The generalized colored overpartition generating-function identity

From papers

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.

Progress summary

Open

No public discussion or published progress on this proposed identity was found.

No public discussion or published progress concerning this identity was found in the retrieved sources.

Current status (as of August 2026): The identity appears open, with no recorded public activity establishing, disproving, or substantially advancing it.

Sources & referencesView supporting material

Primary source

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

1 week ago

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.

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Shivam Patel · 19 hours ago