---
title: spt-Crank-Type Partitions Overview
url: https://www.emergentmind.com/topics/spt-crank-type-partitions
type: topic
---

# spt-Crank-Type Partitions Overview

Searching arXiv for recent and foundational papers on spt-crank-type partitions.
Searching arXiv for “spt-crank partitions”, “higher order spt-functions”, and “spt-crank for overpartitions”.
spt-crank-type partitions are partition-theoretic objects and refinements built to explain smallest-parts functions by means of crank-like or rank-like statistics. In the ordinary partition setting, the starting point is Andrews’ \( \operatorname{spt}(n) \), the total number of occurrences of the smallest part in all partitions of \(n\); from there the subject expands to \(S\)-partitions, marked and doubly marked partitions, overpartitions, Bailey-pair constructions, higher symmetrized moment differences, and related functions such as \(\operatorname{ospt}(n)\) and \(\operatorname{spt}_\omega(n)\). The literature uses the phrase in a broad sense: sometimes for combinatorial classes carrying an explicit spt-crank, sometimes for two-variable generating functions whose specialization at \(z=1\) yields an spt-type function, and sometimes for moment-difference constructions that generalize the basic relation between smallest-parts counts and rank/crank moments [1707.04369; 1501.06843].

## 1. Foundational framework

The basic smallest-parts function is defined by
\[
\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),
\]
where \(n_s(\lambda)\) is the multiplicity of the smallest part in a partition \(\lambda\) of \(n\). Its generating function is
\[
\sum_{n=1}^{\infty}\operatorname{spt}(n)q^n
=\sum_{n=1}^{\infty}\frac{q^n}{(1-q^n)^2(q^{n+1};q)_\infty}.
\]
The surrounding theory depends on the classical rank and crank distributions \(N(m,n)\) and \(M(m,n)\), together with their moments and symmetrized moments [1707.04369].

A central structural point is that spt-type functions are typically realized as differences between rank-like and crank-like quantities. In the ordinary partition case, the smallest-parts function is tied to second rank and crank moments. In higher-order and generalized settings, the same pattern persists: one introduces a refined crank statistic or a two-variable generating function, then proves that the resulting spt-type series is a moment difference, a rank–crank difference, or a root-of-unity specialization of a two-variable partition series [1008.1207].

This framework supports several distinct but related enterprises. One is combinatorial: construct partition classes and a statistic whose residue classes split an spt-type function into equinumerous families. Another is analytic: derive generating functions, dissections, asymptotics, and modular or mock modular structure. A third is arithmetic: explain congruences of Ramanujan type and prove positivity or parity results.

## 2. Classical spt-crank for ordinary partitions

The original spt-crank of Andrews, Garvan, and Liang is defined on \(S\)-partitions. Let \(\mathcal D\) be the set of partitions into distinct parts and \(\mathcal P\) the set of all partitions. Then
\[
S=\{(\pi_1,\pi_2,\pi_3)\in \mathcal D\times\mathcal P\times\mathcal P:
\pi_1\neq\emptyset,\ s(\pi_1)\le \min\{s(\pi_2),s(\pi_3)\}\}.
\]
For \(\pi=(\pi_1,\pi_2,\pi_3)\in S\), the spt-crank is
\[
r(\pi)=\ell(\pi_2)-\ell(\pi_3),
\]
and the sign is \(\omega(\pi)=(-1)^{\ell(\pi_1)-1}\). The net count
\[
N_S(m,n)=\sum_{\substack{|\pi|=n\\ r(\pi)=m}}\omega(\pi)
\]
refines \(\operatorname{spt}(n)\) and gives combinatorial explanations of the congruences modulo \(5\) and \(7\) [1305.2116].

A major development was the conversion of this signed vector-partition theory into signless ordinary partition models. Chen, Ji, and Zang introduced doubly marked partitions \((\lambda,s,t)\), defined by Ferrers-diagram conditions \(1\le s\le D(\lambda)\), \(s\le t\le \lambda_1\), and \(\lambda'_s=\lambda'_t\). For such an object,
\[
g(\lambda,s,t)=\lambda'_s-s+1,\qquad
c(\lambda,s,t)=g(\lambda,s,t)-\lambda_g+t-s
\]
is the spt-crank. They proved that \(N_S(m,n)\) equals the number of doubly marked partitions of \(n\) with spt-crank \(m\), and established a bijection between marked partitions and doubly marked partitions, thereby solving the problem of defining the spt-crank directly on ordinary partition data [1308.3012].

The distribution theory of \(N_S(m,n)\) is closely related to the cumulative distributions of the ordinary rank and crank. Andrews, Dyson, and Rhoades conjectured that \(\{N_S(m,n)\}_m\) is unimodal; Chen and collaborators proved this by introducing the \(m\)-Durfee rectangle symbol and constructing explicit injections. In that setting,
\[
N_S(m,n)-N_S(m+1,n)=\frac12\bigl(N_{\le m}(n)-M_{\le m}(n)\bigr),
\]
so unimodality is equivalent to an inequality between cumulative rank and crank counts [1305.2116].

A further refinement is the “nearly equal distribution” theorem. For every \(n\), there is a reordering \(\tau_n\) of the partitions of \(n\) such that
\[
|\operatorname{crank}(\lambda)|-|\operatorname{rank}(\tau_n(\lambda))|=0\ \text{or}\ 1.
\]
This gives a new combinatorial interpretation of
\[
\operatorname{ospt}(n)=\sum_{m\ge0}mM(m,n)-\sum_{m\ge0}mN(m,n):
\]
it counts those partitions \(\lambda\) for which \(\operatorname{crank}(\lambda)-\operatorname{rank}(\tau_n(\lambda))=1\) [1704.00882].

## 3. Moment-difference theory and higher orders

A decisive extension came from the introduction of positive moments. For \(r\in\mathbb N\),
\[
M_r^+(N)=\sum_{m>0}m^rM(m,N),\qquad
N_r^+(N)=\sum_{m>0}m^rN(m,N).
\]
Bringmann and Mahlburg proved that these two families are asymptotically equal, but that the crank moments are asymptotically larger:
\[
M_r^+(N)\sim N_r^+(N)\sim \gamma_r N^{(r-1)/2}e^{\pi\sqrt{2N/3}},
\]
while
\[
M_r^+(N)-N_r^+(N)\sim \sigma_r N^{(r-2)/2}e^{\pi\sqrt{2N/3}}.
\]
In particular, \(M_r^+(N)>N_r^+(N)\) for sufficiently large \(N\) and all \(r\ge1\) [1205.2155].

The first positive-moment difference is the ospt-function,
\[
\operatorname{ospt}(N)=M_1^+(N)-N_1^+(N).
\]
Andrews, Chan, and Kim gave it a combinatorial interpretation in terms of certain “even and odd strings” in partitions, ensuring \(\operatorname{ospt}(N)>0\) and weak monotonicity in \(N\). Bringmann and Mahlburg obtained its asymptotic behavior,
\[
\operatorname{ospt}(N)\sim \frac{1}{16\sqrt{3}N}p(N),
\]
and proved the parity relation
\[
\operatorname{ospt}(N)\equiv \operatorname{spt}(N)\pmod 2.
\]
They also determined when \(\operatorname{ospt}(N)\) is odd: precisely when \(24N-1=\ell^{2a+1}m^2\) for some prime \(\ell\equiv23\pmod{24}\) with \((m,\ell)=1\) [1205.2155].

Garvan’s higher-order theory packages these ideas in terms of symmetrized moments. For \(k\ge1\),
\[
\operatorname{spt}_k(n)=\mu_{2k}(n)-\eta_{2k}(n),
\]
where \(\mu_{2k}(n)\) and \(\eta_{2k}(n)\) are the \(2k\)-th symmetrized crank and rank moments. The associated generating function is a nested multiple series, and the difference admits a weighted combinatorial interpretation over partitions. This yields, for all \(n\ge1\) and \(k\ge1\),
\[
M_{2k}(n)>N_{2k}(n),
\]
together with explicit congruences for \(\operatorname{spt}_2(n)\), \(\operatorname{spt}_3(n)\), and \(\operatorname{spt}_4(n)\) [1008.1207].

## 4. Overpartitions and other restricted partition classes

The spt-crank program extends naturally to overpartitions and related classes. In this setting one encounters \(\overline{\operatorname{spt}}(n)\), \(\overline{\operatorname{spt}}_1(n)\), \(\overline{\operatorname{spt}}_2(n)\), and \(M2\operatorname{spt}(n)\), where the smallest part may be constrained by parity, overlining, or the requirement that odd parts not repeat. Bringmann, Lovejoy, and Osburn showed that the generating functions of these spt-overpartition functions are quasimock theta functions, and Jennings-Shaffer defined spt-cranks on vector partitions that explain their simple congruences [1311.3680].

For overpartitions, the two-variable series
\[
S(z,q),\quad S_1(z,q),\quad S_2(z,q),\quad S_M(z,q)
\]
play the role of spt-crank generating functions. They satisfy rank–crank difference identities of the form
\[
(1-z)(1-z^{-1})S(z,q)
=\sum_{n,m}\bigl(\overline N(m,n)-\overline M(m,n)\bigr)z^mq^n,
\]
with analogous formulas for the other families. By specializing \(z\) to roots of unity and using Bailey’s Lemma together with overpartition rank-difference formulas, one obtains combinatorial refinements of congruences such as
\[
\overline{\operatorname{spt}}(3n)\equiv0\pmod3,\qquad
\overline{\operatorname{spt}}_2(5n+3)\equiv0\pmod5,
\]
as well as equidistribution statements on marked overpartitions [1311.3680].

A more specialized example is \(\overline{\operatorname{spt}}_2(n)\), the total number of occurrences of the smallest parts among overpartitions of \(n\) where the smallest part is even and not overlined. Jennings-Shaffer gave another spt-crank for this function using the \(M_2\)-rank and a residual crank, and proved a 3-dissection at \(z=\zeta_3\) in which the coefficients of \(q^{3n}\) and \(q^{3n+1}\) vanish. This yields a combinatorial refinement of
\[
\overline{\operatorname{spt}}_2(3n)\equiv
\overline{\operatorname{spt}}_2(3n+1)\equiv0\pmod3
\]
[1406.5458].

The higher-order theory also has overpartition analogues. For overpartitions, overpartitions with smallest part even, and partitions with smallest part even and no repeated odd parts, the higher-order spt-functions are again defined as differences of symmetrized crank and rank moments, and each has an explicit multiple-sum generating function. In these settings the corresponding crank moments dominate the rank moments, yielding nonnegativity of the higher-order spt-type functions and new congruences [1402.5098].

| Family | Statistic or series | Representative property |
|---|---|---|
| Overpartitions | \(S(z,q)\), \(S_1(z,q)\), \(S_2(z,q)\), \(S_M(z,q)\) | Combinatorial refinements of simple mod \(3\) and mod \(5\) congruences |
| Overpartitions with even smallest part | \(M_2\)-rank and residual crank | 3-dissection explaining \(\overline{\operatorname{spt}}_2(3n)\) and \(\overline{\operatorname{spt}}_2(3n+1)\) |
| Partitions without repeated odd parts | \(M2\operatorname{spt}(n)\), \(M2\operatorname{spt}_2(n)\) | Congruences derived from exact relations between rank and residual crank moments |

For partitions without repeated odd parts, Jennings-Shaffer used quasimodular forms on \(\Gamma_0(4)\) to derive exact relations between the \(M_2\)-rank and three residual cranks. These identities imply congruences for \(M2\operatorname{spt}(n)\) and \(M2\operatorname{spt}_2(n)\), including
\[
M2\operatorname{spt}(3n+1)\equiv0\pmod3,\qquad
M2\operatorname{spt}(5n+1)\equiv M2\operatorname{spt}(5n+3)\equiv0\pmod5
\]
[1404.1883].

## 5. Bailey pairs, generalized spt-crank-type families, and mock modularity

A large branch of the subject constructs spt-crank-type functions from Bailey pairs. For an appropriate Bailey pair \(X\), Garvan and collaborators define
\[
S_X(z,q)=\frac{P_X(q)}{(z,z^{-1};q)_\infty}
\sum_{n=1}^\infty (z,z^{-1};q)_n\beta_n^Xq^n,
\]
whose specialization \(S_X(1,q)\) is an spt-type generating function. In this framework the Bailey pairs \(A1,A3,A5,A7,C1,C5,E2,E4\) yield both old and new spt-type functions, each admitting Ramanujan-type congruences explained by the two-variable crank series. The corresponding \(S_X(z,q)\) have representations as infinite products or as Hecke–Rogers-type double sums, and root-of-unity dissections identify vanishing coefficients that force congruences [1501.06843; 1408.5395].

A related extension uses conjugate Bailey pairs and two four-variable \(q\)-hypergeometric functions \(F(\rho_1,\rho_2,z;q)\) and \(G(\rho_1,\rho_2,z;q)\). Their specializations produce new spt-crank-type functions such as \(S_{G1}\), \(S_{G2}\), \(S_{F1}\), \(S_{G3}\), \(S_{L7}\), \(S_{L9}\), and \(S_{L12}\), together with mod \(3\) and mod \(5\) congruences. The same formalism also specializes to many previously known spt-crank-type functions, so \(F\) and \(G\) act as master templates for a substantial part of the theory [1506.05344].

The function \(\operatorname{spt}_\omega(n)\) occupies a particularly explicit mock-theoretic niche. Here \(\operatorname{p}_\omega(n)\) counts partitions of \(n\) such that all odd parts are smaller than twice the smallest part, with
\[
\sum_{n\ge1}\operatorname{p}_\omega(n)q^n=q\,\omega(q),
\]
where \(\omega(q)\) is Ramanujan’s third order mock theta function. Garvan and Jennings-Shaffer introduced a crank \(N_{C_1}(m,n)\) whose residue classes explain
\[
\operatorname{spt}_\omega(5n+3)\equiv0\pmod5.
\]
For fixed \(m\), \(N_{C_1}(m,n)\) is asymptotically positive, and the differences \(N_{C_1}(m,n)-N_{C_1}(m+1,n)\) have an alternating sign pattern for large \(n\). The generating series \(S_\omega(q)\) is the holomorphic part of a weight \(3/2\) harmonic weak Maass form [1603.05608].

This broad Bailey-pair literature makes clear that “spt-crank-type” is not restricted to a single combinatorial model. Rather, it denotes a recurrent mechanism: an spt-type function is embedded in a two-variable \(q\)-series, root-of-unity evaluations detect equidistribution, and the resulting dissections yield explicit congruences.

## 6. Arithmetic, finite analogues, equidistribution, and recent directions

The theory also admits finite analogues. For the restricted partition function \(p(n,N)\), where the largest part is at most \(N\), Dixit and collaborators define
\[
\operatorname{spt}(n,N)=\text{the total number of appearances of the smallest parts in all partitions of }n\text{ with largest part}\le N.
\]
They introduce finite rank and crank generating functions on vector partitions, define finite moments \(N_{k,N}(n)\) and \(M_{k,N}(n)\), and prove the exact identity
\[
\operatorname{spt}(n,N)=M_{2,N}(n)-N_{2,N}(n).
\]
As in the unrestricted case, one has
\[
M_{2,N}(n)>N_{2,N}(n)\qquad (n\ge1),
\]
and the higher-even-moment analogue is conjectured [1812.01424].

A different algebraic direction studies partition polynomials. Using equidistribution criteria, Bringmann, Gomez, Rolen, and Tripp proved that the spt-crank Laurent polynomials satisfy
\[
\Phi_5(w)\mid spt\text{-}crank_{5n+4}(w),\qquad
\Phi_7(w)\mid spt\text{-}crank_{7n+5}(w),
\]
and that the quotient coefficients are non-negative. As corollaries, one recovers
\[
\operatorname{spt}(5n+4)\equiv0\pmod5,\qquad
\operatorname{spt}(7n+5)\equiv0\pmod7.
\]
The same work shows that the roots of the principal spt-crank polynomials become equidistributed on the unit circle as \(n\to\infty\) [2209.15114].

Bias phenomena among rank and crank classes also remain relevant. Bringmann and Pandey proved detailed inequalities among rank classes, crank classes, and \(p(11n+d)\) modulo \(11\), confirming Borozenets’ conjectures. The paper is not itself an spt-crank construction, but it supplies explicit orderings among residue classes that the authors describe as groundwork for analogous investigations of spt-crank-type distributions modulo \(11\) [2308.02327]. This suggests that the arithmetic study of residue-class biases can continue to feed into spt-refinement problems.

A recent higher-moment development concerns the fourth symmetrized crank. A smallest-parts function \(SPT^{-}(n)\) attached to the fourth symmetrized crank is constructed so that
\[
SPT^{-}(n)=\frac{5}{72}M_4(n)-\frac16 n^2p(n)+\frac1{36}np(n)-\mu_4(n),
\]
and it satisfies
\[
\operatorname{spt}_2(n)=SPT^{+}(n)-SPT^{-}(n),\qquad
SPT^{+}(n)>SPT^{-}(n).
\]
The paper also proves congruences such as
\[
SPT^{-}(7n)\equiv0\pmod7,\qquad
SPT^{-}(11n)\equiv0\pmod{11},\qquad
SPT^{-}(7n+5)\equiv0\pmod7,
\]
together with asymptotics showing that \(SPT^{+}(n)\) and \(SPT^{-}(n)\) have the same leading exponential growth [2511.05857].

Taken together, these developments show that spt-crank-type partitions form a research program rather than a single construction. Its recurring themes are explicit combinatorial models, two-variable generating functions, rank–crank differences, Bailey-pair machinery, modular and mock modular structures, and arithmetic refinement through congruences, parity, asymptotics, and equidistribution.

Source: https://www.emergentmind.com/topics/spt-crank-type-partitions