Open research questions in Advanced Combinatorial Mathematics

What the literature leaves open

• 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 involution 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 produced by involution identity and relation to integer partitions · 2026 · DOI

• The lack of a unified analysis of suborbital graphs arising from wreath product actions of classical permutation groups. The need for a complete combinatorial description of the suborbit structure.

  Rank, Subdegrees, and Primitivity in Product-Action Wreath Products Acting on the Cartesian Power of a Base Set · 2026 · DOI

• The need to derive generating functions that capture the distribution of these statistics. The challenge of solving the functional equations to obtain the desired generating functions.

  Statistics in Weak Ascent Sequences · 2026 · DOI

• The lack of a method to count various statistics in weak ascent sequences. The need for a generating function approach to study weak ascent sequences. The gap in the prior work by Bényi et al. which only enumerated weak ascent sequences according to the number of weak ascents.

  Statistics in Weak Ascent Sequences · 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

• One possible direction for further research is: Problem 6.1. Explore the q, t-symmetry of rC⃗kpq, tq and rCKpq, tq using techniques from polyhedral 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.

  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

• 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 previous algebraic technique had limitations, particularly in obtaining a convergence rate. The emptiness problem for two-dimensional SFTs is closely related to the halting problem.

  Explicit entropy bounds for symmetric nearest-neighbor subshifts · 2026

• As follow-up of this work, several open problems appear interesting to us.

  Max-tree for d-permutations and pattern avoidance · 2026

• Further investigation could be carried out to determine whether or not Aunu polynomials have a simple combinatorial interpretation in terms of some other permutation statistic.

  Aunu Polynomials: Excedance Number in Aunu Involutions · 2026 · DOI

• The combinatorial half of Godsil's moment theorem was not previously formalized. The paper fills this gap by establishing a bijection between tree-like walks and Godsil's path tree.

  Walks that Forget the Cycles: A Machine-Checked Bijection between Tree-Like Walks and Godsil's Path Tree in Lean 4 · 2026 · DOI

• The study is limited to finite abelian groups. The computation of S(G,k) is limited to k ≤ 3. The study relies on SAT computations, which may have limitations in terms of scalability.

  Sum-Free Partitions in Finite Abelian Groups and Multiplicative Schur Numbers · 2026 · DOI

• The paper suggests future research on the use of machine-checked proofs in mathematics. The paper suggests future research on the application of Godsil's moment theorem to other areas of mathematics.

  When Walks Become a Spectrum: A Machine-Checked Proof of Godsil's Moment Theorem in Lean 4 (Part IV) · 2026 · DOI

• The paper identifies the need for a geometric-series cancellation and reflection bookkeeping approach to prove the theorem.

  When Walks Become a Spectrum: A Machine-Checked Proof of Godsil's Moment Theorem in Lean 4 (Part IV) · 2026 · DOI

• The evaluation of infinite sums involving binomial coefficients with non-integer scaling or complex shifts is challenging. Prior works do not provide a unified analytic framework for evaluating these sums.

  Generalizing Pascal’s row sums to complex orders: A Poisson summation approach · 2026 · DOI

• While the statistics and exchanging bijection we obtain for k = 3 can be written in closed forms, we expect many natural solutions for larger k to be inherently algorithmic (e.g, the bounce statistic, defined in Appendix A).

  Mapping Uncharted Symmetries: Machine Discovery in Combinatorics · 2026

• The paper aims to clarify this connection and provide a conservative bridge between rational partition states, semicircle geometry, beta-gamma normalization, and Catalan combinatorics.

  Two Skeletons of the Thales Semicircle: Pythagorean Rational States and Catalan Moment Structure · 2026 · DOI

• To explore the connections to communication complexity and circuit lower bounds.

  De Morgan Formula Size Catalogs for Small Boolean Functions · 2026 · DOI

• Further study of the applications of wreath product actions in algebraic graph theory and combinatorial design. Investigation of the connections to character theory and other areas of mathematics.

  Rank, Subdegrees, and Primitivity in Product-Action Wreath Products Acting on the Cartesian Power of a Base Set · 2026 · DOI

• To apply the approach to other areas of symbolic dynamics. To extend the approach to more general subshifts. To improve the estimates of the convergence rate.

  Explicit entropy bounds for symmetric nearest-neighbor subshifts · 2026

• The paper identifies a gap in the understanding of biharmonic functions and their zero sets. The paper identifies a gap in the understanding of the Boggio-Hadamard conjecture.

  Nested nodal loops of biharmonic functions · 2026

• Other future directions include proving or disproving the branch-switch congruence in general, explaining why all observed shared obstruction primes satisfy p ≡ 1 (mod 4), classifying the admissible shared-prime set, explaining the dominance of prime 13, extending the computation beyond 1.01 × 109, comparing results across different branch-switch orientations, and developing a stronger theoretical connection…

  A Branch-Switch Obstruction Phenomenon in Primitive Euler Bricks: Computational Verification Through 1.01 Billion (With test code on manuscript it self) Primes · 2026 · DOI

• To study sum-free sets in other contexts. To use SAT solvers to solve other combinatorial problems. To study the properties of the introduced OEIS sequences.

  Sum-Free Partitions in Finite Abelian Groups and Multiplicative Schur Numbers · 2026 · DOI

Most-cited papers in Advanced Combinatorial Mathematics

• Algorithm 448: number of multiply-restricted partitions · Communications of the ACM · 1973 · 717 citations
• Permutation Methods: A Basis for Exact Inference · Statistical Science · 2004 · 677 citations
• The Number of Partitions of a Set · American Mathematical Monthly · 1964 · 656 citations
• On Sets of Acquaintances and Strangers at any Party · American Mathematical Monthly · 1959 · 385 citations
• Algorithm 235: Random permutation · Communications of the ACM · 1964 · 347 citations
• Log‐Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometrya · Annals of the New York Academy of Sciences · 1989 · 328 citations
• Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators · Operations Research · 1999 · 300 citations
• Selected Selection Theorems · American Mathematical Monthly · 1956 · 275 citations
• A Combinatorial Property of q-Eulerian Numbers · American Mathematical Monthly · 1975 · 135 citations
• The Problem of Apollonius · American Mathematical Monthly · 1968 · 100 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
• A survey on bent partitions · Cryptography and Communications · 2026
• Alguns problemas mais elaborados em análise combinatória elementar, I: Enumeração de funções · REMAT Revista Eletrônica da Matemática · 2026
• Tight cylindric partitions · Proceedings of the American Mathematical Society · 2026
• Valuative invariants of Catalan matroids · Proceedings of the American Mathematical Society · 2026
• Homology of matching complexes and representations of symmetric groups · Proceedings of the American Mathematical Society · 2026
• Laurent polynomials and deformations of non-isolated Gorenstein toric singularities · Advances in Mathematics · 2026
• Bounded Littlewood Identities with Fixed Number of Odd Rows or Odd Columns · The Electronic Journal of Combinatorics · 2026
• A Local Framework for Proving Combinatorial Matrix Inversion Theorems · The Electronic Journal of Combinatorics · 2026

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