Papers

We give a Graham-positive expansion for the product of two double Schubert polynomials indexed by two inverse Grassmannian permutations. Surprisingly, the nonzero structure constants are double Schubert polynomials in two disjoint sets of equivariant variables. We also give a positive expansion for the product of two single Schubert polynomials indexed by a $321$-avoiding permutation (e.g., a Grassmannian permutation) and an inverse Grassmannian permutation. Unexpectedly, the nonzero structure constants are Edelman–Greene coefficients.

Treewidth is the standard measure for how “tree-like” a graph is. This paper studies how the treewidth of a product graph depends on the treewidth of its factors. Kozawa, Otachi, and Yamazaki [2014] and Hickingbotham and Wood [2025] independently showed that $\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)\text{had}(H)-1$ for all graphs $G$ and $H$, where $\text{had}(H)$ is the Hadwiger number of $H$. We improve this bound to $\text{tw}(G\boxtimes H)\geq (\text{tw}(G)+1)(\text{tw}(H)+1)-1$, thereby solving an open problem of Hickingbotham and Wood. We also prove analogous product inequalities for pathwidth, Cartesian products, and strict bramble number, which is a parameter that is tied to treewidth. As an application of our results, we show that products of expanders have large subgraphs that are expanders.

The classical Ingleton inequality is known to hold for entropic points under specific exact conditional independence constraints. Recently, Matveev and Romashchenko (2026) investigated the stability of these implications, quantifying the extent to which the Ingleton inequality can be violated when a group of conditional mutual information terms is small but non-zero. While their proofs relied fundamentally on the complex framework of tropical probability spaces, we revisit these stability results using a completely tropicalization-free approach. By developing an alternative framework, we significantly streamline the underlying concepts and proofs, derive explicit error terms, and improve some estimates. Furthermore, we resolve an open problem posed in prior work by exhibiting a new infinite family of entropy inequalities that establishes the stability of the sum of two Ingleton expressions.

We establish absolute and relative almost-linear Zarankiewicz bounds for semilinear relations in valued vector spaces. For every fixed arity and description complexity, a $K_{t,\ldots,t}$-free semilinear $r$-partite hypergraph has at most \[ O\!\left(n^{r-1}(\log n)^c\right) \] edges, where $c$ depends only on the arity and the number of valuative literals. In the bipartite case a separate arbitrary-trace argument gives the explicit bound $O(n(\log n)^{2s})$ for description complexity $(\rho,s)$. We also prove a relative extension theorem: intersecting any relation with a hereditary almost-linear profile by $s$ affine moving-radius comparisons increases the logarithmic exponent by at most $2s$. For the additive affine-valuative structures on $\mathbb Q_p$ and $\mathbb C_p$, quantifier elimination converts these semilinear results into bounds for all definable relations. Finally, over every valued field with infinite value group, we construct $K_{2,2}$-free semilinear point–box graphs of description complexity $(1,4)$ with $\Omega(n\log n/\log\log n)$ edges.

In this paper, we use the spectral surplus $\lambda(G) - \sqrt{m}$ to measure how far $G$ lies above the Nosal threshold, and prove the following edge-spectral supersaturation results for triangles and books. (a) Every graph $G$ with $m\ge 3$ edges and $\lambda(G) \ge 1 + \sqrt{m-2}$ contains at least $m-2$ triangles, with equality if and only if $G = K_3 \vee \tfrac{m-3}{3} K_1$. This can be viewed as the third-layer supersaturation in the jump phenomenon, after the first layer $t(G) \ge \lfloor \tfrac{1}{2}(\sqrt{m}-1) \rfloor$ proved by Ning and Zhai, and the second layer $t(G) \ge \tfrac{m-1}{2}$ by Zhang and Zhai. (b) Every $m$-edge graph $G$ satisfies $t(G) \ge m\bigl(\lambda - \sqrt{m}\,\bigr)$, with equality if and only if $G$ is complete bipartite. Consequently, $\lambda(G) \ge \sqrt{m} + q$ forces $t(G) > q m$ for every real $q > 0$. This is an edge-spectral counterpart of the Lovász–Simonovits theorem, and it improves the Bollobás–Nikiforov bound $t(G) \ge \tfrac13 \lambda(\lambda^2 - m)$ in the range $\sqrt m \le \lambda(G) \le 1.3\sqrt m $. (c) Every $m$-edge Nosal graph $G$ contains a book of size greater than $\tfrac14 \sqrt{m}$. This improves two recent results on the booksize constant: $\tfrac{1}{24}$ proved by Li, Liu and Zhang, and $\tfrac19$ by Zhai, Li and Lou. This narrows the gap toward the conjectured optimal constant $\tfrac13$. (d) Every $m$-edge Nosal graph $G$ contains at least $\bigl(\tfrac{1}{8} - o(1)\bigr) m$ copies of the kite $C_4^+=B_2$, and the constant $\tfrac18$ is best possible. This determines the sharp asymptotic constant for counting $C_4^+$ and strengthens the $\Omega(m)$ bound of Li, Liu and Zhang.

Two graphs are degree-similar if their adjacency matrices and degree matrices are simultaneously similar. Godsil and Sun asked whether non-isomorphic degree-similar unicyclic graphs exist. We prove that they do not exist. Every graph degree-similar to a unicyclic graph is isomorphic to it. The proof uses a symbolic leaf-peeling algorithm to recover the rooted trees attached to the unique cycle, and a rigidity lemma for colored cycles to identify automorphisms of the cycle.

In this article, we develop an $r-$uniform analogue of the classical Kövári–Sós–Turán inequality. This yields an alternative proof of Erdős's classical upper bound for complete $r$-partite $r$-uniform hypergraphs. More precisely, we establish that for finite non-empty sets $A_{1},\ldots,A_{r}$ with $|A_{1}|\leq\cdots\leq|A_{r}|$ and sufficiently large positive integer $n$, \[\mathrm{ex}(n,\mathbb{K}^{(r)}[A_{1},\ldots,A_{r}])=O\left(n^{r-\frac{1}{|A_{1}|\ldots|A_{r-1}|}}\right).\] Our proof develops an analytic counting framework based on repeated applications of Hölder's inequality and the enumeration of configurations through multiple sums. The argument combines the principle of inclusion–exclusion with a discrete analogue of Fubini's theorem to obtain recursive estimates for extremal quantities. This provides a unified analytic perspective on the generalized Erdős box problem.

Euler's partition theorem was an important step in the establishment of Integer Partitions as a field in its own right. It has attracted many great mathematicians' attention, including Sylvester, and has foreshadowed a number of further results. Another deep result in Partition Theory is the beautiful Lecture Hall Partition Theorem, which is a finite version of Euler's identity. In this work, we introduce the lecture hall length to partitions into distinct parts, and devise a bijection from partitions into odd parts to partitions into distinct parts, which unifies many results together. Various recent theorems regarding the minima excludant and block index become immediate consequences of this bijection. We also pose an interesting conjecture which may uncover a deep relation between Euler's partition theorem and lecture hall partition theorem.

Let $\ell(\Mat_n(F))$ denote the length of the full matrix algebra for a field $F$, i.e. the largest of the least word length needed to span $\Mat_n(F)$, over all generating sets $S$ of $\Mat_n(F)$. Šitov proved the general estimate $$ \ell(\Mat_n(F)) \leq 2n\log_2 n+4n-4. $$ The purpose of this paper is to obtain a log-log saving, and prove that for every $n>1$, $$ \ell(\Mat_n(F)) \leq 2n\log_2 n-2n\log_2\log_2 n+5n. $$ A theorem of Specht gives a word-criterion for unitary similarity of complex $n\times n$ matrices. The trace argument of Freedman–Gupta–Guralnick, as used by Pappacena, shows that any upper bound on $\ell(\Mat_n(F))$ can be used to bound the terseness $\tau(n)$, i.e. the least upper bound for the length of words needed in Specht's theorem. Thus, for $n> 1$, $$ \tau(n)\leq 4n\log_2 n-4n\log_2\log_2 n+10n+1. $$

For a set $A\subseteq\mathbb{N}_0$, let $r_1(A,n)$ denote the number of solutions of the equation $a+a^{\prime}=n$ with $a,a^{\prime}\in A$, and let $r_2(A,n)$ denote the number of such solutions subject to $a\le a^{\prime}$. These functions are called additive representation functions (as first considered by Erdős, Sárközy and Sós). In this paper, we resolve two problems posed by Sárközy and Sós in 1997. First, if $A$ is infinite and $r_2(A,2m+1)\ge r_2(A,2m)$ for every sufficiently large $m$, then the complement of $A$ is finite. This gives a negative answer to Problem 3.1 in~\cite{SarkozySos1997}. Secondly, there exist an arithmetic function $f$ satisfying $f(n) \to \infty$, $f(n+1) \ge f(n)$ for $n > n_0$, and $f(n) = o\left(\frac{n}{(\log n)^2}\right)$, and a set $A$ such that \( |r_1(A,n) - f(n)| = o((f(n))^{1/2}) \) holds on a sequence of integers $n$ whose density is $1$. This gives a positive answer to Problem 3.3 in~\cite{SarkozySos1997}.

The Erdős–Rado sunflower conjecture asserts that, for every fixed integer $r\ge 3$, there is a constant $K(r)$ such that every $\ell$-uniform family with more than $K(r)^{\ell}$ members contains an $r$-sunflower. We prove this conjecture for families of bounded VC-dimension.

Let $a(n)$ be the sequence A028342 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)x^n/n! = \prod_{i\ge1}(1-x^i)^{-1/i}$. Equivalently, $a(n)$ counts permutations of an $n$-element labeled set in which every cycle is assigned one divisor of its length, where a cycle of length $m$ has $d(m)$ choices, $d(m)$ being the number of positive divisors of $m$. We prove a family of congruences for $a$, conjectured by Peter Bala. They state that $k \mid a(n+k)+a(n)$ for odd $k$, that $k \mid a(n+k)-a(n)$ for $k\equiv 0,2,6 \pmod 8$, and that $k \mid 2(a(n+k)-a(n))$ for $k\equiv 4\pmod 8$. The proof first establishes a product congruence $a(n+k)\equiv a(n)a(k)\pmod k$, and then computes $a(p^r)\bmod p^r$ for each prime power by counting the colored permutations fixed by a subgroup of order $p$.

Let $L(t)$ denote the largest number of edges induced by $t$ vertices from two vertex layers of a hypercube. We show that $$\frac14 t\log_2 t+\frac18 t\log_2\log_2 t-O(t) \leq L(t) \leq \frac14 t\log_2 t+ (1+o(1))t\log_2\log_2 t.$$

We study the behaviour of heat semigroups on quantum graphs under Benjamini-Schramm convergence. For quantum graphs with uniformly bounded geometry, equipped with continuity and Kirchhoff vertex conditions, we show that the heat semigroup, transported to a common Hilbert space by a canonical breadth-first identification of the edges, depends continuously on the underlying rooted quantum graph with respect to a local Benjamini-Schramm-type metric. As a consequence, root-averaged pairings of the semigroup converge along Benjamini-Schramm convergent sequences of finite quantum graphs. Combining this with a Trotter-Kato-type approximation of the semigroup by semigroups on metric balls, we obtain a double-limit theorem interchanging the truncation radius and the graph limit.

Fulek defined the $0$-$1$ matrix \[ L_3=\begin{pmatrix} 1&0&0&1&0\\ 0&0&0&0&1\\ 0&1&1&0&0 \end{pmatrix} \] and asked whether $\text{ex}(n,L_3) = O(n)$. We prove that every $r\times s$ $0$-$1$ matrix avoiding $L_3$ has at most $27r+2s$ $1$ entries. Fulek's general lower bound construction has $6n-8$ $1$ entries, so \[ 6n-8\leq \text{ex}(n,L_3)\leq29n \] for $n\geq5$. The same argument applies to an infinite family. If $Q_{a,b,k,\ell}$ is the light three-row matrix with column word $1^a3^k1^b2^\ell$, where $a,b,\ell\geq1$ and $k\geq2$, then \[ \text{ex}(r,s,Q_{a,b,k,\ell}) \leq\bigl(5(k-1)(4b+1)+a+b+\ell-1\bigr)r+2s. \] This verifies a conjecture of Pettie and Tardos on linear light patterns for an infinite family that includes the previously unresolved weight-five pattern $L_3$. The proof assigns matrix entries to edges of a bar $1$-visibility hypergraph, cuts gaps to control the multiplicity of these edges, and charges the cuts to a noncrossing graph on the rows.

Let $G=(V,E)$ be a finite simple graph of order $n\geq 1$, and let $\ell:V\to\{0,1\}$ be a prescribed parity labeling. A set $S\subseteq V$ is called $\ell$-admissible if $d_S(v)\equiv \ell(v)\pmod 2$ for every $v\in S$, where $d_S(v)=|N_G(v)\cap S|$. Let $h_\ell(G)$ be the maximum order of an $\ell$-admissible set and let $f_{\rm oe}(G)=\min_\ell h_\ell(G)$. For $x\in\mathbb R$, define the weighted counting polynomial $$ M_{\ell,x}(G)=\sum_{S\in {\cal A}_\ell(G)}x^{|S|}, $$ where ${\cal A}_\ell(G)$ is the collection of all $\ell$-admissible sets in $G$. For $R\subseteq V$, let $z_\ell(R)$ be the number of vertices $v\in V\setminus R$ for which $d_R(v)\equiv\ell(v)\pmod 2$. We prove the exact identity $$ M_{\ell,x}(G) =2^{-n}\sum_{R\subseteq V} x^{|R|}(2+x)^{z_\ell(R)}(2-x)^{n-z_\ell(R)-|R|}. $$ If $G$ has no isolated vertices, then, for every $\ell$ and every $x\in(0,2)$, $ M_{\ell,x}(G)>x^{n/2}(4-x^2)^{n/4}. $ Combining this estimate with a binary-entropy upper bound and optimizing $x$ gives $$ f_{\rm oe}(G)>c_*n>\frac{2n}{21}, $$ where $c_*\approx0.095862615$. Ferber and Krivelevich (Adv. Math. 2022) proved that $h_{\mathbf{1}}(G)\ge 10^{-4}n$, where $\mathbf{1}$ is the all-one labeling. Since $h_{\mathbf{1}}(G)\ge f_{\rm oe}(G)$, our result improves coefficient in their bound by almost three orders of magnitude, and does so simultaneously for every labeling.

In the damage variant of Cops and Robber, the damage number \(\dmg(G)\) is the number of distinct vertices damaged by the robber under optimal play, with one cop minimizing and the robber maximizing this number. We introduce the zombie damage number \(\zdmg(G)\), obtained by requiring the pursuer to move at every turn along a shortest path toward the survivor. The parameter therefore measures the cost of geodesic pursuit when the objective is containment rather than capture alone. We prove that \(\dmg(G)\leq\zdmg(G)\) and characterize the graphs with \(\zdmg(G)=0\). Geodesic pursuit incurs no additional damage on trees, and we determine the parameter exactly for paths, cycles, split graphs, and complete multipartite graphs. In particular, \(\zdmg(C_n)=n\) for \(n\geq5\), while \(\zdmg(K_{n_1,\ldots,n_k})=n_1+n_2-2\) when \(n_i\geq2\) for every \(i\). We also develop a nonbacktracking trace argument for sparse graphs. If \(G\) is connected, \(\delta(G)\geq2\), and \(g(G)\geq5\), then every vertex is damaged, and hence \(\zdmg(G)=n(G)\). The same argument gives the sharp bound \(\zdmg(G)\geq g(G)\) for every connected graph of finite girth at least five. It follows that fully subdividing each edge of a connected graph of minimum degree at least two produces a graph with maximum zombie damage. These results yield an unbounded separation from the ordinary damage number. Most of the questions leading to these results, as well as the open conjectures, arose through Theo-Conjecture, an advisor-supervised discovery loop combining automated conjecturing, exact computation, language-model-assisted exploration, and human mathematical judgment.

The cycle double cover conjecture states that every bridgeless graph has a list of cycles such that every edge is in exactly two of them. In July 2026, OpenAI announced a proof. This exposition presents the proof with slight modifications intended to make it more accessible.

We show that the Erdős-Rogers function $f_{s,s+1}(n)$ satisfies $$f_{s,s+1}(n) = \Theta( \sqrt{n \log n} )$$ for every $s \ge 2$. More precisely, we construct a $K_{s+1}$-free graph on $n$ vertices in which every set of at least $C(s)\sqrt{n \log n}$ vertices contains a copy of $K_s$ for some constant $C(s)$, which implies the upper bound. The matching lower bound follows from a theorem of Joret, Micek, Reed and Smid on the clique chromatic number of a graph.

We prove that every $n$-vertex simple planar graph with no copy of $C_8$ has at most \[ \frac{69}{25}(n-2) \] edges, for every $n\ge 8$. This improves the best known bound \[ \frac{323}{108}n-6 \qquad \text{for every } n\ge 27. \]

We study the flip dynamics on perfect matchings of graphs, where a flip consists of replacing the edges of a perfect matching along an even cycle with the complementary alternating edges. In particular, we want to bound the minimum length of cycles such that any two perfect matchings are related by flips of such cycles. Given a finite graph $G$, we consider the families of decorated graphs obtained by replacing each vertex of $G$ with a decoration satisfying suitable conditions on the existence of perfect matchings in its subgraphs. We prove a general upper bound for this family. We then obtain stronger bounds for two families of decorations: clique decorations and Fisher decorations, the latter under the assumption that the underlying graph is planar. In both cases, the bounds are independent of the sizes of the decorations.

Parking functions correspond with preferences of $n$ cars which enter sequentially to park on a one-way street where (1) each car parks in the first available spot greater than or equal to its preference and (2) all cars successfully park. We generalize parking functions to parking completions: Here, we are given that some cars have already parked in a set of spots, which are indexed in a sequence $\mathbf{t}$. We then consider a preference list $\mathbf{c}$, where length of $\mathbf{t}$ + length of $\mathbf{c}$ = $n$. If all cars can park, we say that $\mathbf{c}$ is a parking completion. Adeniran et al. (2020) state an open problem which proposes a connection between the number of parking completions to the volumes of Pitman-Stanley polytopes by explicit computation on small values of $n$. In this paper, we provide a solution to this open problem by proving a theorem which explains that the set of parking completions is the set of $\mathbf{x}$-parking functions.

Motivated by the celebrated Lonely Runner Conjecture, we study a variant in which the runners have time-dependent velocities. Let $n \geq 3$ runners start from the same point on the unit circle, where each runner $i\in[n]$ has a locally integrable velocity function $\nu_i\in L^1_{\mathrm{loc}}(\mathbb{R}_{>0})$. Assume that their velocities are strictly ordered almost everywhere and that the relative distance between every pair diverges. We prove that each of the slowest and fastest runners is at a distance strictly larger than $2^{-n+1}$ from every other runner at some time. Moreover, we show that the distance $2^{-n+1}$ is optimal. On the other hand, we construct examples in which every intermediate runner remains arbitrarily close to another runner at all times. As a consequence, we also obtain a sharp nonlinear analogue of a classical theorem of Schoenberg on billiard ball motion in the unit cube.

An $n \times n$ matrix $W$ with exactly $w$ non-zero entries taken from the set of $k^{\rm th}$ complex roots of unity in each row and column satisfying $WW^{\ast} = wI_n$ is a complex generalised weighing matrix $CGW(n,w;k)$. We study such matrices through the centraliser algebras of monomial representations of finite groups. Using an exhaustive search over the linear characters of Schur covers, we classify, up to monomial equivalence, the complex generalised weighing matrices admitting a primitive group of rank at most five and degree at most $80$ acting by strong automorphisms, for coefficient orders $k \leq 6$, with partial results for larger degrees $100$. The census recovers known infinite families related to projective and affine finite geometries, describes infinite families related to Hamming schemes and settles the existence of some small open cases enumerated in the literature. We construct quantum error-correcting codes from the these matrices and determine their minimum distances exactly in all cases.

Let $D_n$ be the disjointness graph on the nonempty subsets of $[n]$, whose independent sets are exactly the intersecting families on $[n]$. We study the weighted independent-set polynomial $W(n)=\sum_F\prod_{S\in F}w(S)$, the sum running over these families, for the doubly exponential weight $w(S)=2^{2^{n-|S|}}-1$. The kernel-bearing (trivial) part $Z_\cap(n)$ is exact by inclusion-exclusion and satisfies $Z_\cap(n)\sim n\cdot 2^{3^{n-1}}$. For the kernel-free remainder we prove the exact prefactor $R(n)=(3/4+o(1))n\cdot 2^{3^{n-1}-2^{n-1}+2}$, whence $\log_2(Z_\cap(n)/R(n))=2^{n-1}-2+\log_2(4/3)+o(1)$, an additive $o(1)$, not merely a leading-order one. The engine is a second-level extremal theorem: among kernel-free maximal linked systems other than the $n$ one-flip stars, the largest weight exponent is $3^{n-1}-3\cdot 2^{n-2}+6$, a fixed gap $2^{n-2}-4$ below the maximum, with the extremisers classified exactly. None of this is special to the weight: for $w_B(S)=B^{B^{n-|S|}}-1$ with integer $B\ge 2$ the same stars dominate, the near-extremal families sit a gap $B^{n-2}-B^2$ below, and the prefactor is $1-B^{-B}$. The combinatorial input is the $p$-biased extremal problem for non-trivial intersecting families: $M_2(n,p)=p-pq^{n-1}+qp^{n-1}$ for all $n\ge 3$, $0<p\le 1/2$, $q=1-p$. This first level is essentially known: the extremal family is the Wheel coterie of Peleg and Wool, and at $p=1/Q$ the statement, with its maximiser classification, is the case $r=n$ of Borg's Hilton-Milner theorem for signed sets (2013). We give a short self-contained Erdős-Ko-Rado proof, uniform in real $p\in(0,1/2]$, whose layer-two rigidity feeds the second level. The novelty claimed lies at the second level and in the prefactor, where the classification cannot be read off the layer profile alone: at $n=5$ one profile carries two non-isomorphic types of extremisers.

For an explicitly represented finite empirical model, deciding whether the contextual fraction is strictly below one is NP-complete, while the standard exact linear program has one column for every global assignment. We identify a permutation-transport class in which this global problem collapses to a fixed-point calculation. Let a connected permutation gain graph act on a finite state set $O$, let $H \leq{ \rm Sym}(O)$ be its holonomy subgroup, let $F = {\rm Fix}(H)$, and let $p$ be an $H$-invariant root distribution. For the induced empirical model, \[ {\rm NCF}(e)=p(F),\qquad {\rm CF}(e)=1-p(F). \] Consequently, compatibility, $F$, and ${\rm CF}(e)$ are computable in $O(|O|(|V|+|E|))$ arithmetic and table operations. For every finite simple $2$-edge-connected graph, any deterministic exact algorithm in the explicit permutation-table query model requires at least $(|O|-1)|E|$ probes in the worst case, making the dependence on the input tables optimal up to constant factors. With a fixed spanning tree, chord insertions and deletions require $O(|O|)$ worst-case time, or time proportional to the moved-set representation, while compatibility and contextual-fraction queries take $O(1)$ time. Finally, for common-marginal realizable binary constraint languages, the support threshold ${\rm CF} < 1$ is polynomial-time equivalent to the associated finite-domain constraint-satisfaction problem and therefore inherits the Bulatov–Zhuk dichotomy. The results identify a query-optimal and dynamically maintainable tractability island inside the general contextual-fraction problem.

For a graph $G$, let $ \lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of $G$. We prove that for all looped graphs $G$ on $n$ vertices, \[\lambda_1(G) + \lambda_2(\overline{G}) \le \frac87 n. \] Our method also gives a short new proof of the Nordhaus-Gaddum result for the spectral radius proved by Terpai that $\lambda_1(G) + \lambda_1(\overline{G}) \le \frac43n - 1$. We investigate the asymptotic maximum of \[ \lambda_i(G)+\lambda_j(\overline G) \] for fixed $i$ and $j$. We prove general bounds on $\lambda_i(G) + \lambda_{j}(\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\lambda_{n-i+1}(G) + \lambda_{n-j+1}(\overline{G})$ for fixed $i$ and $j$. We also show the close relation of these Nordhaus-Gaddum type problems to recent work on the maximum spectral gaps of graphs by Brooks, Linz and Lu.

Let $T(n,k)$ be the minimum total surface area of $k$ axis-aligned boxes with integer sides and pairwise distinct dimension multisets whose union is the cube $[0,n]^3$. We determine the column $k=6$ completely: $T(n,6)=8n^2+2n+12$ for $5\le n\le 9$, and $T(n,6)=8n^2+2n+6$ for all $n\ge 10$, together with the exceptional values $T(3,6)=100$ and $T(4,6)=148$. The threshold $n=10$ equals $1+2+3+4$, the least possible sum of four distinct stick lengths, and the general law holds: for every $k\ge 4$ and every $n\ge (k-2)(k-1)/2$, $T(n,k)=8n^2+2n+2(k-3)$, with thresholds at the triangular numbers. Three structural results support and extend these values. First, a fault-free law: the minimum internal interface of a partition of the cube into six boxes with no fault plane is exactly $2n^2+n$ for all $n\ge 3$ (OEIS A014105), proved by an exact accounting of spanning pieces, floating pieces and cube corners. Second, a reduction theorem: within an explicit range, the three-dimensional problem collapses to a two-dimensional one, $I(n,k)=n^2+W^*(n,k-1)$, where $W^*(n,m)$ is the minimum internal wall of a tiling of the $n\times n$ square by $m$ rectangles of pairwise distinct dimensions; the key ingredient is an unconditional slab lemma. Third, a doubling law in the middle regime of the 2D problem: $W^*(n,4)=n+4$ for $n=4,5$ and $W^*(n,5)=n+6$ for $4\le n\le 9$, proved by finite case trees; via the reduction theorem this gives computer-free proofs of the middle regimes of the columns $k=5$ and $k=6$. The lower bound for the main family does not use the distinctness of the pieces.

Every polynomial with real non-negative coefficients yields a finite probability distribution after normalization. The Ehrhart $h^*$-polynomial of a lattice polytope $P$ is a non-negative integer polynomial that encodes the integer-point counts for positive integer dilations of $P$. We study the corresponding finite distributions, which we call $h^*$-distributions. We determine the mean and variance of these distributions, establish a connection between higher moments and Ehrhart polynomial coefficients, and study their cluster points in the $d$-dimensional probability simplex. We consider the special case of real-rooted $h^*$-distributions, applying existing tail bounds to obtain new linear inequalities for the coefficients of real-rooted $h^*$-polynomials arising from reflexive polytopes. We conclude by establishing sufficient conditions under which a sequence of real-rooted $h^*$-distributions is asymptotically normal, and we apply our results to various families of polytopes, including zonotopes and Pitman-Stanley polytopes.

We study the class $\#\mathsf{L}$ of functions counting accepting paths of non-deterministic log-space Turing machines and construct methods to prove containment in $\#\mathsf{L}$. We prove that a large number of classical combinatorial and number theoretic functions belong to this class: classical functions from enumerative combinatorics (multinomial coefficients, Catalan numbers, linear extensions of trees, Stirling numbers, etc), algebraic combinatorics (number of standard Young tableaux, etc), discrete geometry, number theoretic functions, representation theoretic multiplicities in a large class of cases. We show that $\mathrm{GL}_2$-plethysm coefficients of bounded length outer partition can be counted by log$^2$-space polytime verifiers. We pose numerous questions and conjectures on $\#\mathsf{L}$ containment and its generalizations, that suggest venues for conditionally disproving $\#\mathsf{P}$-completeness. While studying which combinatorial functions are in $\#\mathsf{P}$ provides a formal way of (dis)proving the existence of combinatorial interpretations, the lower class $\#\mathsf{L}$ serves as an analogue for functions computable in polynomial time.