Thejitha–Fathima overcolored partition congruence conjecture

From papers

Let an overcolored partition of $n$ be a partition in which even parts may appear in one of $r$ colors and odd parts may appear in one of $s$ colors, with the first occurrence of each part optionally overlined. Let $\bar{a}_{r,s}(n)$ denote the number of such partitions, where $k\geq 1$ and $i,j\geq 0$ . Thejitha–Fathima congruence conjecture. For all $n\geq 0$ ,

\bar{a}_{2^{k+1}j+2^k-1,2i+1}(3n+2)\equiv 0 \pmod{2^{k+1}},

\bar{a}_{2^{k+1}j+2^k-1,2i+1}(9n+3)\equiv 0 \pmod{2^{k+2}},

\bar{a}_{2^{k+1}j+2^k-1,2i+1}(9n+6)\equiv 0 \pmod{2^{k+2}}.

These congruences are Ramanujan-type divisibility results for overcolored partitions and were proposed by Thejitha and Fathima as families of congruences modulo powers of $2$ . The present paper states that it provides an elementary proof using classical $q$ -series manipulations and properties of Ramanujan's theta function; the supplied text does not specify whether that proof establishes all three displayed congruences, so the database status remains open.

Progress summary

Solved

A March 2026 preprint claims to prove the conjecture, but its displayed argument appears to cover only a narrower family of parameters, so the original claim is not verified.

Thejitha and Fathima proposed three divisibility families for overcolored partition numbers, with odd-color parameter $s=2i+1$ . Hussain, Ghoshal, and Jana announced an elementary proof in March 2026.

Known results

The earlier paper defines the generating function $\sum_{n\geq0}\bar{a}_{r,s}(n)q^n=f_2^{3s-2r}/(f_1^{2s}f_4^{s-r})$ and records the first displayed family as Conjecture 6.1.

March 2026 claimed proof

Hussain, Ghoshal, and Jana state that all three congruences hold and present $q$ -series and theta-function arguments. However, the displayed proof substitutes $s=2^k i+1$ , whereas the conjecture permits every $s=2i+1$ ; no correction or independent verification of this parameter mismatch was found.

Current status (as of August 2026): The preprint’s proof claim is public but unverified; the narrower case $s=2^k i+1$ is addressed, while the full family with $s=2i+1$ remains unresolved.

Sources

Sources & referencesView supporting material

Primary source

Imdadul Hussain, Suparno Ghoshal and Arijit Jana, “Proof of a Conjecture on Overcolored Partition Restricted by Parity of the Parts”, arXiv:2603.12401 (2026).

1 week ago

The first claimed congruence fails for infinitely many admissible parameter choices, despite the paper's claim that all three congruences were proved.

Write

f_d(q)=\prod_{m\ge1}(1-q^{dm}).

The generating function is

\sum_{n\ge0}\overline a_{r,s}(n)q^n = \frac{f_2(q)^{3s-2r}} {f_1(q)^{2s}f_4(q)^{s-r}}.

Modulo $q^3$ ,

\begin{aligned} f_1(q)^{-2s} &=(1-q)^{-2s}(1-q^2)^{-2s}+O(q^3)\\ &=1+2sq+(2s^2+3s)q^2+O(q^3),\\ f_2(q)^{3s-2r} &=1+(2r-3s)q^2+O(q^4),\\ f_4(q)^{r-s}&=1+O(q^4). \end{aligned}

Therefore, for every $r,s\ge1$ ,

\boxed{\overline a_{r,s}(2)=2(r+s^2).}

Now take any $k\ge4$ and $j\ge0$ , and choose $i=1$ and $n=0$ in the conjecture. Its parameters become

r=2^{k+1}j+2^k-1, \qquad s=2i+1=3.

Thus

\overline a_{r,3}(3n+2) = \overline a_{r,3}(2) = 2^{k+2}j+2^{k+1}+16 \equiv16\not\equiv0\pmod{2^{k+1}}.

The smallest instance is

(k,j,i,n)=(4,0,1,0), \qquad (r,s)=(15,3),

for which

\overline a_{15,3}(2)=48\equiv16\not\equiv0\pmod{32}.

More generally, the initial instance of the first asserted congruence holds exactly when

2^k\mid s^2-1.

The discrepancy in the claimed proof is that the conjecture and theorem allow every odd $s=2i+1$ , whereas their proof immediately replaces this by the narrower assumption $s=2^ki+1$ . The narrower original conjecture is unaffected, but the stated unrestricted theorem and congruence are false.

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Shivam Patel · 23 hours ago