Papers

Given a sequence $\mathbf{a}=(a_i)_{i\geq 1}$ of positive real numbers, we say that a colouring $c:E(K_n)\to \mathbb{N}$ is $\mathbf{a}$-bounded if the proportion of edges coloured $i$ is at most $a_i$, for all $i$. We write \[ \mathbf{a}\, \to\, K_k \] if, for all sufficiently large $n$, all $\mathbf{a}$-bounded colourings $c:E(K_n)\to \mathbb{N}$ contain a monochromatic $K_k$. We say that the sequence $\mathbf{a}$ is Decay-Ramsey if $\mathbf{a}\to K_k$ for all $k\ge 2$. In this short paper we introduce the question of determining which sequences are Decay-Ramsey and present some initial results.

We introduce invariants we call $k$-divisibility and $d$-dominance, which are combinatorial constraints on the minimal generators of monomial ideals. We then prove that $k$ (respectively, $d$) is an upper bound (respectively, lower bound) for the projective dimension of a $k$-divisible (respectively, $d$-dominant) monomial ideal. This leads us to an explicit minimal resolution of $k$-divisible, $d$-dominant ideals when $d=k-1$ via a Lyubeznik resolution. This framework allows us to prove a conjecture made by Alesandroni. Using these invariants, we also provide a characterization of monomial ideals whose minimal free resolution is attainable from the Taylor resolution by performing exactly one consecutive cancellation. We also prove that for all $q,k,d$ in the range $1\leq d<k \leq q-1$, there exist monomial ideals that are $k$-divisible and $d$-dominant with $q$ minimal generators.

For $H$ a hypergraph, the Hall ratio $\rho(H)=\text{max}_{X\subseteq V(H)}\frac{\lvert{X}\rvert}{\alpha(H[X])}$ is a natural lower bound for its chromatic number, and among non-empty $k$-uniform hypergraphs the smallest possible Hall ratio is $k/(k-1)$. For graphs ($k = 2$), having minimum Hall ratio $2$ forces chromatic number $2$. Nešetřil, Rödl, and Sales asked whether a similar phenomenon holds at higher uniformities. We answer this question negatively in a strong form, showing that for every $k \geqslant 3$ there are $k$-uniform hypergraphs with Hall ratio exactly $k/(k-1)$ and arbitrarily large fractional chromatic number.

The $k$-color Ramsey number of a poset $P$ is the smallest dimension $d$ such that any $k$-coloring of the $d$-dimensional Boolean lattice contains a monochromatic induced copy of $P$. For $ j\ge 2$ the $j$-diamond is the poset that consists of a maximum element, a minimum element and an antichain of size $j$. We prove that the $k$-color Ramsey number of the $j$-diamond is at most $jk$. Colorings of the Boolean lattice that assign sets of the same size the same color provide matching lower bounds for $j \in \{2,3,4\}$.

Edge deletion problems ask for the fewest edges whose deletion gives a graph with a specified property. We study two such problems on block graphs. First making the graph claw-free, and second making it a line graph. Since every line graph is claw-free, the second problem is at least as difficult as the first. We show that on block graphs the two optimum values are always equal, and both are attained by the line graph of a forest. The main step is a completion argument, in which starting from any claw-free solution, we put back edges to get some nice structure inside each block still maintaining the claw-free property. This reduces both edge-deletion problems to the same problem of selecting vertices within the blocks of the input graph. We solve it by dynamic programming in $O(n+m)$ time, which also produces an optimal deletion set and a forest whose line graph is what remains after the completion argument. We also extend the method to $K_{1,r+1}$-free edge deletion for every fixed $r$.

We establish topological recursion for the spin enumeration of hypermaps, or, in the dual language, spin dessins d'enfants. This problem has remarkable features distinguishing it from other applications of topological recursion to enumerative combinatorics. First, the spectral curve is elliptic rather than rational, as in most cases studied before. Second, it contains degenerate ramification points, and one has to apply a version of degenerate topological recursion. Finally, the meromorphic correlator differentials have "moving" poles on the antidiagonals between pairs of distinct arguments, and one has to adjust the loop equations to this kind of degeneracy as well. The computations involve concepts of different kinds, such as BKP integrability, Gaussian integrals, the x-y swap, and symplectic duality.

A loose hypertree is a hypergraph whose incidence graph is a tree. Let $b\ge3$, $n\equiv1\pmod{b-1}$, and $s:=(n-1)/(b-1)$. We determine the largest $t$-intersecting families of loose spanning $b$-uniform hypertrees on $[n]$, simultaneously for every $1\le t\le s$, provided $n$ exceeds an explicit constant depending only on $b$. The extremal families consist of all such hypertrees containing a fixed $t$-edge loose forest whose component sizes are as equal as the divisibility condition permits. In particular, when $bt\le n$, the extremal forests are matchings. We also prove a Hilton–Milner-type bound: if $n\ge C_b(t+1)$, where $C_b$ depends only on $b$, every nontrivial $t$-intersecting family has at most $(s+1)r_{t+1}(n)$ members. Here $r_j(n)$ is the maximum possible number of loose spanning $b$-uniform hypertrees containing a fixed $j$-edge loose forest. For fixed constants $b,t$, this bound is asymptotically sharp and is $O_{b,t}(n^{2-b})r_t(n)$. The proofs combine exact forest enumeration with the intersection theorems of Kupavskii for spread families.

Let $G$ be the terminal graph of the random triangle-removal process started from $K_n$. We prove that $e(G)/n^{3/2}\to1/(2\sqrt2)$ in $L^2$. More generally, we obtain the analogous result for any fixed strictly $k$-balanced $k$-graph. Our main tool is a more general theorem on random greedy matchings in nearly regular uniform hypergraphs with few short cycles under a stability assumption on the survival equations.

A perfect 1-factorization of the complete graph $K_{2t}$ is a partition of its edges into perfect matchings such that the union of any two of them is a Hamilton cycle. Kotzig conjectured in 1963 that $K_{2t}$ has a perfect 1-factorization for every $t \ge 2$, but apart from some infinite families and a list of sporadic orders the conjecture is open, and $92$ was one of the nine orders up to $100$ for which no perfect 1-factorization was known. In this paper we construct two non-isomorphic perfect 1-factorizations of $K_{92}$.

For a graph $G$, let $\mathrm{tw}(G)$, $\mathrm{tree}\text{-}\alpha(G)$, and $\mathrm{tree}\text{-}\chi(G)$ denote its treewidth, tree-independence number, and tree-chromatic number, respectively. We prove that every nonempty graph $G$ satisfies $\mathrm{tw}(G)+1\leq\mathrm{tree}\text{-}\alpha(G)\mathrm{tree}\text{-}\chi(G)(1+\lfloor\log_2(\mathrm{tree}\text{-}\alpha(G))\rfloor)$. This improves the quadratic bound of Koutsoutis, Krause, Liu, Redzic, and Ueckerdt. For every pair of integers $n,k\geq2$, we construct a graph $G$ with $\mathrm{tree}\text{-}\alpha(G)=n$, $\mathrm{tree}\text{-}\chi(G)=k$, and $\mathrm{tw}(G)=\Omega(nk\log_2 n)$. Thus the upper bound is best possible up to a constant factor.

The first major step in the theory of graph minors of Robertson and Seymour was the grid theorem, a fundamental duality result describing that any graph of sufficiently high treewidth must contain a large grid as a minor. Since this pioneering work, multiple grid theorems have been established in structural graph theory, linking various width parameters of combinatorial objects with the non-existence of grid-like obstructions. In particular, Kawarabayashi and Kreutzer [STOC'15] proved a variant of the grid theorem for directed graphs that links directed treewidth with the non-existence of topological minors of large cylindrical walls. On the other hand, to study immersions of undirected graphs $-$ another containment relation that focuses on edge-disjointness, rather than vertex-disjointness $-$ Wollan [JCTB'15] introduced a parameter, tree-cutwidth, and proved that it is dual to immersions of large walls. In this work, we present a version of Wollan's parameter for directed graphs, the directed tree-cutwidth. The main result is a grid theorem for directed tree-cutwidth, establishing that it is dual to directed immersions of large cylindrical walls.

Kemp and Ramgoolam proposed a basis of monomials for a commutative semisimple algebra with generators \(C_1,\ldots, C_L\), constructed from a degeneracy graph which records how the eigenspaces of \( C_1\) are successively split by \( C_2,\ldots, C_L\). They showed that the basis property is equivalent to the non-vanishing of the determinant of the matrix of values of these monomials on the projector basis, and conjectured a factorised formula for this determinant, as a product of powers of eigenvalue differences. We prove this formula for any number of generators. The proof uses graph analogues of Newton polynomials. Their evaluation matrix is not triangular in general, but expanding the rows of the children of each node of the penultimate layer in a Newton basis of their eigenvalues makes it block triangular, with diagonal blocks given by smaller degeneracy graphs in which the final layer is removed. The formula then follows by induction on the number of layers. We show that the exponents in the determinant are overlap volumes of higher-dimensional Young diagrams attached to the nodes of the graph: intervals, ordinary Young diagrams, plane partitions and their higher analogues. Finally, the block triangular structure gives an explicit recursive inversion of the evaluation matrix, expressing the projectors in the monomial basis.

The problem of maximizing vertex-degree-based topological indices over trees with a prescribed order and maximum degree $\Delta$ has traditionally relied on ad-hoc combinatorial arguments. In this paper, we establish a unified linear programming approach that systematically resolves these extremal problems. By formulating the optimization over degree-pair multiplicities, the search for an extremal tree reduces to minimizing a discrete dual penalty. Treating vertex degrees as continuous variables yields an analytical single-variable test condition that guarantees global optimality. We apply this framework across a diverse collection of invariants. We prove that the standard basic feasible solution is optimal for the Tangent Sombor, $SO_1$, $SO_2$, Albertson, and $\sigma$-irregularity indices for all $\Delta \ge 3$. However, the dual optimality conditions establish exact analytical thresholds at which the standard basis loses optimality, necessitating a transition to alternative basic feasible solutions. For the Arithmetic-Geometric ($AG$) index, the standard basis is optimal strictly for $\Delta \ge 9$; for smaller maximum degrees ($3 \le \Delta \le 8$), optimization shifts to an alternative edge basis, allowing us to explicitly construct and characterize all extremal trees across congruence classes, including chemical trees ($\Delta \in \{3, 4\}$). Conversely, for the Triangle Volume index ($SO_{11}$), the standard basis is optimal exclusively for $\Delta \le 8$, while for $\Delta \ge 9$ we analytically derive a novel, degree-dependent optimal basis to explicitly construct its extremal configurations.

In this paper we describe the quotients of several plactic-like monoids by the least congruences containing the relations $a^{\sigma(a)} = a$ for $\sigma(a) \geq 2$ for every generator $a$. The starting point of this paper is the recent paper of Reutenauer and the first author about the stylic monoid. This monoid is the quotient of the plactic monoid by the relations $a^2 = a$ for every generator $a$. The plactic-like monoids considered here are the plactic monoid itself, the Chinese monoid, the hypoplactic monoid, the sylvester monoid, the metasylvester and hyposylvester monoids. In each case we describe a set of normal forms and the idempotents. We obtain formulae for their cardinalities and we describe their isomorphism types.

For each positive integer $N$, let $p_i$ be the $i$-th prime for $i\in\{1,\ldots,N\}$, and set $$ \Phi_N=\left\{\prod_{i=1}^{N}p_i^{a_i}:a_i\in\{0,1,\ldots,N-1\}\right\}. $$ Write $f(N)$ for the largest cardinality of a subset of $\Phi_N$ containing no two distinct legs of an integral right triangle. We prove $$ N^{N-1}(\log\log\log N)\ll f(N). $$

We introduce a new model of parking functions, called $\rho$-Naples parking functions, in which each car has a certain number of spots to check preceding its preferred one before looking for available spots by moving forward. This notion, where each car has its own number of previous spots to check, generalizes that of Naples parking function, where this number is the same for all cars. We start by showing that, for a given sequence of preferences for each car, a successful parking strategy (that is, an assignment of how many backward spots each car is allowed to check) remains successful if we increase the number of backwards spots to check for some of the cars. This means that the set of successful strategies for a given parking preference is an up-set in the lattice of parking strategies, which raises the intriguing question of characterizing such up-sets. Though we are not able to completely solve this problem, we provide a characterization of those successful strategies which minimize the total number of allowed backward steps for a given parking preference, and we determine a lower bound for the number of cars which are allowed to move backwards (also showing that it can be attained). To conclude, we analyze the specific situation in which each car can take at most one backward step, and we completely characterize the set of successful parking preferences in this case.

This paper develops a natural extension of the multipoint spherical semidefinite method of Liao and Yu~\cite{LY}, studying the interaction between semidefinite programming and strongly regular graph counting. Starting from the positive semidefinite kernels furnished by Gegenbauer polynomials, we express weighted kernel sums and counting identities in the same labelled induced-subgraph counts. Eliminating the counting equations gives a semidefinite feasibility problem. Numerical dual solutions guide the construction of rational certificates, whose contradiction identities are verified exactly. Our aim is a systematic method that treats many parameter sets through the same construction. For least eigenvalue $-3$, it gives new counting certificates for $(351,140,73,44)$, $(550,162,75,36)$, $(703,182,81,35)$ and $(1344,221,88,26)$, previously excluded by Liao and Yu~\cite[Theorem~1]{LY}, and for six additional parameter sets: \[ \begin{array}{@{}lll@{}} (375,102,45,21), & (405,132,63,33), & (441,88,35,13),\\ (476,133,60,28), & (540,147,66,30), & (575,112,45,16). \end{array} \] Seven certificates use Gegenbauer positivity and counting information alone at the displayed orders. The cases $v=375,441,575$ additionally use proved short-vector restrictions, unique vertex extensions, and integer common-neighbour inequalities. The proofs end in displayed inequalities with incompatible signs. Their justification is given by the dual identities and the proved positivity and counting relations; accompanying code automates the longer rational arithmetic.

We introduce the parity module of a matroid with coefficients in a field $\mathbb{k}$. When $\mathbb{k}$ has characteristic zero, we show that the parity module of a matroid is isomorphic to its intersection cohomology module, yielding a new intrinsic characterization of the latter. The structure and dimension of the parity module can vary depending on the characteristic, giving rise to characteristic-dependent generalizations of the Kazhdan-Lusztig and Z-polynomials of matroids. When the matroid is realizable over the complex numbers, the parity module is isomorphic to the hypercohomology of the parity sheaf supported on the arrangement Schubert variety of any such realization. We also introduce the notion of pure modules over any field $\mathbb{k}$ and show that they satisfy a matroid analogue of the Beilinson-Bernstein-Deligne-Gabber decomposition theorem: Every pure module over the graded Möbius algebra decomposes as a direct sum of shifts of the parity modules of matroids obtained by restricting to flats.

A recent graph-theoretic criterion for a Hamiltonian to exhibit a free-parafermion spectrum is that its frustration graph be an indifference graph. We show that an arbitrary finite indifference graph can be realized as a frustration graph of the quantized six-vertex model at odd roots of unity, by a suitable choice of the defining wiring diagram and parameters. This provides a concrete realization of the criterion through the quantum Frobenius property of the quantized six-vertex model, recently established from its three-dimensional integrable structure.

A graph $H$ is called common if the number of monochromatic copies of $H$ in a $2$-edge-coloring of a large complete graph is asymptotically minimized by the random coloring with an equal proportion of each color. The house graph and $K_4$ with one edge subdivided once are known to be common by the flag algebra method. Grzesik, Lee, Lidický, and Volec asked for “human-friendly” proofs of the commonality of these two graphs. In this short note, we give flag-algebra-free proofs that both graphs are common.

In 2019, Liu proved that the cop number of any $(P_k, claw)$-free graph is at most $k-3$. In 2026, Clow and Meger improved and generalized the result of Liu by showing that every connected $(P_k, E)$-free graph has cop number at most $\lceil \frac{k-1}{2} \rceil + 3$, where the graph $E$ is obtained by subdividing two edges of a claw. In this paper, we show that every $(P_k, claw)$-free graph $G$ has cop number at most $\lceil \frac{k-1}{2} \rceil$. This improves the result of Clow and Meger on claw-free graphs. We further show that this bound can be improved to $\lceil \frac{k}{3}\rceil$, when $G$ contains a hole of length at least $k-1$. We show that the cop number of any $(P_k, fork)$-free graph is at most $\lceil \frac{2(k-1)}{3} \rceil$. This also improves the result of Clow and Meger on fork-free graphs for some values of $k$. We further show that every $(P_7, fork)$-free graph has cop number at most $3$, and this bound is tight.

Motivated by Mader's conjecture on connectivity keeping trees, we study the corresponding problem in triangle-free graphs. For positive integers $k,m$, we first prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\left \lceil \frac{m}{2} \right \rceil $ contains a path $P$ of order $m$ such that $G-V(P)$ remains $k$-connected. Moreover, if $\kappa(G)=k$, then for every end $F$ of $G$ and every vertex $v\in F$, the path $P$ can be chosen in $G[F]$ with $v$ as an endvertex. If $\kappa(G)\geq k+1$, then any vertex of $G$ can be chosen as an endvertex of $P$. This result confirms the conjecture of Luo, Tian and Wu on connectivity keeping trees in bipartite graphs for all paths, and it also sharpens Fujita's path theorem to the optimal minimum degree bound. For an integer $i$ with $1\leq i\leq m-1$, let $P_m^+(i)$ denote the tree obtained from a path of order $m-1$ by adding one pendant vertex adjacent to its $i$th vertex. Let $(X,Y)$ be the bipartition of $P_m^+(i)$. We further prove that every $k$-connected triangle-free graph $G$ with $\delta(G)\geq k+\max\{|X|,|Y|\}$ contains a subgraph $T\cong P_m^+(i)$ such that $G-V(T)$ remains $k$-connected. The minimum degree bound is sharp, and the result confirms the bipartite conjecture for all pendant extensions of paths.

Define $\nabla(n)$ as the least possible number of cells that must be removed from the $n\times n$ square board in order to leave no cycles. Put $L_n=\left\lceil\left((n-1)^2+1\right)/3\right\rceil$. The results of Luccio, Madelaine-Stewart, and Lien-Fu-Shih imply that $\nabla(n)=L_n$ in case $n\equiv 1\pmod 6$ and $n\equiv 4\pmod 6$, while for the remaining $4$ residue classes $\nabla(n)=L_n$ or $\nabla(n)=L_n+1$. In this paper, we provide optimal constructions in the odd residue classes $n\equiv3,5\pmod6$, (proving $\nabla(n)=L_n$) and a geometric obstruction showing that equality in lower bound is impossible when $n\equiv0,2\pmod6$ (yielding $\nabla(n)=L_n+1$). Starting with $n=3$, the resulting sequence is thus given by \[ \nabla(n)=\bigg{\lceil}\frac{n^2}{3}-\frac{n}{6}+\frac{4}{3}\bigg{\rceil}-\bigg{\lceil}\frac{n}{2}\bigg{\rceil}. \]

A finite Gödel chain generates not one integer sequence but a large family of them. The combinatorial mechanism is developed in a companion paper; here the sequences themselves are the primary objects. For $V_m=\{v_0<\cdots<v_{m-1}\}$, let $g_{j,n}^{(m)}$ count outputs equal to $v_j$, aggregated over all full bracketings on $n$ variables and all $m^n$ valuations. The root-pair refinement produces $N_{i,j}^{(m)}(n)=\sum_{r=1}^{n-1}g_{i,r}^{(m)}g_{j,n-r}^{(m)}$, with generating functions $N_{i,j}^{(m)}(x)=G_i^{(m)}(x)G_j^{(m)}(x)$. We prove that the initial coefficients of these products uniquely identify the unordered root-value pair. Consequently, for each fixed $m$, there are exactly $\binom{m+1}{2}$ distinct root-pair generating functions. We catalogue the factories for $m=4,5,\ldots,10$, giving the first ten terms of every base sequence and every symmetric root-pair sequence. This range contains 49 base sequences and 210 symmetric pair classes, yielding 259 indexed sequence families before additional aggregates. The catalogue provides reproducible data for integer-sequence analysis and comparison with the On-Line Encyclopedia of Integer Sequences (OEIS).

For all sufficiently large $n$, we prove that every Latin square of order $n$ can be made to avoid any prescribed partial Latin square of order $n$ by permuting only its rows and columns. The proof combines Hall's theorem with elementary estimates for a random permutation.

Extremal constructions play a central role in the design and analysis of algorithms. Integrality-gap instances expose the limitations of relaxations, while hard configurations reveal structural sources of computational difficulty. Such constructions, however, are still typically discovered through a slow, manual process. This paper investigates how machine learning can accelerate their discovery to support algorithm design and analysis. We build on PatternBoost, a framework alternating local search with generation from a trained transformer, using its Axplorer implementation to investigate three problems: integrality gaps for maximum independent set of rectangles, integrality gaps for stabbing unit squares with axis-parallel lines, and hard instances for recursive guillotine cutting. By exploring candidate representations, local search operations, and surrogate models used to rank candidates before exact verification, we carefully adapt the framework to these combinatorial geometry problems.While naive representations and search strategies fail to find non-trivial instances, our tailored adaptation discovers numerous non-trivial constructions. For stabbing unit squares, the method reaches a gap of $20/13\approx 1.54$, improving on the literature's best known lower bound of $3/2$. Purely automated search converges rapidly and then plateaus, prompting the question of what is required to push the bound higher still. We find that initial instance diversity plays an important role: with modest expert input via a structured staircase construction, the system reaches an improved integrality gap of $14/9\approx 1.555$. These results demonstrate that machine learning and domain expertise can be combined to discover new extremal constructions that advance algorithmic lower bounds.

Association schemes induced on the shells of the Grassmann scheme $Gr_q(N,a)$ form an important family in the study of multivariate $P$- and $Q$-polynomial structures. For $a<N/2$, the scheme induced on the furthest shell $X_a$ is the known bivariate $P$- and $Q$-polynomial scheme arising from attenuated spaces. In this paper, we prove that, for every $1\leq i\leq a-1$, the association scheme induced on the interior shell $X_i$ is both trivariate $P$-polynomial and trivariate $Q$-polynomial. The two structures share a common index set consisting of the lattice points of a bounded non-simplicial polyhedron. We establish the $P$-polynomial property through combinatorial interpretations of the intersection numbers. Using Dunkl's coordinate description of $X_i$ and the addition theorem, together with recurrence relations for the $q$-Hahn and $q$-Krawtchouk polynomials, we obtain the required vanishing and nonvanishing conditions for the Krein parameters, and hence the $Q$-polynomial structure. Motivated by this index set, we extend the notion of an $A_M$-Leonard pair from simplicial to polyhedral domains and show that every interior shell carries a polyhedral $A_3$-Leonard pair.

For integers $r\ge1$ and $n\ge2$, let $s_r(n)$ be the least $N$ such that every sequence of $N$ vectors in $\Z^r$, each nonzero modulo $n$, contains $n$ terms whose sum is zero modulo $n$ but nonzero modulo $n^2$. Let $w_r(n)$ be the corresponding least $N$ when any nonempty subsequence is allowed. These constants were introduced as zero-sum problems of new types by Z.-W. Sun. We prove $s_r(n)\ge(2^{r+1}-2)(n-1)+1$ by constructing an explicit sequence from a lattice polytope, disproving his conjecture $s_2(n)=4n+1$ for $n\ge4$. We obtain $w_r(2)=2r+1$ by classifying all sequences avoiding such nonempty subsequences modulo $4$. We also prove Sun's conjecture $w_2(n)=4n-3$ for $n=2^k3^\ell p$, where $p$ is a prime and $k,\ell\ge0$ are integers, by classifying the extremal sequences of length $4p-4$ modulo $p^2$ when $n=p$. For every fixed $n\ge2$, we prove $w_r(n)/s(\Z_n^r)=O_n(r^2/2^r)\to0$ as $r\to\infty$, where $s(\Z_n^r)$ is the classical Erdős-Ginzburg-Ziv constant, with an explicit threshold on $r$ for strict inequality. Thus, for every fixed modulus $n\ge2$, Sun's proposed equality fails for sufficiently large $r$.

Let $A_q(n,d)$ denote the largest size of a code of length $n$ and minimum Hamming distance $d$ over an alphabet of $q$ symbols. For fixed $q\ge2$ and $d\ge4$ we construct codes with redundancy $n-\log_q A_q(n,d)\le(d-3)\log_q n+O(\sqrt{\log n})$ for all large $n$. No algebraic structure is imposed on the alphabet, and the codes are not linear. For $d=4$ and $d=5$ the coefficient of $\log_q n$ matches the Hamming bound, so the redundancy is determined up to lower-order terms. At distance four the codes are provably better than every linear code: by the polynomial-method bound for cap sets, $\mathbb{F}_q$-linear codes with $q>2$ require a strictly larger coefficient in front of $\log_q n$. The construction imposes moment checksums over an auxiliary finite field, as BCH codes do, but one fewer than the distance would normally require. The affine-line theorem from prior work describes the dependencies left by these checks. We exclude them by placing the code positions inside a specially structured subset of the field, built from Behrend's sphere construction with the help of algebraic number theory.

In this paper, we evaluate two companion $q$-series recently suggested for investigation by Andrews and El Bachraoui in the concluding remarks of their work. From these evaluations, we derive the corresponding partition identities and give combinatorial proofs.