# On the Distribution of the spt-Crank

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*Keywords:*partitions; partition crank; partition rank; spt-crank; unimodal

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Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Institute for Advanced Study, School of Natural Sciences, Einstein Drive, Princeton, NJ 08540, USA

Stanford University, Department of Mathematics, Bldg 380, Stanford, CA 94305, USA

Author to whom correspondence should be addressed.

Received: 16 February 2013 / Revised: 10 April 2013 / Accepted: 10 April 2013 / Published: 28 June 2013

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence ${\left\{{N}_{S}(m\text{},\text{}n)\right\}}_{m}$ is unimodal, where ${N}_{S}(m\text{},\text{}n)$ is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of “marked” partitions. A “marked” partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions.

The $\mathrm{spt}$-function, introduced by the first author [1], counts the total number of appearances of the smallest parts in the partitions of n. For example, $\mathrm{spt}\left(4\right)=10$ because the partitions of 4 are $4,3+1,2+2,2+1+1,1+1+1+1$. The first author established the following remarkable congruences
These congruences bear a striking resemblance to Ramanujan’s congruences for the usual partition counting function $p\left(n\right)$, namely

$$\begin{array}{cc}\hfill \mathrm{spt}(5n+4)& \equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}5)\hfill \\ \hfill \mathrm{spt}(7n+5)& \equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}7)\hfill \\ \hfill \mathrm{spt}(13n+6)& \equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}13)\hfill \end{array}$$

$$\begin{array}{cc}\hfill p(5n+4)& \equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}5)\hfill \\ \hfill p(7n+5)& \equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}7)\hfill \\ \hfill p(11n+6)& \equiv 0\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}11)\hfill \end{array}$$

The rank statistic of a partition was defined by the second author [2] as the largest part of the partition minus the number of parts. Let $N(m,n)$ denote the number of partitions of n with rank m and $N(m,t,n)$ denote the number of partitions of n with rank congruent to m modulo t. The second author conjectured, and Atkin and Swinnerton-Dyer [3] proved, that
Therefore, the rank provides a combinatorial interpretation of Ramanujan’s congruences modulo 5 and 7. Moreover, the second author observed that the rank is not sufficient to decompose Ramanujan’s congruence modulo 11, and he conjectured the existence of a statistic called the “crank” that would explain all three congruences.

$$\begin{array}{cc}\hfill N(k,5,5n+4)& =\frac{p(5n+4)}{5}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 4\hfill \\ \hfill N(k,7,7n+5)& =\frac{p(7n+5)}{7}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 6\hfill \end{array}$$

Garvan [4] found the crank statistic for vector partitions and together with the first author [5] presented a definition for the crank of a ordinary partition, namely
where $o\left(\lambda \right)$ is the number of 1s in the partition λ and $\mu \left(\lambda \right)$ is the number of parts of λ strictly larger than $o\left(\lambda \right)$. Let $M(m,n)$ be the number of partitions of n with crank m and $M(m,t,n)$ be the number of partitions of n with crank congruent to m modulo t. Garvan proved [4]
Hence, the crank provides a combinatorial interpretation of all three of Ramanujan’s congruences for the partition counting function.

$$\mathrm{crank}\left(\lambda \right):=\left\{\begin{array}{cc}\phantom{\rule{4.pt}{0ex}}\text{largest}\phantom{\rule{4.pt}{0ex}}\text{part}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\lambda \hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}o\left(\lambda \right)=0\hfill \\ \mu \left(\lambda \right)-o\left(\lambda \right)\hfill & \phantom{\rule{4.pt}{0ex}}\text{else}\hfill \end{array}\right.$$

$$\begin{array}{cc}\hfill M(k,5,5n+4)& =\frac{p(5n+4)}{5}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 4\hfill \\ \hfill M(k,7,7n+5)& =\frac{p(7n+5)}{7}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 6\hfill \\ \hfill M(k,11,11n+6)& =\frac{p(11n+6)}{11}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 10\hfill \end{array}$$

Recently Garvan, Liang and the first author [6] defined the $\mathrm{spt}$-crank and used it to provide a combinatorial interpretation of the $\mathrm{spt}$-congruences modulo 5 and 7. To describe the $\mathrm{spt}$-crank we introduce the set of vector partitions, denoted by V. Then V is the Cartesian product
where $\mathcal{D}$ is the set of partitions into distinct parts and $\mathcal{P}$ is the set of all integer partitions. For $\pi =({\pi}_{1},{\pi}_{2},{\pi}_{3})\in V$, let $\left|\pi \right|=\left|{\pi}_{1}\right|+\left|{\pi}_{2}\right|+\left|{\pi}_{3}\right|$, where $\left|\xb7\right|$ is the sum of the parts of a partition. If $\left|\pi \right|=n$ we say that π is a vector partition of n. The crank of a vector partition is defined as $\#\left({\pi}_{2}\right)-\#\left({\pi}_{3}\right)$, where $\#(\xb7)$ is the number of parts in an integer partition.

$$V=\mathcal{D}\times \mathcal{P}\times \mathcal{P}$$

To define the $\mathrm{spt}$-crank we introduce the set of S-partitions. Let
where $s\left(\pi \right)$ is the smallest part in the partition. For $\pi \in S$ define a weight ${\omega}_{1}$, by ${\omega}_{1}\left(\pi \right)={(-1)}^{\#\left({\pi}_{1}\right)-1}$. If $\pi \in S$ has crank m, then we refer to the $\mathrm{spt}$-crank as ${\omega}_{1}\left(\pi \right)$. Define
and ${N}_{S}(m,t,n)={\sum}_{k\equiv m\phantom{\rule{4.44443pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}t)}{N}_{S}(k,n)$. Garvan, Liang and the first author [6] establish the following combinatorial interpretation of the $\mathrm{spt}$-congruences modulo 5 and 7
In a second paper [7], they prove a number of basic results about these values. For instance,
and, surprisingly,
Later, a simpler proof of this result was given by the second author [8].

$$S:=\{\pi =({\pi}_{1},{\pi}_{2},{\pi}_{3})\in V:1\le s\left({\pi}_{1}\right)<\infty \phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}s\left({\pi}_{1}\right)\le min\left(s\left({\pi}_{2}\right),s\left({\pi}_{3}\right)\right)\}$$

$${N}_{S}(m,n):=\sum _{\pi \in S,\left|\pi \right|=n,\mathrm{crank}\left(\pi \right)=m}{\omega}_{1}\left(\pi \right)$$

$$\begin{array}{cc}\hfill {N}_{S}(k,5,5n+4)& =\frac{\mathrm{spt}(5n+4)}{5}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 4\hfill \\ \hfill {N}_{S}(k,7,7n+5)& =\frac{\mathrm{spt}(7n+5)}{7}\phantom{\rule{72.26999pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}0\le k\le 6\hfill \end{array}$$

$${N}_{S}(m,n)={N}_{S}(-m,n)$$

$${N}_{S}(m,n)\ge 0$$

Table 1 suggests that the sequence ${\left\{{N}_{S}(m,n)\right\}}_{m}$ is (weakly) unimodal. Precisely, we give the following conjecture.

$${N}_{S}(m,n)\ge {N}_{S}(m+1,n)$$

$n\backslash m$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | |||||||||

1 | 1 | |||||||||

2 | 1 | 1 | ||||||||

3 | 1 | 1 | 1 | |||||||

4 | 2 | 2 | 1 | 1 | ||||||

5 | 2 | 2 | 2 | 1 | 1 | |||||

6 | 4 | 4 | 3 | 2 | 1 | 1 | ||||

7 | 5 | 4 | 4 | 3 | 2 | 1 | 1 | |||

8 | 7 | 7 | 6 | 5 | 3 | 2 | 1 | 1 | ||

9 | 10 | 9 | 8 | 6 | 5 | 3 | 2 | 1 | 1 | |

10 | 13 | 13 | 11 | 10 | 7 | 5 | 3 | 2 | 1 | 1 |

11 | 17 | 16 | 15 | 12 | 10 | 7 | 5 | 3 | 2 | 1 |

12 | 24 | 24 | 21 | 18 | 14 | 11 | 7 | 5 | 3 | 2 |

13 | 31 | 29 | 27 | 23 | 19 | 14 | 11 | 7 | 5 | 3 |

14 | 40 | 40 | 36 | 32 | 26 | 21 | 15 | 11 | 7 | 5 |

15 | 53 | 51 | 48 | 41 | 35 | 27 | 21 | 15 | 11 | 7 |

16 | 69 | 68 | 62 | 56 | 46 | 38 | 29 | 22 | 15 | 11 |

This property is not true for the ordinary rank or crank statistic. For example,
for all $n>2$ and a similar statement holds for the crank. Our first statement reinterprets this conjecture in terms of the rank and crank. Define the cumulative density functions of the rank and crank as follows:

$$N(n-1,n)=N(n-3,n)=1\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}N(n-2,n)=0$$

$${N}_{\le m}\left(n\right):=\sum _{\left|r\right|\le m}N(r,m)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{M}_{\le m}\left(n\right):=\sum _{\left|r\right|\le m}M(r,m)$$

$${N}_{S}(m,n)\ge {N}_{S}(m+1,n)$$

$${N}_{\le m}\left(n\right)\ge {M}_{\le m}\left(n\right)$$

$${N}_{\le m}\left(n\right)\ge {M}_{\le m}\left(n\right)$$

This theorem leads to a good heuristic reason to believe that the $\mathrm{spt}$-crank is unimodal. Define the moments of the rank and crank statistic by

$${N}_{2\ell}\left(n\right):=\sum _{m}{m}^{2\ell}N(m,n)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{M}_{2\ell}\left(n\right):=\sum _{m}{m}^{2\ell}M(m,n)$$

Garvan [12] conjectured that
for each $\ell >0$ and n, despite the fact that
This conjecture says that while the rank and crank are distributed asymptotically the same, the crank distribution is slightly “wider” for any fixed n. The first author [1] proved that $\mathrm{spt}\left(n\right)=\frac{1}{2}({M}_{2}\left(n\right)-{N}_{2}\left(n\right))$, which yields the $\ell =1$ case of Garvan’s conjecture. Garvan [13] later proved his own conjecture by introducing higher order $\mathrm{spt}$-functions. As a result, we expect that
which is by Theorem 1.2 is equivalent to Conjecture 1.1.

$${M}_{2\ell}\left(n\right)>{N}_{2\ell}\left(n\right)$$

$${N}_{2\ell}\left(n\right)\sim {M}_{2\ell}\left(n\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n\to \infty $$

$${N}_{\le m}\left(n\right)\ge {M}_{\le m}\left(n\right)$$

The next theorem provides an asymptotic result supporting Conjecture 1.1.

$${N}_{\le m}\left(n\right)\sim {M}_{\le m}\left(n\right)\sim \frac{(2m+1)\pi}{48\sqrt{2}{n}^{\frac{3}{2}}}exp\left(\pi \sqrt{\frac{2n}{3}}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty $$

$$\left({N}_{\le m}\left(n\right)-{M}_{\le m}\left(n\right)\right)\sim \frac{(2m+1){\pi}^{2}}{192\sqrt{3}{n}^{2}}exp\left(\pi \sqrt{\frac{2n}{3}}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}n\to \infty $$

$${N}_{S}(m,n)>{N}_{S}(m+1,n)$$

We close this section by giving an asymptotic for the distribution of the numbers ${N}_{S}(m,n)$. Let
be the moments of the $\mathrm{spt}$-crank.

$${N}_{S,k}\left(n\right):=\sum _{m}{m}^{k}{N}_{S}(m,n)$$

To define the asymptotic result we give the following definitions: Define
where ${B}_{n}\left(x\right)$ is the nth Bernoulli polynomial. Define the Kloosterman sum
where ${\omega}_{h,k}:=exp\left(\pi i\phantom{\rule{4pt}{0ex}}s(h,k)\right)$ with
the Dedekind sum, and
is the sawtooth function.

$$\gamma (a,b,c):=\frac{\left(2\right(a+b+c\left)\right)!}{(a+1)!\left(2b\right)!\left(2c\right)!}{B}_{2b}\left(\frac{1}{2}\right){B}_{2c}\left(\frac{1}{2}\right){(-1)}^{a+c}{4}^{-a-c}{\pi}^{-a}({3}^{a+1}-1)$$

$${K}_{k}\left(n\right):=\sum _{{\scriptstyle \begin{array}{c}0\le h<k\\ (h,k)=1\end{array}}}{\omega}_{h,k}{e}^{-\frac{2\pi ihn}{k}}$$

$$s(h,k):=\sum _{\mu \phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}k)}\left(\left(\frac{\mu}{k}\right)\right)\left(\left(\frac{h\mu}{k}\right)\right)$$

$$\left(\left(x\right)\right):=\left\{\begin{array}{cc}x-\lfloor x\rfloor -\frac{1}{2}\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}x\in \mathbb{R}\backslash \mathbb{Z}\hfill \\ 0\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}x\in \mathbb{Z}\hfill \end{array}\right.$$

$${N}_{S,2\ell}\left(n\right)=\frac{1}{2}\sum _{k<\sqrt{n}}\frac{{K}_{k}\left(n\right)}{k}\sum _{a+b+c=\ell}{k}^{a+1}\gamma (a,b,c){(24n-1)}^{c+\frac{a}{2}-\frac{1}{4}}{I}_{\frac{1}{2}-a-2c}\left(\frac{\pi \sqrt{24n-1}}{6k}\right)+O\left({n}^{2\ell +\u03f5}\right)$$

$${N}_{S,2\ell}\left(n\right)=\frac{\sqrt{3}}{\pi}{(-1)}^{\ell}{B}_{2\ell}\left(\frac{1}{2}\right){\left(24n\right)}^{\ell -\frac{1}{2}}exp\left(\pi \sqrt{\frac{2n}{3}}\right)\left(1+O\left(\frac{1}{\sqrt{n}}\right)\right)$$

$$\frac{{N}_{S,2\ell}\left(n\right)}{\mathrm{spt}\left(n\right){\left(6n\right)}^{\ell}}\sim ({2}^{2\ell}-2)\left|{B}_{2\ell}\right|$$

$$\frac{{N}_{2\ell}\left(n\right)}{p\left(n\right){\left(6n\right)}^{\ell}}\sim \frac{{M}_{2\ell}\left(n\right)}{p\left(n\right){\left(6n\right)}^{\ell}}\sim ({2}^{2\ell}-2)\left|{B}_{2\ell}\right|$$

In Section 2 we prove Theorem 1.2. In Section 3 we use the results of Bringmann, Mahlburg, and the third author [16] on the moments of the rank and crank statistics to establish Theorem 1.4. In Section 4 we use the circle method to calculate the asymptotics of Theorem 1.3. Finally, in Section 5 we discuss the $\mathrm{spt}$-crank in terms of ordinary integer partitions. It seems a challenging and interesting problem to find an interpretation of the $\mathrm{spt}$-crank in terms of ordinary integer partitions.

In this section we prove Theorem 1.2. Garvan, Liang and the first author (Corollary 2.5 of [6]) give
where ${N}_{V}(m,n)$ is the number of vector partitions with crank m. Note that ${N}_{V}(m,n)=M(m,n)$ for $n>1$. Formal q-series manipulations lead to the following: for any $m\ge 0$ we have

$$\sum _{n\ge 1,m\in \mathbb{Z}}{N}_{S}(m,n){z}^{m}{q}^{n}=\frac{{z}^{-1}}{{(1-{z}^{-1})}^{2}}\sum _{n=0}^{\infty}\sum _{m\in \mathbb{Z}}\left({N}_{V}(m,n)-N(m,n)\right){z}^{m}{q}^{n}$$

$$\sum _{n=1}^{\infty}{N}_{S}(m,n){q}^{n}=\sum _{n>1}^{\infty}\sum _{\ell \ge m}(\ell -m)\left({N}_{V}(\ell ,n)-N(\ell ,n)\right){q}^{n}$$

For example, when $m=0$ we obtain
where
The $\mathrm{ospt}$ function is the difference of “first” moments of the crank and rank distributions, see [17]. From (2.1) we have

$$\sum _{n=1}^{\infty}{N}_{S}(0,n){q}^{n}=\sum _{n=1}^{\infty}\mathrm{ospt}\left(n\right){q}^{n}$$

$$\mathrm{ospt}\left(n\right)=\sum _{\ell \ge 0}\ell \left(M(\ell ,n)-N(\ell ,n)\right)$$

$$\begin{array}{c}\hfill \sum _{n=1}^{\infty}({N}_{S}(m,n)-{N}_{S}(m+1,n)){q}^{n}=\sum _{n>1}\sum _{\ell >m}({N}_{V}(\ell ,n)-N(\ell ,n)){q}^{n}\end{array}$$

Using the symmetry of the rank and crank statistics, and the fact that ${\sum}_{m}N(m,n)={\sum}_{m}M(m,n)=p\left(n\right)$ we have
This establishes Theorem 1.2

$$\begin{array}{cc}\hfill \sum _{n>0}({N}_{S}(m,n)& -{N}_{S}(m+1,n)){q}^{n}=\sum _{n>0}\sum _{\ell >m}({N}_{V}(\ell ,n)-N(\ell ,n)){q}^{n}\hfill \\ \hfill =& \sum _{n=1}^{\infty}\frac{1}{2}\left(\sum _{\ell}{N}_{V}(\ell ,n)-N(\ell ,n))+\sum _{-m\le \ell \le m}(N(m,n)-{N}_{V}(m,n))\right){q}^{n}\hfill \\ \hfill =& \frac{1}{2}\sum _{n=1}^{\infty}\sum _{-m\le \ell \le m}(N(m,n)-{N}_{V}(m,n)){q}^{n}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{(2.3)}\end{array}$$

In this section we will calculate the asymptotic for the moments of the $\mathrm{spt}$-crank statistic. This calculation uses the results of [16] and establishes Theorem 1.4. For details see [16].

Let
where $C(x;q)$ is the crank generating function and $R(x;q)$ is the rank generating function. Notice that
By the symmetry of the statistic we have ${\sum}_{m}{m}^{k}{N}_{S}(m,n)=0$ for all n when k is odd. We define ${S}_{k}\left(q\right)={\sum}_{n=1}^{\infty}\left({\sum}_{m}{m}^{k}{N}_{S}(m,n)\right){q}^{n}$ to be the S-crank moment generating functions and ${N}_{S,k}\left(n\right)={\sum}_{m}{m}^{k}{N}_{S}(m,n)$ to be S-crank moments weighted by ${\omega}_{1}$.

$$S(x;q)=\sum _{n=1}^{\infty}\sum _{m}{N}_{S}(m,n){x}^{m}{q}^{n}=-\frac{1}{(1-x)(1-{x}^{-1})}\left(C(x;q)-R(x;q)\right)$$

$$S({e}^{2\pi iu};q)=\sum _{k\ge 0}\frac{{\left(2\pi iu\right)}^{k}}{k!}\sum _{n=1}^{\infty}\left(\sum _{m}{m}^{k}{N}_{S}(m,n)\right){q}^{n}$$

The proof of Theorem 1.4 follows in a straightforward way from the results of [16] and a simple modification of some of the lemmas therein.

Throughout the remainder of this section let $z\in with$ Re $\left(z\right)>0$ and $0\le h<k$ with $(h,k)=1$ We define ${\left[a\right]}_{b}$ the inverse of $a$ modulo $b$. Moreover, for fixed $h$ and $k$ we let $q={e}^{\frac{2\pi i}{k}\left(h+iz\right)}$. Define $\chi (h,{[-h]}_{k},k)$ to be the multiplier of the Dedekind eta-function. In particular,

$$\chi (h,{[-h]}_{k},k)={i}^{-\frac{1}{2}}{\omega}_{h,k}^{-1}{e}^{-\frac{\pi i}{12k}\left({[-h]}_{k}-h\right)}$$

Finally, we define

$${f}_{\nu}(u;z):={e}^{\frac{\nu \pi {u}^{2}}{z}}\frac{sin\left(\pi u\right)}{sinh\left(\frac{\pi u}{z}\right)}$$

$$C({e}^{2\pi iu};q)=-{i}^{\frac{3}{2}}{e}^{\frac{\pi i}{12k}(h-{[-h]}_{k})}{\chi}^{-1}(h,{[-h]}_{k},k){e}^{\frac{\pi}{12k}\left(\frac{1}{z}-z\right)}{z}^{-\frac{1}{2}}{f}_{k}(u;z)+\sum _{r=0}^{\infty}\frac{{a}_{r}\left(z\right){u}^{r}}{r!}$$

$$R({e}^{2\pi iu};q)=-{i}^{\frac{3}{2}}{e}^{\frac{\pi i}{12k}(h-{[-h]}_{k})}{\chi}^{-1}(h,{[-h]}_{k},k){e}^{\frac{\pi}{12k}\left(\frac{1}{z}-z\right)}{z}^{-\frac{1}{2}}{f}_{3k}(u;z)+\sum _{r=0}^{\infty}\frac{{a}_{r}\left(z\right){u}^{r}}{r!}$$

Combining Propositions 3.1 and 3.2 and (3.1) we have the following lemma.

$$S({e}^{2\pi iu};q)=-\frac{1}{4}{i}^{\frac{3}{2}}{e}^{\frac{\pi i}{12k}(h-{[-h]}_{k})}{\chi}^{-1}(h,{[-h]}_{k},k){e}^{\frac{\pi}{12k}\left(\frac{1}{z}-z\right)}{z}^{-\frac{1}{2}}\frac{{e}^{\frac{3\pi k{u}^{2}}{z}}-{e}^{\frac{\pi k{u}^{2}}{z}}}{sin\left(\pi u\right)sinh\left(\frac{\pi u}{z}\right)}+\sum _{r=0}^{\infty}\frac{{a}_{r}\left(z\right){u}^{r}}{r!}$$

Taylor expanding the expression in Lemma 3.3 with respect to u and using (3.2) give asymptotics for ${S}_{k}\left(q\right)$. The circle method can now be used to turn those asymptotics for the generating functions into asymptotics for the coefficients. Applying the following theorem gives Theorem 1.4. The theorem is a general circle method result, which is a slight modification of Theorem 4.1 of [16].

$${F}_{r,\ell}\left({e}^{\frac{2\pi i}{k}(h+iz)}\right)=\sum _{n}{c}_{r,\ell}\left(n\right){e}^{\frac{2\pi i}{k}(h+iz)}$$

$${F}_{r,\ell}\left({e}^{\frac{2\pi i}{k}(h+iz)}\right)=-{i}^{\frac{3}{2}}{e}^{\frac{\pi i}{12k}(h-{[-h]}_{k})}{\chi}^{-1}(h,{[-h]}_{k},k){e}^{\frac{\pi}{12k}\left(\frac{1}{z}-z\right)}\sum _{a+b+c=\ell}C(a,b,c){z}^{-p(a,b,c)}+{E}_{r,\ell ,k}\left(z\right)$$

$${c}_{r,\ell}\left(n\right)=2\pi \sum _{k\le \sqrt{n}}\frac{{K}_{k}\left(n\right)}{k}\sum _{a+b+c=\ell}C(a,b,c){(24n-1)}^{\frac{p(a,b,c)}{2}-\frac{1}{2}}{I}_{\frac{1}{4}-p(a,b,c)}\left(\frac{\pi \sqrt{24n-1}}{6k}\right)+O\left({n}^{2\ell +\u03f5}\right)$$

In this section we consider the cumulative density functions of the rank and crank. We show that these generating functions are partial theta functions times the partition generating function. Obtaining an asymptotic expansion for the coefficients of such a generating function via the circle method is classical (see [18], for example). We have the following well known generating functions for $N(m,n)$ and $M(m,n)$
and
Fine [19] showed that
Similarly, we have from (4.2)

$$\begin{array}{cc}\hfill \sum _{n\ge 0}N(m,n){q}^{n}=& \frac{1}{{\left(q\right)}_{\infty}}\sum _{n>0}{(-1)}^{n+1}{q}^{\frac{n(3n-1)}{2}+\left|m\right|n}(1-{q}^{n})\hfill \end{array}$$

$$\begin{array}{cc}\hfill \sum _{n\ge 0}M(m,n){q}^{n}=& \frac{1}{{\left(q\right)}_{\infty}}\sum _{n>0}{(-1)}^{n+1}{q}^{\frac{n(n-1)}{2}+\left|m\right|n}(1-{q}^{n})\hfill \end{array}$$

$${R}_{\le m}\left(q\right)=\sum _{n}{N}_{\le m}\left(n\right){q}^{n}=\frac{1}{{\left(q\right)}_{\infty}}\left(2\sum _{n=0}^{\infty}{(-1)}^{n}{q}^{\frac{3{n}^{2}+n}{2}+mn}-1\right)$$

$${C}_{\le m}\left(q\right)=\sum _{n}{N}_{\le m}\left(n\right){q}^{n}=\frac{1}{{\left(q\right)}_{\infty}}\left(2\sum _{n=0}^{\infty}{(-1)}^{n}{q}^{\frac{{n}^{2}+n}{2}+mn}-1\right)$$

Note that
So we have
where $\left(\frac{-4}{\xb7}\right)$ is the Kronecker symbol. Similarly, we have
where
We set $q={e}^{-s}$ and consider the asymptotic as $s\to {0}^{+}$.

$${q}^{\frac{{n}^{2}+n}{2}+mn}={q}^{\frac{1}{2}({n}^{2}+2mn+n)}={q}^{\frac{1}{2}({n}^{2}+2mn+n+m+\frac{1}{4})-\frac{1}{2}\left({m}^{2}+m+\frac{1}{4}\right)}={q}^{\frac{1}{2}{\left(m+n+\frac{1}{2}\right)}^{2}-\frac{1}{2}{\left(m+\frac{1}{2}\right)}^{2}}$$

$${C}_{\le m}\left(q\right)=\frac{1}{{\left(q\right)}_{\infty}}\left(2{(-1)}^{m}{q}^{-\frac{1}{8}{\left(2m+1\right)}^{2}}\sum _{n>2m}^{\infty}\left(\frac{-4}{n}\right){q}^{\frac{{n}^{2}}{8}}-1\right)$$

$${R}_{\le m}\left(q\right)=\frac{1}{{\left(q\right)}_{\infty}}\left(2{q}^{-\frac{1}{24}{\left(2m+1\right)}^{2}}\sum _{n>2m}^{\infty}{\chi}_{m}\left(n\right){q}^{\frac{{n}^{2}}{24}}-1\right)$$

$${\chi}_{m}\left(n\right)=\left\{\begin{array}{cc}+1\hfill & n\equiv 2m+1\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}12)\hfill \\ -1\hfill & n\equiv 2m+7\phantom{\rule{10.0pt}{0ex}}(mod\phantom{\rule{0.277778em}{0ex}}12)\hfill \\ 0\hfill & \phantom{\rule{4.pt}{0ex}}\text{else}\hfill \end{array}\right.$$

The following proposition is a slight variation of a proposition of Lawrence and Zagier [20]. Since the proof is analogous and standard, we do not include it here.

$$\sum _{n>m}^{\infty}C\left(n\right){e}^{-{n}^{2}t}\sim \sum _{r=0}^{\infty}{L}_{m}(-2r,C)\xb7\frac{{(-t)}^{r}}{r!}$$

$${L}_{m}(-r,C)=-\frac{{M}^{r}}{r+1}\sum _{n=(m+1)}^{m+M}C\left(n\right){B}_{r+1}\left(\frac{n}{M}\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}(r=0,1,\dots )$$

$$\begin{array}{cc}\hfill \sum _{n>2m}^{\infty}\left(\frac{-4}{n}\right){q}^{\frac{{n}^{2}}{8}}& \sim {(-1)}^{m}\left(\frac{1}{2}+\left(2{m}^{2}-\frac{1}{2}\right)\frac{(-s)}{8}+\left(8{m}^{4}-12{m}^{2}+\frac{5}{2}\right)\frac{{(-s)}^{2}}{{8}^{2}\xb72}+\cdots \right)\hfill \\ \hfill \sum _{n>2m}^{\infty}{\chi}_{m}\left(n\right){q}^{\frac{{n}^{2}}{24}}& \sim \frac{1}{2}+\left(2{m}^{2}-4m-\frac{5}{2}\right)(-s)+\frac{1}{2}(2m-5)(2m+1)(4{m}^{2}-8m-41)\frac{{s}^{2}}{2}+\cdots \hfill \end{array}$$

Using ${\left(q\right)}_{\infty}^{-1}=\sqrt{\frac{s}{2\pi}}{e}^{\frac{{\pi}^{2}}{6s}-\frac{s}{24}}(1+O\left({s}^{N}\right))$ for any $N>0$ (this follows from Euler–Maclaurin summation formula or the modularity of the Dedekind eta-function, see [21] page 53), we see that
and
A standard application of the circle method (see, for instance, Wright [18] for a similar situation) gives the theorem.

$${C}_{\le m}\left({e}^{-s}\right)\sim \sqrt{\frac{s}{2\pi}}{e}^{\frac{{\pi}^{2}}{6s}-\frac{s}{24}}\left(\frac{(2m+1)}{4}s+\frac{(2m+1)}{16}{s}^{2}+\cdots \right)$$

$${R}_{\le m}\left({e}^{-s}\right)\sim \sqrt{\frac{s}{2\pi}}{e}^{\frac{{\pi}^{2}}{6s}-\frac{s}{24}}\left(\frac{(2m+1)}{4}s+3\frac{(2m+1)}{16}{s}^{2}+\cdots \right)$$

This section collects some observations concerning the values of ${N}_{S}(m,n)$. In particular, we are concerned with defining the $\mathrm{spt}$-crank in terms of partitions (perhaps with their parts marked by the multiplicity).

A marked partition means a pair $(\lambda ,k)$ where λ is a partition and k is an integer identifying one of its smallest parts. If there are s smallest parts then the $k=1,2,\cdots ,s$. Evidently, a good first approximation for the $\mathrm{spt}$-crank is
where p is the number of parts in λ greater than or equal to k. If $T(n,m)$ is the number of marked partitions of n with $F(\lambda ,k)=m$, then the difference
is zero for most of the possible values of n and m. Table 2 and Table 3 give the values of $T(n,m)$ and $D(n,m)$ for small $n\le 12$.

$$F(\lambda ,k):=\left\{\begin{array}{cc}p-k\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}p>0\hfill \\ 1-k\hfill & \phantom{\rule{4.pt}{0ex}}\text{if}\phantom{\rule{4.pt}{0ex}}p=0\hfill \end{array}\right.$$

$$D(n,m):=T(n,m)-{N}_{S}(n,m)$$

$n\backslash m$ | −9 | −8 | −7 | −6 | −5 | −4 | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1 | ||||||||||||||||||

2 | 1 | 1 | 1 | ||||||||||||||||

3 | 1 | 1 | 1 | 1 | 1 | ||||||||||||||

4 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | ||||||||||||

5 | 1 | 1 | 2 | 3 | 1 | 2 | 2 | 1 | 1 | ||||||||||

6 | 1 | 1 | 2 | 4 | 4 | 3 | 4 | 3 | 2 | 1 | 1 | ||||||||

7 | 1 | 1 | 2 | 3 | 5 | 5 | 3 | 4 | 4 | 3 | 2 | 1 | 1 | ||||||

8 | 1 | 1 | 2 | 3 | 6 | 8 | 6 | 6 | 6 | 6 | 5 | 3 | 2 | 1 | 1 | ||||

9 | 1 | 1 | 2 | 3 | 5 | 7 | 11 | 8 | 8 | 8 | 8 | 6 | 5 | 3 | 2 | 1 | 1 | ||

10 | 1 | 1 | 2 | 3 | 5 | 8 | 12 | 15 | 10 | 11 | 11 | 11 | 10 | 7 | 5 | 3 | 2 | 1 | 1 |

$n\backslash m$ | −11 | −10 | −9 | −8 | −7 | −6 | −5 | −4 | −3 | −2 | −1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0 | ||||||||||||||||||||||

2 | 0 | 0 | 0 | ||||||||||||||||||||

3 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||

4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||

5 | 0 | 0 | 0 | 1 | −1 | 0 | 0 | 0 | 0 | ||||||||||||||

6 | 0 | 0 | 0 | 1 | 0 | −1 | 0 | 0 | 0 | 0 | 0 | ||||||||||||

7 | 0 | 0 | 0 | 0 | 1 | 1 | −2 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||||

8 | 0 | 0 | 0 | 0 | 1 | 2 | −1 | −1 | −1 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||||

9 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | −1 | −2 | −1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||||

10 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 4 | −3 | $-2$ | −2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||||

11 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 4 | 4 | −2 | −4 | −2 | −1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ||

12 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 2 | 6 | 4 | −5 | −4 | −3 | −1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

The differences $D(n,m)$ are non-zero and have a simple regular behavior in the central angle between the two lines
and zero everywhere else. Additionally, the numbers become periodic on the boundaries. (This is hard to tell from the Table 2, but easy to see from a larger table.) The left side boundary has period 2 and the right hand boundary has period 3. After removing those periodic parts, there are two more boundary lines
which separate the regions where the numbers are periodic from the regions where they are not. So it is easy to conjecture that there is a series of boundaries $N=km+c$, $N=-km+c$, for each integer k, separating regions with period $k-1$ from regions with period k.

$$n=3m+5\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}n=-2m+2$$

$$n=-3m+3\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}n=4m+11$$

Finally, we speculate that a definition of the $\mathrm{spt}$-crank may be different depending on the size of the smallest part. It remains a challenge to find a definition of the $\mathrm{spt}$-crank for ordinary partitions.

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