spt-Crank-Type Partitions Overview

• spt-Crank-Type partitions are partition-theoretic objects that refine the smallest-parts function by incorporating crank-like and rank-like statistics.
• They utilize two-variable generating functions and Bailey pair techniques to uncover combinatorial, analytic, and arithmetic insights, including explicit congruences and asymptotics.
• Recent developments extend the theory to overpartitions and restricted partitions, linking these refinements with mock modular forms and advanced moment-difference constructions.

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 (Chen, 2017, Garvan et al., 2015).

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

$n$0

The surrounding theory depends on the classical rank and crank distributions $n$1 and $n$2, together with their moments and symmetrized moments (Chen, 2017).

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 (Garvan, 2010).

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 $n$3-partitions. Let $n$4 be the set of partitions into distinct parts and $n$5 the set of all partitions. Then

$n$6

For $n$7, the spt-crank is

$n$8

and the sign is $n$9. The net count

$S$0

refines $S$1 and gives combinatorial explanations of the congruences modulo $S$2 and $S$3 (Chen et al., 2013).

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 $S$4, defined by Ferrers-diagram conditions $S$5, $S$6, and $S$7. For such an object,

$S$8

is the spt-crank. They proved that $S$9 equals the number of doubly marked partitions of $\operatorname{ospt}(n)$0 with spt-crank $\operatorname{ospt}(n)$1, 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 (Chen et al., 2013).

The distribution theory of $\operatorname{ospt}(n)$2 is closely related to the cumulative distributions of the ordinary rank and crank. Andrews, Dyson, and Rhoades conjectured that $\operatorname{ospt}(n)$3 is unimodal; Chen and collaborators proved this by introducing the $\operatorname{ospt}(n)$4-Durfee rectangle symbol and constructing explicit injections. In that setting,

$\operatorname{ospt}(n)$5

so unimodality is equivalent to an inequality between cumulative rank and crank counts (Chen et al., 2013).

A further refinement is the “nearly equal distribution” theorem. For every $\operatorname{ospt}(n)$6, there is a reordering $\operatorname{ospt}(n)$7 of the partitions of $\operatorname{ospt}(n)$8 such that

$\operatorname{ospt}(n)$9

This gives a new combinatorial interpretation of

$\operatorname{spt}_\omega(n)$0

it counts those partitions $\operatorname{spt}_\omega(n)$1 for which $\operatorname{spt}_\omega(n)$2 (Chen et al., 2017).

3. Moment-difference theory and higher orders

A decisive extension came from the introduction of positive moments. For $\operatorname{spt}_\omega(n)$3,

$\operatorname{spt}_\omega(n)$4

Bringmann and Mahlburg proved that these two families are asymptotically equal, but that the crank moments are asymptotically larger: $\operatorname{spt}_\omega(n)$5 while

$\operatorname{spt}_\omega(n)$6

In particular, $\operatorname{spt}_\omega(n)$7 for sufficiently large $\operatorname{spt}_\omega(n)$8 and all $\operatorname{spt}_\omega(n)$9 (Bringmann et al., 2012).

The first positive-moment difference is the ospt-function,

$z=1$0

Andrews, Chan, and Kim gave it a combinatorial interpretation in terms of certain “even and odd strings” in partitions, ensuring $z=1$1 and weak monotonicity in $z=1$2. Bringmann and Mahlburg obtained its asymptotic behavior,

$z=1$3

and proved the parity relation

$z=1$4

They also determined when $z=1$5 is odd: precisely when $z=1$6 for some prime $z=1$7 with $z=1$8 (Bringmann et al., 2012).

Garvan’s higher-order theory packages these ideas in terms of symmetrized moments. For $z=1$9,

$$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$0

where $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$1 and $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$2 are the $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$3-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 $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$4 and $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$5,

$$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$6

together with explicit congruences for $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$7, $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$8, and $$\operatorname{spt}(n)=\sum_{\lambda\in P(n)} n_s(\lambda),$$9 (Garvan, 2010).

4. Overpartitions and other restricted partition classes

The spt-crank program extends naturally to overpartitions and related classes. In this setting one encounters $n_s(\lambda)$0, $n_s(\lambda)$1, $n_s(\lambda)$2, and $n_s(\lambda)$3, 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 (Garvan et al., 2013).

For overpartitions, the two-variable series

$n_s(\lambda)$4

play the role of spt-crank generating functions. They satisfy rank–crank difference identities of the form

$n_s(\lambda)$5

with analogous formulas for the other families. By specializing $n_s(\lambda)$6 to roots of unity and using Bailey’s Lemma together with overpartition rank-difference formulas, one obtains combinatorial refinements of congruences such as

$n_s(\lambda)$7

as well as equidistribution statements on marked overpartitions (Garvan et al., 2013).

A more specialized example is $n_s(\lambda)$8, the total number of occurrences of the smallest parts among overpartitions of $n_s(\lambda)$9 where the smallest part is even and not overlined. Jennings-Shaffer gave another spt-crank for this function using the $\lambda$0-rank and a residual crank, and proved a 3-dissection at $\lambda$1 in which the coefficients of $\lambda$2 and $\lambda$3 vanish. This yields a combinatorial refinement of

$\lambda$4

(Jennings-Shaffer, 2014).

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 (Jennings-Shaffer, 2014).

Family	Statistic or series	Representative property
Overpartitions	$\lambda$5, $\lambda$6, $\lambda$7, $\lambda$8	Combinatorial refinements of simple mod $\lambda$9 and mod $n$0 congruences
Overpartitions with even smallest part	$n$1-rank and residual crank	3-dissection explaining $n$2 and $n$3
Partitions without repeated odd parts	$n$4, $n$5	Congruences derived from exact relations between rank and residual crank moments

For partitions without repeated odd parts, Jennings-Shaffer used quasimodular forms on $n$6 to derive exact relations between the $n$7-rank and three residual cranks. These identities imply congruences for $n$8 and $n$9, including

$n$00

(Jennings-Shaffer, 2014).

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 $n$01, Garvan and collaborators define

$n$02

whose specialization $n$03 is an spt-type generating function. In this framework the Bailey pairs $n$04 yield both old and new spt-type functions, each admitting Ramanujan-type congruences explained by the two-variable crank series. The corresponding $n$05 have representations as infinite products or as Hecke–Rogers-type double sums, and root-of-unity dissections identify vanishing coefficients that force congruences [(Garvan et al., 2015); (Jennings-Shaffer, 2014)].

A related extension uses conjugate Bailey pairs and two four-variable $n$06-hypergeometric functions $n$07 and $n$08. Their specializations produce new spt-crank-type functions such as $n$09, $n$10, $n$11, $n$12, $n$13, $n$14, and $n$15, together with mod $n$16 and mod $n$17 congruences. The same formalism also specializes to many previously known spt-crank-type functions, so $n$18 and $n$19 act as master templates for a substantial part of the theory (Jennings-Shaffer, 2015).

The function $n$20 occupies a particularly explicit mock-theoretic niche. Here $n$21 counts partitions of $n$22 such that all odd parts are smaller than twice the smallest part, with

$n$23

where $n$24 is Ramanujan’s third order mock theta function. Garvan and Jennings-Shaffer introduced a crank $n$25 whose residue classes explain

$n$26

For fixed $n$27, $n$28 is asymptotically positive, and the differences $n$29 have an alternating sign pattern for large $n$30. The generating series $n$31 is the holomorphic part of a weight $n$32 harmonic weak Maass form (Jang et al., 2016).

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 $n$33-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 $n$34, where the largest part is at most $n$35, Dixit and collaborators define

$n$36

They introduce finite rank and crank generating functions on vector partitions, define finite moments $n$37 and $n$38, and prove the exact identity

$n$39

As in the unrestricted case, one has

$n$40

and the higher-even-moment analogue is conjectured (Dixit et al., 2018).

A different algebraic direction studies partition polynomials. Using equidistribution criteria, Bringmann, Gomez, Rolen, and Tripp proved that the spt-crank Laurent polynomials satisfy

$n$41

and that the quotient coefficients are non-negative. As corollaries, one recovers

$n$42

The same work shows that the roots of the principal spt-crank polynomials become equidistributed on the unit circle as $n$43 (Folsom et al., 2022).

Bias phenomena among rank and crank classes also remain relevant. Bringmann and Pandey proved detailed inequalities among rank classes, crank classes, and $n$44 modulo $n$45, 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 $n$46 (Bringmann et al., 2023). 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 $n$47 attached to the fourth symmetrized crank is constructed so that

$n$48

and it satisfies

$n$49

The paper also proves congruences such as

$n$50

together with asymptotics showing that $n$51 and $n$52 have the same leading exponential growth (Patkowski, 8 Nov 2025).

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.