The generalized colored overpartition generating-function identity

Let $x_1,\dots,x_k,m,n\in\mathbb{N}$ and $j\in\{1,\dots,k\}$ . Define $\overline{S_j}(x_1,\dots,x_k;m,n)$ as the colored overpartition count from Definition~, with the fourth condition replaced by requiring that the $s$ smallest parts are overlined, where

s \coloneqq \sum_{\substack{r=0\r \neq j-1}}^{k-1} V_{2^r},

and $V_{2^r}$ is the number of parts $\lambda_i$ such that $v(c_i)=2^r$ . Generalized colored overpartition identity. In this setting,

\sum_{\substack{x_1, \dots, x_k, m, n \geq 0}} \overline{S_j}(x_1,\dots,x_k;m,n) y_1^{x_1} \cdots y_k^{x_k} d^m q^n = \frac{(-y_1 q;q)_\infty \cdots (-y_k q;q)_\infty}{(y_j d q;q)_\infty}.

This identity is proposed as a further generalization of the generating-function interpretation established in the paper; its status is not resolved in the supplied text.

References

Primary source

Laure Velenik, “An iterative-bijective approach to asymmetric generalizations of Schur's theorem”, arXiv:2510.00846 (2025).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A posted calculation claims the proposed formula fails in every case beyond the already known first case, but the calculation has not been independently checked.

Velenik’s paper proves the $j=1$ specialization and presents the broader formula as Conjecture 3.2. The conjecture is therefore a proposed extension of an established identity, not a theorem in the source.

Known results

$j=1$ , any $k$ : established by Velenik, 2025, as Theorem 1.22.

Posted attempt

A complete disproof is claimed for every $k\ge2$ and $2\le j\le k$ , using the coefficient $[y_1y_jdq^2]$ : the proposed product gives $1$ , while the defined overpartition count is claimed to give $0$ . The first case is $k=j=2$ , and the attempt concludes that the identity holds if and only if $j=1$ . This reader-written attempt has not been independently verified.

Current status (as of August 2026): The case $j=1$ is established, while the claimed disproof for all $j\ge2$ is unverified; absent confirmation, the remaining cases are unresolved.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

Conjecture 3.2 is false for every previously unproved case. More precisely, it holds exactly when $j=1$ , the case already established by Theorem 1.22 of the source.

Fix any

k\ge2,\qquad 2\le j\le k,

and consider the coefficient

[y_1y_jd q^2].

On the proposed product side,

\frac{\prod_{r=1}^k(-y_rq;q)_\infty} {(y_jdq;q)_\infty},

there is exactly one contributing choice:

(y_1q)(y_jdq)=y_1y_jdq^2.

Hence the proposed right-hand coefficient is $1$ .

An object contributing to the left-hand side would have total size $2$ , exactly one nonoverlined part, and primary-color multiplicities

x_1=x_j=1,\qquad x_r=0\quad(r\ne1,j).

There are only two possible underlying partitions.

One part. The sole part must be the nonoverlined composite-colored part

2_{\,1+2^{j-1}}.

Its least primary color is

v(1+2^{j-1})=1\ne2^{j-1}.

Consequently Definition 3.1 gives $s=1$ , requiring its unique smallest part to be overlined. This contradicts the requirement of one nonoverlined part.

Two parts. Both parts have size $1$ , with primary colors $1$ and $2^{j-1}$ . Again $s=1$ , so the lower part must be overlined. The source's difference condition then requires

0=\lambda_1-\lambda_2 \ge \omega(c_1)+\delta(c_1,c_2) -\mathbf1_{\{\lambda_2\text{ nonoverlined}\}} = 1+\delta(c_1,c_2) \ge1,

an impossibility.

Thus the actual left-hand coefficient is $0$ , whereas the conjectured right-hand coefficient is $1$ :

\boxed{[y_1y_jdq^2]\,\mathrm{LHS}=0 \ne 1=[y_1y_jdq^2]\,\mathrm{RHS}.}

This disproves the proposed identity for every $k\ge2$ and every $j\ge2$ .

The first counterexample is $k=j=2$ . Its origin is visible by comparison with the source's valid Theorem 1.20(b): that theorem counts parts of pure color $1$ in the overlining restriction, whereas the conjecture counts every part whose least primary color is $1$ , incorrectly including composite color $3$ . The valid $j=1$ case is exactly Theorem 1.22. Therefore the complete classification is

\boxed{\text{The proposed identity holds if and only if }j=1.}

Source: L. Velenik, arXiv:2510.00846, Definitions 1.11, 1.21 and 3.1; Theorems 1.20(b), 1.22; Conjecture 3.2.