Two-colored generalized Frobenius partitions and minimal-excludant sums over bipartitions

Abstract: Let $\cpsi_{2,a}(n)$ denote the number of $(2,a)$-colored Frobenius partitions of weight $n$, where the two rows have prescribed length difference. We study the two cases $a=0$ and $a=1$ and connect them with minimal-excludant statistics on bipartitions. Let $σ\mex_2(n)$ be the sum of the Lin--Liu bipartition minimal excludants over all bipartitions of $n$, and let $E_2(n)$ be the number of bipartitions whose two component minimal excludants are equal. For all $n\geq 0$, we give a combinatorial proof of [ \cpsi_{2,0}(n)=2σ\mex_2(n) \qquad\text{and}\qquad \cpsi_{2,1}(n)=2σ\mex_2(n)-E_2(n). ] These identities give direct combinatorial interpretations of two-colored Frobenius partition functions in terms of bipartition minimal-excludant sums.

• The paper presents explicit combinatorial identities linking (2,a)-colored Frobenius partition counts to minimal-excludant sums over bipartitions.
• It employs Durfee rectangle decompositions and bijections to map colored partitions to bipartitions with precise mex adjustments.
• The findings optimize partition enumeration techniques and open avenues for studies on congruences and multipartition generalizations.

Analysis of Two-colored Generalized Frobenius Partitions and Minimal-Excludant Sums Over Bipartitions

Introduction

The paper "Two-colored generalized Frobenius partitions and minimal-excludant sums over bipartitions" (2606.19696) establishes combinatorial identities linking $(2,a)$-colored generalized Frobenius partition counts with minimal-excludant statistics defined over bipartitions. The focus is on strict combinatorial proofs connecting the partition theory constructs originating from Andrews, Lin-Liu, and Jiang-Rolen-Woodbury. The authors present exact formulas for the enumeration of colored Frobenius partitions based on bipartition minimal excludant sums and detail conditions under which correction terms emerge.

Definitions and Framework

Colored generalized Frobenius partitions extend the classical Frobenius symbol to incorporate elements from $k$ colored copies of nonnegative integers. The $(2,a)$-colored partitions considered require two rows whose length difference is prescribed by $a$. This allows for combinatorial manipulations not available in the classical framework.

Minimal excludant (mex) statistics, first studied by Andrews and Newman, quantify the least integer not present as a part in a partition. Their bipartition analogue, as defined by Lin-Liu, computes $$\text{mex}_2(\mathbf{T}) = \min(\text{mex}(T_1), \text{mex}(T_2))$$ for the bipartition $\mathbf{T} = (T_1, T_2)$. The associated sums $\omega \text{mex}_2(n)$ aggregate these minimal excludants over all bipartitions of $n$, and $E_2(n)$ counts bipartitions where both components share the same minimal excludant.

Main Identities and Proof Structure

The principal result is the explicit combinatorial equivalence:

• $\text{cy}_{2,0}(n) = 2\omega \text{mex}_2(n)$
• $k$0

Here, $k$1 and $k$2 denote counts of $k$3-colored Frobenius partitions for $k$4 and $k$5, respectively.

The proof strategy proceeds via:

1. Reduction of colored Frobenius symbol arrays to pairs of ordinary partitions using Durfee rectangle decompositions.
2. Explicit computation of weight adjustments corresponding to row length differences.
3. Construction of bijections between colored Frobenius partition arrays and bipartitions, parametrized by indices $k$6 satisfying row length constraints.
4. Aggregation over all possible $k$7 yields sums expressible in terms of bipartition counts, which are then matched with the minimal excludant sums.
5. Correction terms arising in the $k$8 case directly relate to the equal-mex bipartition counts, producing the subtraction of $k$9 in the identity.

Numerical and Structural Claims

The paper delivers precise enumeration formulas, notably:

• Establishing a factor of $(2,a)$0 in the map from $(2,a)$1-colored Frobenius partitions to bipartition minimal excludant sums.
• Demonstrating that the count for $(2,a)$2-colored partitions must subtract the count of bipartitions with equal component minimal excludants.
• Confirming that all results hold for $(2,a)$3 without restriction.

These claims are supported via bijective arguments and explicit calculation of multiplicities involving staircase partitions.

Implications and Outlook

The explicit combinatorial translation between colored Frobenius partition counts and bipartition minimal excludant statistics sharpens the understanding of the algebraic and combinatorial interplay in partition theory. This correspondence deepens the link between partition enumeration techniques (Durfee decompositions, colored augmentations) and the study of gap statistics (mex), potentially enabling new approaches to congruence and generating function analysis.

Practically, these identities enable efficient computation of colored partition functions via minimal excludant aggregation, reducing complexity in explicit enumeration. Theoretically, the results open avenues for further generalization, including extension to higher $(2,a)$4-color cases, multipartite partitions, and links with modular forms and Jacobi representations.

Future developments may focus on:

• Generalizing the bipartition minimal excludant statistics to multipartition scenarios.
• Investigating congruences and modularity phenomena for colored partition functions parameterized by minimal excludant data.
• Applying these combinatorial interpretations to analytic partition theory, particularly in contexts related to automorphic and Jacobi forms.

Conclusion

The paper rigorously establishes combinatorial identities linking $(2,a)$5-colored generalized Frobenius partition counts to minimal-excludant statistics over bipartitions, with explicit correction terms determined by equal-mex bipartitions. The results provide exact enumerative formulas, enhance structural understanding of partition combinatorics, and lay groundwork for further explorations in partition theory and associated algebraic structures.