Papers

We study six Eulerian-type polynomial families. We prove that the descent polynomials of derangements are real-rooted, settling the derangement part of a conjecture of S.~Fu, Z.~Lin, and J.~Zeng. The proof uses a compatible-pair recursion and finite-symbol stability. We also resolve the real-rootedness conjecture in OEIS \oeis{A335340}, strengthen the known rowwise real-rootedness of an even-top descent family to consecutive strict interlacing, and prove real-rootedness, consecutive interlacing, and real-rooted gamma-polynomials for U.~Shankar's super-Eulerian polynomials. A differential recurrence gives consecutive weak interlacing for ternary words counted by increasing runs. Finally, we prove stability of the peak-value refinement and consecutive interleaving of its positive weighted diagonals, settling a conjecture of P.~Alexandersson and O.~Nabawanda.

The independent set polytope, or stable set polytope, of a graph $G$ is the 0/1-polytope defined by the convex hull of the characteristic vectors of all independent sets of $G$. We present a simple algorithm for computing a Hamilton path on the independent set polytope of a given $n$-vertex graph $G$ with amortized delay $\mathcal{O}(n)$. The independent sets are listed such that two consecutive sets differ either in removing a vertex, or adding a vertex and removing its neighbors from the independent set, i.e., the symmetric difference between two consecutive independent sets induces a star in $G$. As applications of this result, we obtain an algorithm to compute a Hamilton path on the matching polytope of an $m$-edge graph $G$ with worst-case delay $\mathcal{O}(m)$, which lists all matchings of $G$ in such a way that the symmetric difference between two consecutive matchings is a path on at most three edges. Furthermore, we obtain an algorithm to compute a Hamilton path on the chain polytope and order polytope of an $n$-element poset $P$ with amortized delay $\mathcal{O}(n)$, which lists all antichains of $P$ or all ideals of $P$, respectively, by star exchanges. Our algorithms are derived from the generic framework proposed by Merino and Mütze (FOCS'23+SICOMP'24) for computing Hamilton paths on arbitrary 0/1-polytopes, which uses a linear optimization procedure as a black box. Our algorithms bypass solving the computationally intractable maximum weight independent set problem by a simple and purely combinatorial greedy rule.

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $N$ such that every red/blue edge-coloring of the complete graph $K_N$ on $N$ vertices contains a monochromatic copy of $H$. The Ramsey multiplicity $M(H,n)$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_n$. It is called threshold Ramsey multiplicity if $n=r(H)$, and denoted by $m(H)$. The only previously known general infinite family for which $m(H)$ has been determined is stars, due to Harary and Prins (1974). Let $C_k$ denote a cycle on $k$ vertices. Conlon, Fox, Sudakov, and Wei (2022) conjectured that $m(C_k)=(k-1)!/2$ for every sufficiently large odd integer $k$. In this paper, we determine $M(C_k,r(C_k)+\ell)$ for every fixed nonnegative integer $\ell$ and all sufficiently large odd $k$, and characterize all extremal colorings, thereby confirming the conjecture. This is also a second general infinite family for which $m(H)$ has been determined.

Fix $k\ge 3$, and let $f_k(N)$ be the largest harmonic sum of a subset of $[N]$ containing no $k$ distinct integers with a common pairwise least common multiple. We prove that $f_k(N)=(\log N)^{\gamma_k+o(1)}$ for a well-defined exponent $\gamma_k\in(0,1]$. Following the weighted-pressure idea of Chojecki, we give a self-contained proof of variational formulas for $\gamma_k$ in terms of weighted sunflower-free families. We then eliminate the continuous weight: if $M_k(n,r)$ is the largest size of an $r$-uniform $k$-cosunflower-free family on $[n]$, then \[ \gamma_k=\sup_{n\ge1,\ 1\le r\le n}\frac{r}{en}M_k(n,r)^{1/r}. \] This finite-block formula yields a direct transfer principle from uniform set-system constructions, recovers the Tang–Zhang bounds, and gives $0.438899\ldots<\gamma_3\le0.889881\ldots$.

We study external difference families arising from cyclotomic classes in finite fields from the viewpoint of a fixed number of blocks. For families consisting of even-indexed cyclotomic classes, the EDF condition can be expressed in terms of relations among cyclotomic numbers. We first study the two-block case and recover a classical characterization in terms of quadratic forms. Our main result shows that, for a prime $p=12k+1$, the family $\{C_0^6,C_2^6,C_4^6\}$ forms an EDF in $\mathbb{F}_p$ if and only if $k$ is a square. The proofs combine symmetry relations of cyclotomic numbers with their explicit evaluations.

We establish five families of integer-parameter extensions of Ramanujan's partial theta identities. The families follow from a common two-parameter specialization of Andrews' transformation, for which we give an independent proof based on a $q$-difference recurrence and a boundary estimate. Specializations of the integer parameter recover six identities from Ramanujan's lost notebook. As an application, a residue argument applied to the fifth family gives an integer-parameter extension of Lovejoy's residual identity, from which we construct a corresponding family of conjugate Bailey pairs.

Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function $\sum_{n\ge0} a(n)t^n/n! = 1+\sum_{m\ge1}\prod_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\ge1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $\varphi(k)$. We prove the stronger statement that the Carmichael function $\lambda(k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(np^r+i)\equiv a(np^{r-1}+i)\pmod{p^r}$ for every $i\ge0$, every prime $p$, and all $n,r\ge1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\in\mathbb Z[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.

Let $(x_n)_{n\geq1}$ be a sequence of distinct points on the unit circle. An $r$-span is the total length of $r$ consecutive gaps determined by the inserted points. Write $M_n^{(r)}$ and $m_n^{(r)}$ for the largest and smallest $r$-spans after the first $n$ insertions. We prove that there is an absolute constant $c>0$ such that, for every sufficiently large $r$, \[ \limsup_{n\to\infty}\bigl(nM_n^{(r)}-r\bigr) \geq c\sqrt{\log r}, \qquad \limsup_{n\to\infty}\bigl(r-nm_n^{(r)}\bigr) \geq c\sqrt{\log r}, \] and \[ \limsup_{n\to\infty}\frac{M_n^{(r)}}{m_n^{(r)}} \geq 1+\frac{\log r}{100r}. \] Thus all three asymptotic conjectures made by de Bruijn and Erdős in 1949 are resolved. The ratio bound matches the upper bound of Clément and Steinerberger up to an absolute constant and answers a question of Brethouwer. The proofs compare interval counts at nearby times. Pointwise control leads to a one-dimensional sequence-discrepancy argument for the ratio, while averaged control and Halász's planar $L^1$ discrepancy theorem give the two one-sided conclusions.

We present tensor network representations for discrete maximum entropy distributions under expectation constraints. To this end, we introduce Computation-Activation Networks (CompActNets), a tensor network architecture that subsumes exponential families. By leveraging the geometry of the convex polytope of realizable expectation vectors, we represent any maximum entropy distribution in the same architecture. We exploit the fact that proper faces of this polytope correspond to the boundary closure of exponential families, which restricts the distribution's support. We then derive explicit representations for the support within the CompActNet architecture. The proposed framework suggests tensor network ranks as complexity measures for faces. Finally, a case study on Boolean statistics links the geometry of 0/1-polytopes directly to propositional formulas.

A corollary of Hall's marriage theorem is that a sufficient condition for a list $(V_1, \ldots ,V_m)$ of sets to have a system of distinct representatives is that $|V_i|\ge deg_{\{V_1, \ldots ,V_m\}}(v)$ for every $i\in [m]$ and $v \in \bigcup_{i\in [m]}V_i$. This we dub a global condition. A folklore result is that a local condition - that the inequality holds for pairs $i,v$ for which $v \in V_i$ - suffices. These are special cases of a general type of results - large sets, whose elements are sparse in some sense, have a system of representatives that is independent in a related graph. We study two such scenarios, in both of which each $V_i$ is replaced by a $k$-uniform hypergraph $H_i$, the representatives are hyperedges, and distinctness is replaced by disjointness. In one setting the sparsity is measured by the degrees of vertices in the hypergraphs, in the other by the degrees of vertices in the line graph. The proofs use the topological version of Hall's theorem. In particular, we shall use a lower bound on the topological connectivity of the independence complex of a graph, defined by vector representations. We also provide short proofs of the local infinite version, known as the “Milner-Shelah theorem”.

We study near alternating sign matrices with prescribed row and column weights. After recalling the basic definitions and the necessary conditions coming from the Gale–Ryser theorem, we give a graph-theoretic characterization of the existence problem. More precisely, we show that a near alternating sign matrix with prescribed row and column weight sequences exists if and only if there exists a binary array with the same weights whose associated adjacency graph is bipartite. This reformulation shows that the Gale–Ryser conditions are not sufficient in general. We then formulate a natural conjecture up to permutations of the row and column weight sequences, and present some positive results, based on convex binary arrays and composition constructions.

For every $n\geq1$, we classify monotone rotation-invariant real-valued valuations on closed spherical convex sets, without assuming continuity or measurability. On proper sets, namely those contained in an open hemisphere, these are precisely the nonnegative linear combinations of the normalized spherical quermassintegrals. On all closed spherical convex sets, they are precisely the linear combinations of the spherical intrinsic volumes with nonnegative, nondecreasing coefficients. The representations are unique, and all such valuations are continuous and invariant under the full orthogonal group. In hyperbolic space, an isometry-invariant real-valued valuation on compact convex sets is continuous if and only if it is a linear combination of the Euler characteristic and the hyperbolic quermassintegrals. This representation is unique. Monotonicity is equivalent to nonnegative coefficients and implies continuity. If monotonicity is required only between nonempty sets, the Euler coefficient is unrestricted in the proper spherical and hyperbolic cases, whereas the classification on all closed spherical convex sets is unchanged. We also obtain corresponding classifications for valuations on closed convex cones that vanish at the zero cone and monotone classifications on compact projectively convex sets contained in an affine chart of real elliptic space.

Every quadratic Hessian slice of a Lorentzian polynomial yields bounded monomial ratios among the normalized coefficients of the polynomial. We determine exactly for which pairs $(n,d)$ these quadratic-slice ratios generate the full bounded-ratio cone for every $M$-convex support $S\subseteq\Delta_n^d$. For $d\geq 2$, this quadratic local-to-global principle holds universally if and only if \[ n\leq 3,\qquad d=2,\qquad\text{or}\qquad (n,d)=(4,3). \] In every remaining case, the principle fails already for Lorentzian polynomials with full support: for cubics in $n\geq 5$ variables and for polynomials of degree $d\geq 4$ in $n\geq 4$ variables. We identify the two minimal obstructions, at $(n,d)=(4,4)$ and $(n,d)=(5,3)$, and propagate them throughout the failure region by degree and variable aggregation.

A Hadamard difference set (HDS) $D$ of order $u^2$ in an abelian group $G$ satisfies $|\chi(D)|=u$ for every nontrivial character $\chi$ of $G$. We call such a character value naive if it is divisible by $u$, i.e., if it is equal to $u$ times a root of unity. All previously known abelian HDSs only have naive character values. We show that for $d\ge 1$ and $u=3d$, a group $Z_3^2\times H$, with $H$ an abelian group of order $2^{2d+2}$, contains a HDS of order $u^2$ with non-naive character values if and only if $8\le\exp H\le 2^{d+2}$. All difference sets obtained are new. The proof rests on a specific HDS in $Z_3^2\times Z_8\times Z_2$, a covering extended building set on $Z_3^2\times Z_8\times Z_4$, and a variation of the Davis-Jedwab recursive construction.

It was recently shown that every connected graph of order $n \geq 5$ and size $m$ satisfies $\chi(G) \leq \left\lceil \frac{m}{\sqrt{n}} \right\rceil$, and it was asked whether the stronger inequality $\chi(G) \leq \left\lceil \frac{m}{2\sqrt{n}} \right\rceil + 1$ holds for every connected triangle-free graph. In this paper, we answer this question in the affirmative. In fact, we prove that every connected triangle-free graph $G$ with $G \not\cong C_5$ satisfies $\chi(G) \leq \left\lceil \frac{m}{\sqrt{5.5n}} \right\rceil + 1$, where the constant $\sqrt{5.5}$ cannot be replaced by any constant greater than or equal to $\sqrt{6}$, and the equality holds for the Grötzsch graph and for every odd cycle of length between $7$ and $21$.

We study the asymptotically maximal possible number of integer solutions to the linear equation $ax+by+cz = 0$ with a fixed choice of $a, b, c \in \mathbb{Z}$ and variables $x, y, z \in S$ for some finite set $S\subset \mathbb{Z}$, as $|S|\to +\infty$. Define $\gamma_{a, b, c}$ to be the largest constant for which there are arbitrary large finite sets $S\subset \mathbb{Z}$ such that the number of solutions to $ax+by+cz=0$ with $x,y,z\in S$ is $\gamma_{a,b,c}|S|^2-o(|S|^2)$. We prove structural results for general $a, b, c$ and moreover, we show that $5/13\le \gamma_{1,1,-3}\le 1/2-\delta$ for some constant $\delta>0$. In addition we show that the limit as $a \rightarrow \infty$ of $\gamma_{1,1,-a}$ is equal to precisely $1/5$.

A matching $M$ of a $2$-connected graph $G$ is removable if $G-M$ is $2$-connected, extending Halin's classical notion of a removable edge. We prove that for every integer $d\ge 5$, every $2$-connected graph $G$ with $\delta(G)\ge d$ and $|V(G)|\ge 2d$ has a removable $d$-matching. This is best possible, since the complete bipartite graph $K_{d,\,n-d}$ on $n\,(\ge 2d)$ vertices has no $(d+1)$-matching. Consequently, our result gives a complete answer, for every $d\ge 5$, to a question of Li, Zhou, Fujita, and Mao on the maximum size of a removable matching guaranteed by the minimum degree. The previously best known bound, due to Li, Zhou, Fujita, and Mao and to Chu, Kim, and Park, guaranteed a removable $(d-2)$-matching under the same assumptions. Our proof is based on an analysis of minimal non-removable matchings, matchings that are not removable although all of their proper submatchings are.

Hadwiger conjectured in 1943 that every graph with no $K_t$ minor has chromatic number at most $t-1$. Delcourt and Postle proved that every graph with no $K_t$ minor has chromatic number $O(t\log\log t)$. We build on their result to improve this bound to $O(t\log\log\log t)$.

Using the strong discrete Morse theory developed by Fernández, we define the complex $\mathcal{SM}(K)$ of strong discrete Morse matchings on a simplicial complex $K$, as well as the pure subcomplex $\mathcal{SM}_{pure}(K)$ generated by the facets in $\mathcal{SM}(K)$ of maximal dimension. For most complexes $K$ these objects are proper subcomplexes of the complexes ${\mathfrak M}(K)$ and ${\mathfrak M}_{\textrm{pure}}(K)$ defined by Chari–Joswig using all Morse matchings on $K$. The homotopy types of the latter are not well-understood in general, but they are known when $K$ is the path $P_n$ with $n$ edges, the cycle $C_n$ with $n$ edges, the star $S_n$ with $n$ leaves, the $n$-simplex $\Delta^n$ ($n\le 3$), and the boundary $\partial\Delta^n$ ($n\le 3$). in this paper we compute the homotopy types of $\mathcal{SM}_{pure}(C_n)$ and $\mathcal{SM}(K)$ for $K=P_n,S_n,\Delta^n,\partial\Delta^n$ for all $n$. We also compute the homology of $\mathcal{SM}(C_n)$ for $n\le 21$.

Let $F$ be a function field over an algebraically closed field $K$ and $S$ a finite dimensional $K$-subspace of $F$. The square of $S$ is spanned by all products of pairs of elements in $S$. We conjecture that if $\dim S^2 \leq 3 \dim S - 4$, then $S^2$ must contain a Riemann–Roch space of dimension at least $2 \dim S - 1 + g$, where $g$ is the genus of $F$. This generalizes a theorem of Freiman from additive combinatorics, stating that small sumsets must contain long arithmetic progressions. We prove our conjecture in the case that $S$ is contained in a Riemann–Roch space of dimension at most $3/2\dim S + 1$. For the proof we study the annihilator of $S$ and introduce the notion of weight for linear forms.

In secure distributed matrix multiplication, a user wishes to compute the product of two matrices with the assistance of servers, in such a way that any $T$ colluding servers learn nothing about either matrix. Degree tables are a combinatorial tool for constructing polynomial codes for this problem and underlie several state-of-the-art schemes, including $\mathsf{GASP}_r$, $\mathsf{GASP}_{r,s}$, and $\mathsf{DOG}_{r,s}$. We introduce a periodic-gap framework for constructing degree tables that contains these three families as special cases and leads to two new constructions, $\mathsf{SHIFT}_{r,s}$ and $\mathsf{COVER}_r$. We determine their exact recovery thresholds and show that, in many cases, they outperform the current state of the art. We also prove a new lower bound on the recovery threshold of ordinary integer degree tables. In the balanced case, in which the partitioning parameters and the security parameter are equal, we sharpen this bound to match $\mathsf{SHIFT}_{r,s}$ up to lower-order terms, showing that it is asymptotically optimal among degree-table constructions.

We establish a uniform product-difference theorem for dense subsets of $\mathbb Z^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $\delta>0$ there exists an integer $K(\delta)\geq 1$ such that every set $E\subseteq\mathbb Z^2$ with upper Banach density $d^\star(E)\geq\delta$ satisfies \[ K(\delta)\mathbb Z \subseteq \{ab:(a,b)\in E-E\}. \] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\subseteq\mathbb Z$ and $E\subseteq\mathbb Z^2$, respectively, the sets \[ (E_1-E_1)^2-(E_2-E_2)^2 \quad\text{and}\quad \{x^2-y^2:(x,y)\in E-E\} \] contain nontrivial ideals of $\mathbb Z$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of $\mathbb Z^2$.

Let $s^+(G)$ denote the sum of the squares of the positive adjacency eigenvalues of a graph $G$. The square-energy conjecture of Elphick, Farber, Goldberg, and Wocjan, proved by Liu, Tang, and Zhang, gives a lower bound of $n-1$ for any connected graph of order $n$. We strengthen this bound to $s^+(G)\ge n$ for every connected graph $G$ of order $n$ with minimum degree at least two, unless $G$ is a cycle.

For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum \Theta, this gives dim P_a(F^{\square m}) = |m\Theta| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.

For every finite $P\subset\mathbb{R}^2$ we prove $|\{p\cdot q: p,q\in P\}|\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}G^{(2)}/(F^{(2)}G)| \le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: $\max_{p\in P}|p\cdot P| = O(|P|/\sqrt{\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.

We study cyclic orderings of all edges of a complete $k$-uniform hypergraph on $n$ vertices in which the binary incidence sequences of the vertices are cyclic shifts of a common word. The shifts are chosen independently, with no prescribed action on the vertices. For $2\leq k<n$, coprimality $\gcd(n,k)=1$ is known to suffice even when consecutive edges must differ by a single vertex exchange. We recall a short orbit construction and prove the converse for the first two nontrivial uniformities without any adjacency requirement. For $k=2$, an ordering exists exactly when $n=2$ or $n$ is odd; for $k=3$, exactly when $n=3$ or $3\nmid n$. The necessity proofs use reflected convolution identities and pair-intersection counts to constrain the vertex shifts to a torsion coset. For triples, multiplicity-preserving dilation and conditional prime-power capacity bounds complete the argument.

Let $a(n,k)$ be the number of $1324$-avoiding permutations of length $n$ with $k$ inversions. Linusson and Verkama proved $a(n,k)\le a(n+1,k)$ for $k\le2n-7$. We study the obstruction beyond that line: the residuals $\mathcal R_{\delta,n}$, namely the indecomposable, non-almost-decomposable avoiders at defect $\delta=k-2n+7$. Contracting maximal increasing consecutive runs reduces residuality to a quadratic equation on a finite family of skeletons. It follows that, for every fixed $\delta$, the eventual count has the form $|\mathcal R_{\delta,n}|=A_\delta n^2+B_\delta n+C_\delta$. Our central uniform result determines the quadratic coefficient at every defect: with $P(q)=\prod_{j\ge1}(1-q^j)^{-1}$, $\sum_{\delta\ge0}A_\delta q^\delta=4q^3(1+q)P(q)^2/(1-q)^2$. This is a formula for the leading coefficient of the residual count, not for the full count. The same structural estimates give computer-assisted proofs of $a(n,k)\le a(n+1,k)$ for every $n\ge1$ and $k\le2n+6$, and of regions whose width grows with $n$: for $n\ge2^{16},2^{18},2^{20}$ the defect may be as large as $\lfloor\sqrt n/4\rfloor$, $\lfloor\sqrt n/3\rfloor$, $\lfloor\sqrt n/2\rfloor$, respectively. More generally, every fixed $c<\sqrt2\log(5)/\log(68)$ is admissible for all sufficiently large $n$. The three added fixed defects $11,12,13$ use complete catalogue and rational-sum certificates supplied in the accompanying archival supplement. The leading-coefficient theorem is obtained from a complete finite classification of marked rank-three cores and all-parameter extension lemmas. We also determine the exact rank-three stabilization onset, while keeping it separate from the still unknown onset of the complete residual count. The unrestricted Claesson–Jel'inek–Steingr'imsson conjecture, the full residual polynomials, and the sharp global base-length bound remain open.

An old problem in combinatorial group theory asks, given a group $\Gamma$ and a subset $S \subseteq \Gamma$, when does there exist an ordering $s_1, \cdots, s_k$ of the elements of $S$ such that the partial products $\prod_{i = 1}^j s_i$, $1 \leq j \leq k$, are all distinct. If such an ordering exists, we call $S$ rearrangeable. There have been many conjectures about rearrangeable subsets, the most general being that for every group, every subset not containing the identity element is rearrangeable. We prove an asymptotic version of this: For any Group $\Gamma$ and any subset $S \subseteq \Gamma$, there exists a rearrangeable set $S' \subseteq S$ such that $|S'| = |S| - o(|S|)$. To do this we build upon the work of Bucić, Frederickson, Müyesser, Pokrovskiy, and Yepremyan focusing on the following problem of independent interest in Graph Theory. If $G$ is a d-regular properly colored directed graph does there exist a rainbow path of length $d - 1$? We establish an asymptotic version of this, proving that $G$ contains a rainbow path of length $d - o(d)$. This solves two problems given by Bucić, Frederickson, et al. and proves the above result on rearrangeable subsets of groups by considering the Cayley graph of $\Gamma$ with (not necessarily generating) set $S$.

An Eulerian orientation of a $4$-regular undirected graph (simple or multigraph) $G=(V,E)$ with $n=|V|$ vertices is an assignment of directions to its edges such that every vertex $v \in V$ has the same indegree and outdegree. In the present article, we improve the bounds on the number of Eulerian orientations for certain classes of connected, loopless $4$-regular graphs. The previous bound is due to M. Las Vergnas (1983) and is exactly $9\cdot 2^{n-3}$, which is a sharp bound for a certain family of multigraphs with $n\geq 4$. Here, we show that the number of Eulerian orientations for all biconnected $4$-regular multigraphs is at most $2^n+2$, which is also sharp. We exhibit families of graphs that attain this maximum value. For simple graphs, we prove an upper bound of $\mathcal{O}(3^{n/2})$ in the biconnected case and $\mathcal{O}(6^{n/3})$ for the separable case. Additionally, we provide a divide-and-conquer algorithm that leverages structural properties to compute the exact number of Eulerian orientations for separable graphs without exhaustive enumeration. Finally, we analyze the effect of standard inductive construction operations, used to generate $4$-regular graphs from smaller ones, as shown by F.Bories et.al. (1983) for simple graphs and by G.Ding et.al. (2003) for multigraphs, on the number of Eulerian orientations.

For irreducible components $C_\sigma$ of Lusztig's nilpotent varieties of type $A$ with graded dimension $(1,2,\dots,n,n-1,\dots,1)$ arising from permutations $\sigma\in S_n$, we characterize the property that $C_\sigma\oplus C_\sigma$ is an irreducible component combinatorially in terms of $\sigma$. The permutations that occur, which we call pseudosmooth, are described by a recursion on direct sums and deleting suitable corners, whose terminal cases are the permutations obtained from \[ 3412,\quad 4231,\quad 35142,\quad 42513,\quad 45312,\quad 426153,\quad 463152,\quad 526413 \] by inflating the entries into consecutive decreasing blocks. The main new input is a hereditary property valid for decompositions of multisegments with disjoint extreme points.