Crank-type combinatorial interpretation of the spt-function congruences beyond moduli 5 and 7: the mod 13 case and higher prime moduli

The problem originates in Freeman Dyson's 2012 paper 'Partitions and the grand canonical ensemble' (Ramanujan J. 29, 423-429; reprinted as pp. 342-348 of his World Scientific collection Birds and Frogs, 2015), whose second part guesses - and Garvan proves - an explicit inclusion-exclusion formula for $N_S(m,n)$, the net number of S-partitions of $n$ with spt-crank $m$, and hopes it will yield 'a better understanding of the beautiful new congruence properties' of Andrews' smallest-parts function $\mathrm{spt}(n)$. Much of that hope is now realized: Andrews-Garvan-Liang's spt-crank gives equidistribution interpretations of $\mathrm{spt}(5n+4)\equiv 0\pmod 5$ and $\mathrm{spt}(7n+5)\equiv 0\pmod 7$; nonnegativity and unimodality of $N_S(m,n)$ are proved; and Chen-Ji-Zang interpret $N_S(m,n)$ as counting doubly marked partitions, giving an ordinary-partition form of the spt-crank. The surviving core is the third of Andrews' original congruences, $\mathrm{spt}(13n+6)\equiv 0\pmod{13}$, together with Ono's congruences at primes $\ell\ge 5$ and Garvan's prime-power congruences: their analytic theory is essentially complete, yet no crank-type combinatorial interpretation exists for any of them. The problem asks for such a statistic and equidistribution theorem - the mod-13 analogue of $N_S(k,5,5n+4)=\mathrm{spt}(5n+4)/5$ - together with the accompanying structural theory (nonnegativity, unimodality, and a directly defined spt-crank on marked partitions).

Background

An integer partition of $n$ is a nonincreasing sequence of positive integers with sum $n$; $p(n)$ denotes the number of partitions of $n$. Ramanujan's classical congruences $p(5n+4)\equiv 0\pmod 5$, $p(7n+5)\equiv 0\pmod 7$ and $p(11n+6)\equiv 0\pmod{11}$ triggered a long search for 'combinatorial explanations' of partition congruences: a statistic on the counted objects whose distribution splits them into equinumerous classes. Dyson's rank of a partition (largest part minus number of parts) does this for the mod-5 and mod-7 congruences: writing $N(i,t,n)$ for the number of partitions of $n$ with rank congruent to $i$ modulo $t$, the equidistribution identities $N(i,5,5n+4)=p(5n+4)/5$ and $N(i,7,7n+5)=p(7n+5)/7$ were conjectured by Dyson and proved by Atkin and Swinnerton-Dyer. The crank of a partition (the largest part when the partition contains no 1's, otherwise the number of parts exceeding the number of 1's, minus the number of 1's), conjectured by Dyson and constructed by Andrews and Garvan as a weighted count of vector partitions (triples of partitions with a sign), gives the analogous equidistribution modulo 5, 7 and 11.

Andrews introduced the smallest-parts function $\mathrm{spt}(n)$, the total number of occurrences of the smallest part over all partitions of $n$ (for example $\mathrm{spt}(4)=10$), proved that $\mathrm{spt}(n)=n\,p(n)-\tfrac{1}{2}N_2(n)$, where $N_2(n)=\sum_m m^2\,N(m,n)$ (with $N(m,n)$ the number of partitions of $n$ with rank $m$) is the second moment of the rank, and established the Ramanujan-type congruences $\mathrm{spt}(5n+4)\equiv 0\pmod 5$, $\mathrm{spt}(7n+5)\equiv 0\pmod 7$ and $\mathrm{spt}(13n+6)\equiv 0\pmod{13}$. The mod-5 and mod-7 proofs descend from the rank equidistribution identities; the mod-13 proof is analytic, using rank-difference identities modulo 13 (reproduced in the account of Chen-Ji-Zang). On the analytic side the congruence theory is essentially complete: Folsom and Ono connected $\mathrm{spt}$ to harmonic Maass forms (real-analytic modular forms whose differential shadows are half-integral-weight theta functions); Garvan proved congruences modulo all powers of 5, 7 and 13; Ono proved Ramanujan-type congruences modulo every prime $\ell\ge 5$, confirming a conjecture of Garvan. None of these analytic results carries a combinatorial interpretation.

The combinatorial framework is due to Andrews, Garvan and Liang. An S-partition is a triple $(\pi_1,\pi_2,\pi_3)$ in which $\pi_1$ is a nonempty partition into distinct parts whose smallest part is at most the smallest parts of the ordinary partitions $\pi_2$ and $\pi_3$; its sign is $(-1)^{\ell(\pi_1)-1}$, where $\ell$ denotes the number of parts, and its spt-crank is $\ell(\pi_2)-\ell(\pi_3)$. Writing $N_S(m,n)$ for the net (sign-weighted) number of S-partitions of $n$ with spt-crank $m$, they proved the equidistribution theorems $N_S(k,5,5n+4)=\mathrm{spt}(5n+4)/5$ and $N_S(k,7,7n+5)=\mathrm{spt}(7n+5)/7$ - combinatorial interpretations of the mod-5 and mod-7 spt congruences in exactly the rank/crank sense - and the nonnegativity $N_S(m,n)\ge 0$ for all $m$ and $n\ge 1$, which makes the signed count a genuine counting function.

The source of this problem is Dyson's paper 'Partitions and the grand canonical ensemble'. Besides a physics-motivated approximation to the Hardy-Ramanujan asymptotic formula for $p(n)$, it guesses an explicit formula for $N_S(m,n)$ - an inclusion-exclusion recurrence

$$N_S(m,n)=\sum_{k\ge 1}(-1)^{k-1}\sum_{j=0}^{k-1}p\bigl(n-k(m+j)-k(k+1)/2\bigr),$$

proved by Garvan, from which Dyson rederived nonnegativity - and it expresses the hope that the formula and its rank/crank connections may lead to a better understanding of the congruence properties of $\mathrm{spt}$. Andrews, Dyson and Rhoades then conjectured that the sequence $\{N_S(m,n)\}_m$ is unimodal for every $n$ (a sequence that is nondecreasing up to a peak and nonincreasing thereafter), a conjecture equivalent to an inequality between rank and crank counts of ordinary partitions; they proved it asymptotically and posed the challenge of interpreting the spt-crank on ordinary partitions, introducing marked partitions (a partition with one of its smallest parts distinguished). Chen, Ji and Zang resolved the conjecture and, in large part, the challenge: they proved unimodality, and they introduced doubly marked partitions (a partition with two distinguished columns of equal length, the first anchored within its Durfee square, the largest square fitting in the partition's Ferrers diagram - the left-justified array of dots that encodes a partition), showed that $N_S(m,n)$ equals the number of doubly marked partitions of $n$ with spt-crank $m$, and constructed a bijection between marked and doubly marked partitions; an spt-crank defined directly on marked partitions themselves remains unknown.

What survives untouched is the heart of Dyson's hope: the congruences of $\mathrm{spt}$ beyond moduli 5 and 7 - the mod-13 congruence, the prime-power congruences, and Ono's congruences at primes $\ell\ge 5$ - still have no crank-type combinatorial interpretation. The pattern is known to extend to other statistics: spt-cranks explain the congruences of the spt-functions of overpartitions (partitions in which the first occurrence of each part size may be distinguished) and of many 'exotic' spt-functions generated by Bailey pairs (pairs of q-series linked by an identity of Bailey that generates further q-series identities). This makes the missing mod-13 case for $\mathrm{spt}$ itself the conspicuous gap in the program.

Problem Statement

Give a combinatorial interpretation, in the equidistribution sense that Andrews-Garvan-Liang established for the mod-5 and mod-7 congruences, of the Ramanujan-type congruences of the smallest-parts function that remain without one. The sharply defined minimal core is the mod-13 congruence of Andrews:

$$\mathrm{spt}(13n+6)\equiv 0 \pmod{13}.$$

Construct an explicitly defined statistic of crank type on a natural class of partition-like objects (in the spirit of S-partitions, marked partitions, or doubly marked partitions - the class is to be given by the combinatorial structure of its objects and their size, and the statistic is to be a fixed function of the object alone, defined independently of $n$ and not chosen to fit the counts), and prove an equidistribution theorem of the form

$$N^*(k,13,13n+6)=\mathrm{spt}(13n+6)/13,\qquad 0\le k\le 12,\ n\ge 0,$$

where $N^*(k,13,n)$ counts the objects of size $n$ (with the sign weight, if the objects are vector-partition-like) whose statistic is congruent to $k$ modulo 13 - the exact analogue of the theorem $N_S(k,5,5n+4)=\mathrm{spt}(5n+4)/5$; an equivalent directly combinatorial equidistribution statement (for example a bijective splitting into thirteen equinumerous classes) is equally acceptable. The same scientific objective, as one program realizing Dyson's stated hope, further comprises the following related components: (i) extending the interpretation to Ono's congruences modulo primes $\ell\ge 5$ and to Garvan's congruences modulo powers of 5, 7 and 13; (ii) establishing the accompanying structural properties of the new statistic, in parallel with what is known for $N_S(m,n)$ - nonnegativity and unimodality of its distribution in $m$; and (iii) the directly related structural question left open by Chen-Ji-Zang, namely an spt-crank defined directly on marked (ordinary) partitions rather than pulled back through the bijection with doubly marked partitions, together with whatever further structure of $N_S(m,n)$ follows from Dyson's grouped formula. The problem's central question is the mod-13 equidistribution theorem with full proof; components (i)-(iii) are further, related questions of the same program.

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