i) There is a unique root w (0,1) of h (x).
k k
∈
(ii) The unique critical point of g on the positive real axis is given by v such
k k
that e2πivk = w . Furthermore, g′′(v ) < 0.
k k
Proof. (i) Descartes’ Rule of Signs implies that h has either zero or two positive real
k
roots. It is immediate to verify that g (0) = 1,g (1) = 0 and g (3/4) < 0 for k 2,
k k k
≥
so the second root must lie in (0,1) as claimed.
(ii) Next, to identify the critical points of g , we calculate its derivative
k
g′(u) = 4π2u log 1 e2πiu 2πi+log 1 e2πiku 2πi.
k − − − −
This vanishes precisely when
(cid:0) (cid:1) (cid:0) (cid:1)
1 e2πiku
2πiu+log − = 0.
1 e2πiu
(cid:18) − (cid:19)
Exponentiating and writing x := e2πiu then shows that the critical points of g (u)
k
correspond to the roots of h (x).
k
8 KATHRINBRINGMANN,KARL MAHLBURG,ANDKARTHIKNATARAJ
Finally, we calculate the second derivative (again writing x = e2πiu) of g
k
(2πi)2x (2πi)2kxk 1 kxk
g′′(u) = 4π2 + = 4π2 .
k − 1 x − 1 xk − 1 x − 1 xk
− − (cid:18) − − (cid:19)
At the critical point this further simplifies, since 1 wk = w−1(1 w ), which gives
− k k − k
4π2 1 kwk+1
g′′(v ) = − − k .
k k 1 w
(cid:0)− k (cid:1)
We claim that at the critical point 1 kwk+1 > 0. Indeed, the derivative of h is
− k k
h′(x) = (k +1)xk 2, (2.6)
k −
and at the root w , we have h′(w ) < 0. Plugging in w to (2.6), multiplying by w
k k k k k
and substituting wk = 2w 1 then implies that
k k −
0 > (k +1)wk+1 2w = kwk+1 1.
k − k k −
This completes the proof of (ii). (cid:3)
3. Proof of Theorem 1.1
3.1. Analytic proof. WefollowAndrews’ proofofTheorem2in[3]. Firstweobserve
that satisfies the q-difference equation
k
C
k−1
k(z;q) = zjqj(j2+1) k zqj+1;q . (3.1)
C C
j=0
X (cid:0) (cid:1)
The terms on the right result from conditioning on the length of the sequence that
begins with 1. The j = 0 term corresponds to the case where there is no 1, and thus
the smallest part is at least 2; the other terms correspond to the case that there is a
run 1,2,...,j, and no j +1, so the next part is at least j +2. Applying (3.1) twice,
we obtain the relation
k(z;q) zq k(zq;q) = k(zq;q) zkqk(k2+1) k zqk+1;q . (3.2)
C − C C − C
Now consider the double series (cid:0) (cid:1)
( 1)jzkj+rq(r+kj)(r+kj+1)+kj(j−1)
2 2
F (z;q) := − .
k (qk;qk) (q;q)
j,r≥0 j r
X
DISTINCT PARTS PARTITIONS WITHOUT SEQUENCES 9
Expanding this as a series in z, so that F (z;q) =: γ (q)zn, we therefore have
k n≥0 n
( 1)jqr(r+1)+krjP+k(k+1)j2
2 2
γ = − .
n (qk;qk) (q;q)
kj+r=n j r
X
Now we calculate
( 1)jqr(r+1)+krj+k(k+1)j2
(1 qn)γ = − 2 2 (1 qr)+qr 1 qkj
− n (qk;qk) (q;q) − −
kjX+r=n j r (cid:16) (cid:0) (cid:1)(cid:17)
= qnγn−1 q(k+1)(n−k)+k(k2+1)γn−k,
−
where the first term follows from the shift r r + 1, and the second term from
7→
j j + 1. Multiplying by zn and summing over n finally gives the q-difference
7→
equation
Fk(z;q) = (1+zq)Fk(zq;q) zkqk(k2+1)Fk zqk+1;q .
−
As this is equivalent to (3.2), we therefore conclude (cf. [1] and the uniqueness of
(cid:0) (cid:1)
solutions to q-difference equations) that = F , completing the proof of Theorem
k k
C
1.1.
3.2. Combinatorial proof. In this section we follow the approach from Section 3.2
of [9], using a combinatorial decomposition of partitions into simple components that
essentially split the double summation in Theorem 1.1. Denote the size of a partition
λ by λ and write ℓ(λ) for the number of parts, or length. Let be the set of
k
| | D
partitions without k-sequences or repeated parts, and note that with this notation
we have
(z;q) = zℓ(λ)q|λ|.
k
C
λX∈Dk
If λ and ℓ(λ) = m, so that λ = λ + +λ in nonincreasing order, then define
k 1 m
∈ D ···
λ′ by removing a triangular partition (m 1) +(m 2) + + 1, so that the new
− − ···
parts are
λ′ := λ (m j), 1 j m.
j j − − ≤ ≤
The definition of implies that λ′ is a partition in which each part occurs at most
k
D
k 1 times, so
−
zkqk;qk
zℓ(λ′)q|λ′| = 1+zqn +z2q2n + +zk−1qn(k−1) = ∞.
··· (zq;q)
(cid:0) ∞(cid:1)
λX∈Dk nY≥1(cid:0) (cid:1)
10 KATHRINBRINGMANN,KARL MAHLBURG,ANDKARTHIKNATARAJ
Euler’s summation formulas (Corollary 2.2 in [2]) then imply the double series
( 1)jzkjqkj(j+1) zrqr
zℓ(λ′)q|λ′| = − 2 . (3.3)
(qk;qk) (q;q)
λX∈Dk jX,r≥0 j r
To complete the proof, observe that
ℓ(λ) = ℓ(λ′),
ℓ(λ)(ℓ(λ)+1)
λ = λ′ + .
| | | | 2
Plugging in to (3.3), we obtain
( 1)jzkj+rq(kj+r)(kj+r−1)+kj(j+1)+r
2 2
(z;q) = − .
Ck (qk;qk) (q;q)
j,r≥0 j r
X
Theorem 1.1 follows upon simplifying the exponent of q.
Remark. For example, if k = 3 and λ = 15 +12+ 11+ 9+8 +4 +2 +1, then the
associated λ′ is 8+6+6+5+5+2+1+1,which consists of parts that are repeated
at most twice.
3.3. Monotonicity. We close with several additional combinatorial observations on
the monotonicity of the enumeration functions.
Proposition 3.1. For m,n 0 and k 2, we have
≥ ≥
(i) Q (m,n) Q (m,n),
k k+1
≤
(ii) Q (m,n) Q (m,n+1).
k k
≤
Proof. As mentioned in the introduction, (i) follows immediately from the definition.
For (ii), note that if λ is a partition of n with m parts, then
k
∈ D
(λ +1)+λ + +λ
1 2 m
···
is a partition of n+1 with m parts. Furthermore, this new partition remains in
k
since λ +1 > λ > λ > > λ . D(cid:3)
1 1 2 m
···
Remark. Part (ii) has the important consequence that
Q (n) Q (n+1). (3.4)
k k
≤