Papers

We study the computational complexity of several counting problems on graphs. Each of these problems consists of counting orderings of the vertices or edges with adjacency constraints. We show $\#P$-completeness for all of them via a common new technique. Given a counting function $C$ of interest, we define a parameterized family of instances $G_q$, where the parameter $q$ controls the amplification of a simple gadget. After multiplying by an explicit factor $f(q)$, we show that the values of $f(q) \cdot C(G_q)$, for positive integers $q$, agree with a rational function in $q$ whose numerator and denominator can be interpolated in polynomial time. We then recover a $\#P$-hard function by evaluating this rational function symbolically at a limiting value $L \in \mathbb{Q} \cup \{\infty, -\infty\}$. With this methodology, we show $\#P$-completeness for the following counting problems: (a) successive vertex orderings of bipartite graphs, (b) st-numberings of graphs, (c) shellings of bipartite graphs, (d) linear extensions of N-free posets of height $3$, and (e) linear extensions of posets of height $2$. Result (d) settles a conjecture of Felsner and Manneville (2015). Although result (e) was first proved by Dittmer and Pak (2018), we include an alternative proof, using our technique, that does not rely on the result of Brightwell and Winkler (1991) about the hardness of counting linear extensions for general posets.

Andrews and El Bachraoui investigated two-color partitions in which even parts may occur only in blue and obtained several identities relating parity refinements of these partitions to overpartitions and minimal excludants. Some of their proofs are analytic and rely on $q$-series manipulations. In this paper, we construct explicit weight-preserving bijections and parity-reversing involutions that provide combinatorial proofs of these identities. We also give a new combinatorial proof of related overpartition refinements involving $\overline{p}(n)$ and $\overline{p}_o(n)$.

We discuss the problem of characterizing equidistant binary codes of a given length $n$ having largest possible distance and the maximum number of codewords. Such characterizations have been studied by several authors over the years and they involve symmetric BIBDs with certain parameters. In this primarily expository paper, we investigate the history of this problem and give a unified presentation of the main results. Perhaps surprisingly, researchers on this problem were unaware of early relevant work by Marrero and Butson from 1973. Also, it turns out that published results on characterizations of equidistant binary codes have missed one of the possible subcases when $n \equiv 2 \bmod 4$.

We model Herman Kahn's escalation ladder as an impartial combinatorial game. Reindexing each rung by its distance to the nuclear threshold turns the ladder into a subtraction game, the most tractable class in combinatorial game theory, and the doctrinal fact that no side wishes to fire first selects the misere convention. We prove that single-ladder stability is governed by a congruence (Theorem 4.1) and derive a ladder-design corollary that makes the burden of first escalation a function of ladder length and escalation granularity (Corollary 4.2). For simultaneous theaters we show, under normal play, that joint stability is the Nim-sum of the theater-wise escalation distances (Theorem 5.2), a condition that is neither additive nor dominated by the most dangerous theater. We then show the Nim-sum reduction fails under misere play, introduce the misere quotient as its replacement, and prove by exhaustive backward induction that for two-step escalation the quotient is the order-six monoid with generators a, b satisfying a^2=1 and b^3=b, with loss set {a,b^2} (Theorem 6.3). To our knowledge, impartial combinatorial game theory has not previously been applied to nuclear escalation ladders; the existing game-theoretic literature on escalation is classical and payoff-based.

We say that a square matrix $A$ is \emph{$1$-principled} if all its principal minors are equal to $1$. We show that over any well-behaved ring, any power $A^m$ of a $1$-principled matrix $A$ is again $1$-principled. Well-behaved rings include all reduced rings as well as all quotients of commutative rings modulo integrally closed ideals; in particular, all fields and all quotients of $\mathbb{Z}$ are well-behaved. We note that $m$ can be any integer, positive or negative. This generalizes Problem B5 of the 2021 Putnam contest in multiple directions. Over arbitrary commutative rings, we identify a stronger property that is always inherited by powers: We say that a matrix $A = \left(a_{i,j}\right)_{i,j\in\left[n\right]}$ is \emph{$1$-nullcyclic} if all its diagonal entries are $1$ and if all the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish. We show that if $A$ is $1$-nullcyclic, then so is $A^m$ for any integer $m$. Furthermore, every $1$-nullcyclic matrix is $1$-principled over any commutative ring, while the converse holds if the ring is well-behaved. Along the way, we prove analogous results that don't require the diagonal entries to be $1$. These are concerned with principled matrices (those whose principal minors equal the respective products of diagonal entries) and nullcyclic matrices (those whose cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ with $k>1$ and distinct $i_1,i_2,\ldots,i_k$ vanish); their diagonal entries can be arbitrary. A crucial auxiliary result, which holds for any $n\times n$-matrix $A$, is that the cyclic products $a_{i_1, i_2} a_{i_2, i_3} \cdots a_{i_k, i_1}$ (with $k>1$ and distinct $i_1,i_2,\ldots,i_k$) are integral over the ideal generated by the principal minors of $A$ minus the corresponding products of diagonal entries of $A$.

We solve large-$N$ Yang–Mills theory on $\mathbb{Z}^d$, for every $d\geq2$, at strong coupling, for structure group $\mathrm U(N)$ and for the heat-kernel action. More precisely, we prove that normalized Wilson loop expectations have infinite-volume large-$N$ limits, factorize at leading order, and admit an all-order $1/N$-expansion with exponentially local coefficients, whose leading order characterizes the master field. We also prove an area-law upper bound for the heat-kernel master field, with a stronger coefficientwise version. The proof is based on a rooted heat-kernel master loop equation. Unlike the Wilson-action equation or the two-dimensional Makeenko–Migdal equation, this equation does not close on Wilson loop observables alone; it closes on an extended space of loop observables coupled to compactly supported plaquette decorations. We prove a strong-coupling, order-truncated rooted trajectory expansion and then identify its leading term with the master field. The main inputs are the universal finite-$N$ duality formulas developed in the companion paper \cite{Lem26a} and large-$N$ heat-kernel estimates from \cite{LemMai25,LM2}.

A subgraph $H$ of $G$ is said to be $F$-saturated relative to $G$, if $H$ does not contain any copy of $F$, but the addition of any edge $e$ in $E(G)\backslash E(H)$ would create a copy of $F$. The minimum size of an $F$-saturated graph relative to $G$ is denoted by $sat(G,F)$. Let $K_k^n$ be the complete $k$-partite graph with $n$ vertices in each part. In this paper, we determine $sat(K_4^n,C_4)$ for all $ n \geq 2$. Moreover, we determine all extremal configurations of $sat(K_k^n,C_4)$ for all $n\ge 2$ and $k\ge 4 $.

In this paper, we study the so-called log-convergence of graphs defined by Balázs Szegedy (arXiv:1504.00858). We answer his Question 4 affirmatively: the sequence of incidence graphs of projective planes over finite fields log-converges, and the limit coincides with that of a particular random graph model.

We settle the long-standing open question whether there exists a $3$-ladder of cardinality $\aleph_2$. Given a positive integer $n$, an $n$-ladder is a lower finite lattice whose elements have at most $n$ lower covers. In 1984, Ditor proved that every $n$-ladder has cardinality at most $\aleph_{n-1}$, and that this cardinal bound is sharp for $n = 1,2$. He then raised the question of whether the bound is attained for $n\ge 3$ as well. An affirmative answer is known to be consistent with $\mathsf{ZFC}$. We prove, relative to the consistency of a Mahlo cardinal, that the question is independent of $\mathsf{ZFC}$. More precisely, we show that the nonexistence of a $3$-ladder of cardinality $\aleph_2$ is equiconsistent with a Mahlo cardinal.

The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $\sigma$ such that the spectral radius of the signed adjacency matrix $A_\sigma$ satisfies $\rho(A_\sigma)\le 2\sqrt{d-1}$. Bilu and Linial also proved the weaker bound $O(\sqrt{d\log^3 d})$ for graphs of maximum degree $d$. Marcus, Spielman, and Srivastava confirmed the conjecture in the case of $d$-regular bipartite graphs. In this paper, we prove that every graph of maximum degree $d$ has a signing $\sigma$ such that $$\rho(A_\sigma)\le 2\sqrt{3(d-1)}.$$ This removes the polylogarithmic factor from the estimate of Bilu and Linial and gives an explicit $2\sqrt{3(d-1)}$ two-sided spectral bound. The proof builds on the method of interlacing polynomials introduced by Marcus, Spielman, and Srivastava, together with results on mixed characteristic polynomials established by Marcus, Spielman, and Srivastava and by Bownik.

A graph $G$ is said to be $F$-semi-saturated if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. The semi-saturation number $ssat(n,F)$ is the minimum number of edges in an $F$-semi-saturated graph of order $n$. In this paper we investigate the semi-saturation number of $K_r$ on $n$ vertices with maximal degree at most $\Delta$, denoted by $ssat^{\Delta}(n,K_r)$. This investigation was suggested by Erd\H os, Rényi and Sós, who in 1966 considered the graph of diameter 2 with degree restrictions, equivalently $ssat^{\Delta}(n,K_3)$. The following are some of our results. For arbitrary $r \geq 4$, we show that the limit $ \lim_{n \rightarrow \infty} ssat^{cn}(n,K_r)/n$ exists for all $0 < c \leq 1$, except for some sparse values of $c$ contained in a countable and rational sequence $c_i \rightarrow 0$. Moreover, we establish the asymptotic behaviour of this limit for $\frac{r}{r+2} < c <1$ and determine the exact value of $ssat^{\Delta}(n,K_r)$ for some specific $\Delta$. As an application, we determine the relation between the saturation number of the join graph $K_r \vee F$ and that of $F$ for a large class of pairs $(r,F)$.

For a strict partition $\lambda$, let $\mathcal Q_\lambda(X;t)=Q_\lambda[X-tX]$ be the shifted $t$-Schur function arising from the modified Greaves–Jing–Zhu operator on the odd power-sum ring. We study transition matrices between the shifted bases with parameters $t$ and $s$. The relative scaling operator is diagonal in the odd power-sum basis, leading to explicit spectral data, determinant and trace formulas, weighted symmetry, a spin-character formula, and a transition Cauchy identity. For the cyclotomic specialization $C_{\lambda\mu}^{[M]}(t)=C_{\lambda\mu}(t^M,t)$, the relative operator becomes plethystic substitution by $1+t+\cdots+t^{M-1}$. We prove Schur $Q$-positivity and reciprocity, derive factorization and root-of-unity rank formulas, and give an exact computation method. For $M=2$, all one-row transitions are computed explicitly, and the nonzero coefficients are unimodal.

Let $T(x)=\prod_{k=0}^{\infty}(1-x^{2^k})$ be the generating function of the Thue–Morse sequence, and write $T(x)^m=\sum_{n\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \[ \nu_2\bigl(t_5(4n+j)\bigr) =4\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,2,3\}, \] \[ \nu_2\bigl(t_9(8n+j)\bigr) =5\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-2\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,\ldots,7\}. \] These formulas confirm Conjecture~5.2 of Gawron–Miska–Ulas~\cite{ga} for $m=5$ and $m=9$, and imply that $t_5(n)\neq 0$ and $t_9(n)\neq 0$ for every $n\geq 0$. A key structural ingredient is a closed-form formula for the determinant of a family of matrices with binomial-coefficient entries.

We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. Here, quasisymmetric cubical sets are cubical sets equipped with coordinate permutation symmetries that are compatible with faces and degeneracies, but not necessarily with connections. We give a purely combinatorial construction of the left adjoint of the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, and prove that the unit of this adjunction is an objectwise weak equivalence. As a consequence, we obtain a purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets. As an application, we establish the Hurewicz theorem for the discrete homotopy groups of quasisymmetric cubical sets.

We consider the encoding of graph problems as Quadratic Unconstrained Binary Optimization (QUBO) problems, which are solvable by either quantum or classical annealers. Yet, the class of problems encoded as QUBO problems has not previously included nowhere-zero flows. Nowhere-zero flows are related to Tutte's $5$-flow conjecture and appear in many contexts in graph theory. We provide an encoding of nowhere-zero flows as a QUBO Hamiltonian and prove the correctness of the construction. Our construction yields a Hamiltonian $H_{\mathrm{mod},k}$ whose ground state has zero energy if and only if the graph $G$ has a nowhere-zero $\mathbb Z_k$-flow. By Tutte's equivalence theorem, zero ground energy is equivalent to $\varphi(G)\le k$, and the zero-energy degeneracy is given by the flow polynomial $F(G;k)$. In particular, when the ground-state energy is zero, this is also the ground-state degeneracy. The construction uses one-hot variables to represent the edge flow residues modulo $k$ and auxiliary variables to represent the per-vertex modular quotient. We prove that the correctness of the construction is independent of the choice of orientation, root vertex, and positive penalty weights. We verify the construction on $59$ examples of graphs and values of $k$ that include both yes-instances and no-instances. We exhaustively sweep orientations and root choices on selected robustness instances and test a finite suite of positive penalty weights. The resulting Hamiltonian is implemented using the dimod.BinaryQuadraticModel class, which is compatible with the D-Wave Ocean SDK. Quantum-hardware runs and claims about potential speedup using these devices are left to follow-up work.

We investigate the realizations of Coxeter permutahedra which are also Coxeter matroid polytopes; these are polytopes of the form $\mathrm{conv}(W \cdot \mathbf{a})$ where $W$ is a finite Coxeter group acting on $\mathbb{R}^n$ and $\mathbf{a}$ is generic. Our main focus is how the geometric properties of $\mathrm{conv}(W \cdot \mathbf{a})$ change as $\mathbf{a}$ changes, with particular attention to persistent simplices, triangulations, and subdivisions.

In this paper, we study the group randomness of 0-1 real sequences derived from a binary linear code by investigating the spectral behaviour of a suitable normalization of the Gram matrix of a $p \times n$ random matrix whose rows are uniformly drawn from those 0-1 real sequences, where $y=p/n \in (0,1)$ is fixed. We show that as $n \to \infty$, its empirical spectral distribution converges to the Marchenko-Pastur law at a rate at least of the order $n^{-1/4}$ with high probability, and the fluctuation of its largest eigenvalue is asymptotically Gaussian with mean $p+1+y$ and variance $4y$, provided that the dual distance of the code is at least 5.

Let $g \ge1$. A set $\mathcal{A}$ of nonnegative integers is a Sidon set if for each $d>0$ there is at most one pair $(a,b) \in \mathcal{A} \times \mathcal{A}$ with $d=a-b$. If there are at most $g$ pairs, then $\mathcal{A}$ is a $g$-Golomb ruler. We prove that if $\mathcal{A}$ is a $g$-Golomb ruler, then \[\liminf_{n\to\infty} \frac{ | \mathcal{A}\cap[0,n) | }{\sqrt{n/\log n}} \le \frac{2\sqrt g }{\sqrt{\log 2}},\] generalizing and sharpening results of Erdős and Cilleruelo. There is a $g$-Golomb ruler $\mathcal{G}$ with \[\frac{\sqrt g }{\sqrt2} \le \limsup_{n\to\infty} \frac{ | \mathcal{G}\cap[0,n) | }{\sqrt n} \le \sqrt{g } ,\] generalizing a result of Krückeberg.

More than forty years ago, Andersen and Hoffman independently proved that every symmetric Latin rectangle can be extended to a symmetric Latin square with prescribed diagonal entries. We generalize this theorem as follows. Let $k\leq n^2$, and let $M$ be an $n\times n$ array whose top-left $r\times r$ subarray is filled with symbols from $\{1,2,\ldots,k\}$. Suppose that, for each $i\in\{1,\ldots,r\}$ and each symbol, the number of occurrences of that symbol in row $i$ equals its number of occurrences in column $i$, and that each remaining diagonal entry is either empty or already contains a symbol from $\{1,\ldots,k\}$. We establish necessary and sufficient conditions for completing $M$ so that the resulting array is symmetric off the prescribed $r\times r$ subarray, each symbol occurs a specified total number of times in $M$, and, for every symbol, its numbers of occurrences in any two rows (respectively, columns) differ by at most one. Restricted to symmetric arrays, our theorem generalizes results of Cruse (1974), Goldwasser et al. (2015), and Bahmanian and Hilton (2025). It also extends Baranyai's theorem for complete graphs (1973) by characterizing when a partial coloring of $K_r$ with a loop on every vertex can be extended to an almost regular coloring of $K_n$ with a loop on every vertex, where $n\geq r$.

We construct a reduced planar convex body $R$ with thickness $\Delta(R)=1$ and \[\operatorname{area}(R)=0.786215\ldots>0.785398\ldots=\frac{\pi}{4}.\] Thus $R$ is a counterexample to Lassak's conjectured upper bound $\operatorname{area}\le(\pi/4)\Delta^2$ for planar reduced bodies. The construction is given by an explicit support function, and the proofs use only elementary support-function, width, area, and contact-point computations.

Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction to Cayley codes for connected Cayley graphs that maintains code properties such as rate, minimum distance and symmetry. Also, for a given Cayley code, we identify a family of symmetric Cayley codes, each associated with a normal edge-transitive Cayley graph, such that the given Cayley code embeds into the direct sum of the symmetric Cayley codes. We analyse several families of examples, in particular studying the behaviour of the Cayley code construction under forming direct products and cartesian products of Cayley graphs, and we pose a number of open questions.

Interval graphs are a special class of chordal graphs, and hence have connections to commutative algebra via Fröberg's theorem that characterizes linear resolutions of squarefree quadratic ideals. In recent years, several hypergraphic analogues of interval and chordal graphs have been proposed, in part as an effort to extend Fröberg's theorem to ideals generated in higher degree. In this paper, we study two such classes from the literature, cointerval hypergraphs and underclosed complexes, and show that they are in fact equivalent up to complementation. We then consider their place in the broader theory of higher-dimensional chordality, proving that an underclosed clutter is chordal in the sense of Woodroofe. As a consequence, we answer a question of Dochtermann and Engström by showing that the associated Alexander dual complexes are vertex decomposable, implying that the corresponding circuit ideals have linear quotients. We furthermore show that these dual complexes have shellings induced by their underclosed vertex orders.

Let $[0, 1]^{n} \subseteq \mathbb{R}^{n}$ be endowed with its pointwise order, and let $k$ be a positive integer. A subset $A$ of $[0, 1]^{n}$ is said to be a \emph{$k$-antichain} if $\operatorname{card}(A \cap C) \leq k$ for each chain $C \subseteq [0, 1]^{n}$. Letting $\mathcal{H}^{m}$ denote the $m$-dimensional Hausdorff outer measure, Pelekis and Vlasák [Publ.\ Math.\ Debrecen, 2020] conjectured that there exists a $k$-antichain $A \subseteq [0, 1]^{n}$ satisfying $\mathcal{H}^{n-1}(A) = k n$, and proved the special case of this conjecture for $n = 2$, whereas Janzer [Mathematika, 2020] proved the $k = 1$ case of Pelekis and Vlasák's conjecture. This conjecture is motivated by a result due to Erdős on $k$-antichains in $\{ 0, 1 \}^{n}$. We prove Pelekis and Vlasák's conjecture in full generality, thus establishing that their upper bound $\mathcal{H}^{n-1}(A) \leq k n$ is sharp for $k$-antichains $A$ in $[0, 1]^{n}$.

Packing problems form a central theme in graph theory, owing to their relevance in modeling conflict-free resource allocation, network design, and communication constraints. Motivated by applications in wireless networks where each device can participate in at most one communication at a time and simultaneous links must avoid interference we consider a generalization of induced matching known as edge open packing. Two edges of a graph are said to conflict if a third edge connects one endpoint of each; an edge open packing set is a set of edges containing no such conflicting pair. The largest cardinality of such a set is the edge open packing number of a graph. In this work, we study the computational complexity of the Maximum Edge Open Packing Problem. We give a polynomial-time algorithm for the problem in distance-hereditary graphs, exploiting their canonical decomposition via twin-set interactions. We further show that the problem remains polynomial-time solvable on biconvex bipartite graphs, thereby identifying a tractable subclass within bipartite graphs, in contrast to the known NP-hardness of the problem on Eulerian bipartite graphs. Finally, we initiate the parameterized complexity study of the problem and present a fixed-parameter tractable algorithm for chordal graphs, parameterized by the clique number $\omega$, running in $O(2^{\omega}\cdot\mathrm{poly}(n))$ time.

Maranca and Rosenberg (2024) devised a ranking scheme for unlabeled multifurcating rooted trees, in which the trees are bijectively associated with the positive integers. Here, generalizing earlier results for bifurcating trees, we determine, for trees with a fixed number of leaves, which multifurcating trees obtain the maximal and minimal ranks. We identify these maximizing and minimizing trees for each of two sets of unlabeled multifurcating rooted trees: strictly $k$-furcating trees, in which each internal node possesses exactly $k$ descendants, and at-most-$k$-furcating trees, in which internal nodes possess at least 2 and at most $k$ descendants. In both scenarios, we find that a tree that can be regarded as maximally balanced attains the minimal rank, and a minimally balanced tree attains the maximal rank. We deduce recurrences for the maximal and minimal rank for trees with fixed numbers of leaves in both the strictly $k$-furcating and at-most-$k$-furcating cases. The maximal rank on $(n-1)(k-1)+1$ leaves grows with $(k!)^{\frac{1}{k-1}} \beta_k^{(k^n)}$ in the strictly $k$-furcating case, and the maximal rank on $n$ leaves grows with $(k!)^{\frac{1}{k-1}} \gamma_k^{(k^n)}$ in the at-most-$k$-furcating case, where $\beta_k > 1$ and $\gamma_k > 1$ are constants that depend on the value of $k$. We show that $\beta_k$ decreases as the value of $k$ increases, and that $\gamma_k > \beta_k$ for $k \geq 3$. The results contribute to the use of tree encodings for empirical characterization of phylogenies and measurement of tree balance.

We classify Boolean cubic forms in ten variables up to GL(10,2)-equivalence. The catalog contains all 3691560 nonzero orbits. For every orbit we provide a representative with small monomial count, the stabilizer order, and the alternating rank together with an explicit decomposition. The classification is obtained by rank-stratified enumeration. We verify completeness by the Burnside orbit count and independently by the orbit–stabilizer identity. We also provide a fast, complete GL(10,2)-invariant. By polarization, this gives the first complete classification of alternating trilinear forms in dimension 10 over GF(2).

Many combinatorial designs ask for equal distribution of given symbols across the entries of a matrix. The paramount examples are Latin squares, where each symbol from $\{1,\dots,n\}$ appears once per row and column of an $n\times n$ matrix. Generalized Latin rectangles extend this to $\lambda n \times n$ matrices with repeated symbols under controlled column frequencies. In this more general setting, we examine structural properties of pairwise reflection-symmetry, which requires that, on every pair of columns, each ordered symbol pair $(p,q)$ occurs as often as its reversal $(q,p)$. This order-balance is precisely what makes head-to-head comparisons unbiased, i.e., no symbol gains a systematic advantage from the position it occupies relative to another, a fairness demand arising for instance when scheduling tournaments or laying out comparative trials. Existence of such objects for odd $\lambda$ turns out to be remarkably more subtle than for even $\lambda$. After showing that existence holds also for sufficiently large odd $\lambda$, we initiate the search for the smallest possible value of $\lambda$ in this setting. We obtain the insight that a column multiplicity of $\lambda=1$ can be achieved if and only if $n$ is a power of two. We complement the existence results with a direct product construction and add several further observations on the property. Finally, we propose and evaluate a quadratically constrained integer program to computationally search for these objects. The resulting experiments reveal that many of them possess an underlying group-theoretic structure which, as we conjecture, may even be unavoidable in certain settings.

In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the present paper we enumerate these partitions and prove that each block of $p$-weight $w$ contains precisely \[ \binom{w+\frac{p-3}{2}}{w} \] such partitions. More generally, for any integers $d,e>1$, we study and enumerate $d$-balanced $e$-regular partitions – partitions for which every hook of length divisible by $e$ has arm length divisible by $d$. Our first main result is that the crystal (affine) reflections preserve the $d$-balanced property for all $d,e > 1$. It follows that, for fixed $d$, $e$, and $w$, the number of $d$-balanced $e$-regular partitions in a block of $e$-weight $w$ is independent of the $e$-core. We then compute this number by working in RoCK blocks, obtaining an explicit binomial formula valid for every block. We also investigate closely related odd sequences of partitions. Among others, we find the generating function of the number of odd sequences occurring in a block. Alongside their representation-theoretic relevance, we expect these results to be of independent combinatorial interest.

We study the realization of finite groups as automorphism groups of finite posets. Given a finite group $G$, let $\beta(G)$ denote the smallest number of elements in a poset $P$ with $\Aut(P)\cong G$. While $\beta(G)$ is known for several cyclic and small abelian groups, the non-cyclic abelian case is largely open. In this paper we prove that $\beta(\Z_{2}\times\Z_{2^{n}})=2^{\,n+1}+2$ for every $n\ge 3$.

We study the coupon collector's problem in a generalized setting where each draw reveals a fixed number of coupons and the sampling mechanism is required to be fair, meaning that every coupon appears with the same frequency among the admissible draws. Grunbaum and Yaakobi conjectured that, among all fair mechanisms with fixed parameters, the fully random model maximizes the expected time to complete coverage. We disprove this conjecture by exhibiting explicit counterexamples arising from finite geometry. In particular, we show that the line set of the Fano plane yields a fair mechanism whose expected coverage time exceeds that of the full model. Further exact and computational results are obtained for projective planes of higher order. In addition, we analyze a simple infinite family of fair mechanisms, the star mechanism, for which the expected coverage time admits a closed form. Depending on the scaling regime, this mechanism can be asymptotically slower or faster than the full model, showing that no universal extremality principle holds for fair mechanisms without additional structural assumptions.