Open research questions in Advanced Combinatorial Mathematics

What the literature leaves open

• The lack of a super-analogue of the RSK correspondence for super tableaux over a signed alphabet. The need for a geometrical interpretation of the super-RSK correspondence.

  A super Robinson–Schensted–Knuth correspondence with symmetry and the super Littlewood–Richardson rule · 2026 · DOI

• The authors face the challenge of understanding the underlying mechanisms of the coherence phenomenon. The authors must overcome the limitation of their observations being restricted to a certain range. The authors need to address the issue of error in decimal estimates of Fourier coefficients.

  Spectrum sequences associated to sequences of rational numbers, draft 17jul26 v2 · 2026 · DOI

• The study of cusp forms and elliptic curves is limited by the lack of understanding of the coherence phenomenon. The calculation of the minimum moduli among the eigenvalues of Jh,n is challenging. The use of the Anthropic AI model Claude requires further development and testing.

  Spectrum sequences associated to sequences of rational numbers, draft 17jul26 · 2026 · DOI

• The paper faces the challenge of analyzing the complex properties of random matroids. The paper must overcome the difficulty of understanding the phase transitions of matroid properties. The paper needs to address the gap in current knowledge about random matroids.

  Evolution of random representable matroids: minors, circuits, connectivity and the critical number · 2026 · DOI

• Future research can build on the results of this paper to further understand the evolution of random matroids. Future research can explore the applications of the paper's findings to various fields. Future research can investigate other matroid properties and their phase transitions.

  Evolution of random representable matroids: minors, circuits, connectivity and the critical number · 2026 · DOI

• The paper suggests extending the results to non-vexillary permutations. The paper suggests further study of double β-Edelman-Greene coefficients and their properties.

  Tableau formula for vexillary double Edelman–Greene coefficients · 2026 · DOI

• The paper identifies a gap in understanding double β-Edelman-Greene coefficients for non-vexillary permutations. The paper identifies a need for a general formula for double β-Edelman-Greene coefficients.

  Tableau formula for vexillary double Edelman–Greene coefficients · 2026 · DOI

• To prove or disprove Conjecture 2. To evaluate the asymptotic behavior of the proposed formula. To explore the connections between the frozen-square enumeration and other models in combinatorics and statistical mechanics.

  Frozen-Corner Enumeration of Alternating Sign Matrices · 2026 · DOI

• A general formula for the frozen-square enumeration is still needed. The evaluation of the asymptotic behavior of the proposed formula is still out of reach.

  Frozen-Corner Enumeration of Alternating Sign Matrices · 2026 · DOI

• The need for efficient algorithms to find the k-bonacci vector Zeckendorf representation of a vector. The lack of understanding of the properties of k-bonacci vector Zeckendorf representations.

  Properties of Multidimensional Vector Zeckendorf Representations · 2026 · DOI

• The construction is intentionally narrow. The causal language is intended to remain compatible with the relativistic framework established by Einstein.

  Structural Origin of the Fixed Ratios R and S · 2026 · DOI

• The understanding of sparse paving positroids is incomplete. The classification of sparse paving positroids is not well-established.

  Sparse paving positroids · 2026 · DOI

• The earlier expressions for R and S were used as primitive definitions. The present note provides a common mathematical reference for those values.

  Structural Origin of the Fixed Ratios R and S · 2026 · DOI

• This positively answers [Open question 2 of Dutta,Dwivedi,Saxena FOCS’21] and [Problem 8.

  Separated borders: Exponential-gap fanin-hierarchy theorem for approximative depth-3 circuits · 2026 · DOI

• Conde, Gimbert, González, Miller and Miret (2014) established that the cyclotomic conjecture, if true, would imply the non-existence of almost Moore digraphs - a well-known open question concerning the directed degree-diameter problem.

  A proof of the cyclotomic conjecture and the non-existence of almost Moore digraphs · 2026

• For further study of permutation problems with restricted positions the reader is directed to [4], and in particular the relationship of D(n, k,O) to kth order differences of n!jk.

  Matchings, Derangements, Rencontres · 1983 · DOI

• Finding a simple proof for the equality an = |s−1(132, 231)| remains an open problem, as previous proofs used the kernel method and may be more complicated than the proofs for the other two cases.

  Stack-sorting preimages and 0-1-trees · 2026 · DOI

• In [HW25] the first and the last named authors have introduced and studied a family of meromorphic differentials ω(0) n (z1, ..., zn) defined via (1) and an invol- ution identity (2), where ιz = az+b cz−a for a2 + bc 6= 0 is a holomorphic involution of P1. In this paper we settle the so-far open question about the symmetry of the ω(0) n (z1, ..., zn) by its reduction to a combinatorial Theorem 1 about integer partitions. SYMMETRY OF MEROMORPHIC DIFFERENTIALS, AND INTEGER PARTITIONS 29 As also shown in [HW25], the quartic analogue of the Kontsevich matrix model is an example where the ω(0) In [HW23] the first and the last named authors succeeded in proving recursion formulae also for ω(1) n (z1, ..., zn) of genus g = 1 by another strategy (extended loop equations). The proof that also the ω(1) n (z1, ..., zn) are symmetric is left for the future. n (z1, ..., zn) are of this form.

  Symmetry of meromorphic differentials produced by involution identity and relation to integer partitions · 2026 · DOI

• To extend the results to more general cases. To improve the efficiency of word reconstruction algorithms. To apply the results to other areas of language theory.

  A word reconstruction problem for polynomial regular languages · 2026 · DOI

• The lack of a characterization of necessary queries for unique reconstruction. The complexity of the word reconstruction problem for polynomial regular languages.

  A word reconstruction problem for polynomial regular languages · 2026 · DOI

• Future research can focus on determining the monophonic pebbling numbers of other graphs. The study of monophonic pebbling numbers can be extended to other areas of graph theory.

  On monophonic pebbling number · 2026 · DOI

• The gap is the lack of knowledge on monophonic pebbling numbers. The study addresses the gap by determining the monophonic pebbling numbers of several graphs.

  On monophonic pebbling number · 2026 · DOI

• In this article, we establish the q, t-symmetry of rCKpq, tq, which is the q, t-polynomial graded by the pair of statistics (area,depth) on K-Dyck paths. Our proof relies on construct- ing an involution on K-Dyck paths, which swaps the area and depth of a path. However, this involution cannot be used to prove a similar result for CKpq, tq, as it is not generally q, t- symmetric. Additionally, we analyze the q, t-symmetry in the refined case for certain singular ⃗k. The dinv of a ⃗k-Dyck path π is not defined using the area sequence of π, and thus, we do not identify a direct generalization of ddinv for ⃗k-Dyck paths via the depth labeling se- quence. However, using the depthpπq and dinvpωpπqq statistics, can also provide an alternative description of the higher q, t-Catalan polynomials. Similar to the development of the q, t-symmetry of C⃗kpq, tq, in [BHH+24], some results from [XZ25] are reproven using a different perspective. Therefore, one possible direction for further research is: Problem 6.1. Explore the q, t-symmetry of rC⃗kpq, tq and rCKpq, tq using techniques from poly- hedral geometry. Despite the pair of statistics discussed above, one might wonder if there exist other pairs of statistics exhibit q, t-symmetry on ⃗k-Dyck paths or K-Dyck paths. The answer appears to be obvious. We present two types of new q, t-polynomials here. In [LL23], the authors demonstrated that the pair (run, ret) constitutes a q, t-symmetric pair of statistics on classical Dyck paths of composition type α, meaning that for each π P Dn, the lengths of successive North-step runs are determined by the composition α ( n in left-to-right order. In the context of ⃗k-Dyck paths, run represents the sum of the lengths of all S˚ segments occurring before the first W W in π, while ret counts the number of times the path, excluding p0, 0q, intersects the horizontal axis. The distinction between Dn of composition type ⃗k and ⃗k-Dyck paths lies in the fact that, for ⃗k-Dyck paths, successive North-step runs may occur immediately above one another. Consequently, the q, t-polynomial associated with ⃗k-Dyck paths, when graded by the pair (run, ret), can be interpreted as a summation over appropriate families of classical Dyck paths of composition type. The q, t-symmetry of these polynomials follows directly from the result in [LL23]. SYMMETRY OF THE REFINED q, t-CATALAN POLYNOMIALS FOR ⃗k-DYCK PATHS 23 To summarize, for any pair of statistics pstat1, stat2q, if it is q, t-symmetric on classical Dyck paths of composition type α, then it is also q, t-symmetric on ⃗k-Dyck paths and K-Dyck paths. Therefore, our next direction is as follows. Problem 6.2. Find additional pairs of q, t-symmetric statistics pstat1, stat2q on classical Dyck paths of composition type α. Finally, the results in Proposition 5.5 and 5.6 suggest that the q, t-polynomial pC⃗kpq, tq, graded by the pair of statistics (bounce, depth) on ⃗k-Dyck paths, may also exhibit q, t- symmetry for some special cases. For instance, pC⃗kpq, tq is q, t-symmetric for ℓp⃗kq ď 2 or ⃗k “ pa, 1, cq with positive a and c, as in these cases, bouncepπq “ depthpπq for every π. Based on computational data, we make the following observation. Problem 6.3. Prove the following observation: If ⃗k “ pa, 2, cq, pa, 1, 1, dq, pa, 2, 1, dq, or pa, 1, 1, 1, eq, where c, d, and e are positive, then pC⃗kpq, tq is q, t-symmetric. Moreover, pC⃗kpq, tq does not exhibit q, t-symmetric if ℓp⃗kq ě 6. This problem may be approached using techniques from MacMahon’s partition analysis, specifically by applying explicit formulas for bounce and depth. We leave it as an exercise for interested readers. Acknowledgements: The authors would like to thank the anonymous referee for valuable suggestions for improving the presentation. Y. Zhang was supported by the Yunnan Provincial Department of Education Scientific Research Fund Project Grant: 2026J0603.

  Symmetry of the refined q,t-Catalan polynomials for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mover accent="true"> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">→</mml:mo> </mml:mrow> </mml:mover> </mml:math> -Dyck paths · 2026 · DOI

• Future research could explore the application of Lorentzian structures to other areas of combinatorics. The development of new algorithms and methods for constructing and analyzing Lorentzian-weighted incidence matrices could be a fruitful area of research.

  Lorentzian Structures on Matroidal Posets and the Spectral Geometry of Combinatorial Designs · 2026 · DOI

• The traditional study of combinatorial designs has leveraged algebraic methods and graph-theoretic techniques, but the connection to Lorentzian structures is new. The paper identifies a gap in the understanding of the interplay between Lorentzian structures, matroidal posets, and the spectral properties of combinatorial designs.

  Lorentzian Structures on Matroidal Posets and the Spectral Geometry of Combinatorial Designs · 2026 · DOI

Most-cited papers in Advanced Combinatorial Mathematics

• Chains in Power Sets · Mathematics Magazine · 1991 · 8 citations
• Bidimensional Extensions of Cobalancing and Lucas-Cobalancing Numbers · Annales Mathematicae Silesianae · 2023 · 5 citations
• Some remarks on permutation type tests in linear models · Discussiones Mathematicae Probability and Statistics · 2023 · 5 citations
• A task to connect counting processes to lists of outcomes in combinatorics · The Journal of Mathematical Behavior · 2021 · 5 citations
• On the set of many-to-one strongly stable fractional matchings · Mathematical Social Sciences · 2021 · 4 citations
• New Uniform Subregular Parallelisms of PG(3, 4) Invariant under an Automorphism of Order 2 · Cybernetics and Information Technologies · 2020 · 4 citations
• When Walks Become a Spectrum: A Machine-Checked Proof of Godsil's Moment Theorem in Lean 4 (Part IV) · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 3 citations
• Special Issue on Partition – IR 21(3) – Guest Editor Introduction · India Review · 2022 · 3 citations
• Study of a cyclic system of difference equations with maximum · Electronic journal of qualitative theory of differential equations · 2020 · 3 citations
• Walks that Forget the Cycles: A Machine-Checked Bijection between Tree-Like Walks and Godsil's Path Tree in Lean 4 · Zenodo (CERN European Organization for Nuclear Research) · 2026 · 2 citations

Most recent work

• When Walks Become a Spectrum: A Machine-Checked Proof of Godsil's Moment Theorem in Lean 4 (Part IV) · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Walks that Forget the Cycles: A Machine-Checked Bijection between Tree-Like Walks and Godsil's Path Tree in Lean 4 · Zenodo (CERN European Organization for Nuclear Research) · 2026
• Stack-sorting preimages and 0-1-trees · Discrete Mathematics & Theoretical Computer Science · 2026
• Symmetry of meromorphic differentials produced by involution identity and relation to integer partitions · Annales de l’Institut Henri Poincaré D, Combinatorics, Physics and their Interactions · 2026
• Rank, Subdegrees, and Primitivity in Product-Action Wreath Products Acting on the Cartesian Power of a Base Set · International Journal of Development Mathematics (IJDM) · 2026
• Fast Construction of Self-Avoiding Polygons and Efficient Evaluation of Closed Walk Fractions on the Square Lattice · Experimental Mathematics · 2026
• Statistics in Weak Ascent Sequences · Mathematics · 2026
• Generalizing Pascal’s row sums to complex orders: A Poisson summation approach · Notes on Number Theory and Discrete Mathematics · 2026
• A word reconstruction problem for polynomial regular languages · Discrete Applied Mathematics · 2026
• Covering Radii of 3-Zonotopes and the Shifted Lonely Runner Conjecture · Experimental Mathematics · 2026

Related topics in Mathematics