We revisit the theory of Ihara L-functions in the context initially studied by Bass and Hashimoto and more recently by Zakharov. In particular, we study whether the Artin formalism is satisfied by these L-functions. As an application, we give a proof of the analogue of Iwasawa’s asymptotic class number formula for the p-part of the number of spanning trees in branched $\mathbb{Z}_{p}$-towers of finite connected graphs using Ihara zeta and L-functions. Moreover, we relate a generator for the characteristic ideal of the finitely generated torsion Iwasawa module that governs the growth of the p-part of the number of spanning trees in such towers with Ihara L-functions.

We introduce three families of vectors $|\underline {\lambda }^{so}\rangle $, $|\underline {\lambda }^{sp}\rangle $ and $|\underline {\lambda }^{o}\rangle $ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains [33, Section 7] and Betea [6, Section 3] are determinantal with respect to some explicit correlation kernels.

Furthermore, we establish a number of generalized Littlewood identities summed over certain restricted partitions by computing the inner products with elements indexed by one-column partitions or generalized partitions $(0^m)$ in the complete dual Fock space. In particular, for each positive integer n, we obtain generalized Littlewood identities for $(-n)$-asymmetric partitions. We show that these generalized Littlewood identities contain several well-known Littlewood-type identities as special cases. Consequently we also give a new proof of the generalized Littlewood identity [28, (5.25)] for Lie superalgebras.

We study locally finitary realizations of simple transitive module categories of infinite rank over the monoidal category $\mathscr {C}$ of finite-dimensional modules for the complex Lie algebra $\mathfrak {sl}_2$. Combinatorics of such realizations is governed by six infinite Coxeter diagrams. We show that five of these are realizable in our setup, while one (type $B_\infty $) is not. We also describe the $\mathscr {C}$-module subcategories of $\mathfrak {sl}_2$-mod generated by simple modules as well as the $\mathscr {C}$-module categories coming from the natural action of $\mathscr {C}$ on the categories of finite-dimensional modules over Lie subalgebras of $\mathfrak {sl}_2$.

An $[s,t]$-graph is a graph in which every induced subgraph on s vertices contains at least t edges. Li and Wang [‘Hamilton cycles in $2$-connected $[5,3]$-graphs’, J. Inn. Mong. Norm. Univ. Nat. Sci. 35(3) (2006), 285–287] investigated the Hamiltonicity of $[5,3]$-graphs, and more recently, Liu and Liu [‘On the pancyclicity of $2$-connected $[5,3]$-graphs’, Discrete Applied Mathematics 389 (2026), 323–334] studied their pancyclicity. In this note, we continue this line of research by studying the Hamiltonian-connectedness of $[5,3]$-graphs.

We study relations between $(n_4)$ incidence configurations and the classical Poncelet Porism. Poncelet’s result concerns two conics and a sequence of points and lines that inscribes one conic and circumscribes the other. Poncelet’s Porism states that whether this sequence closes up after m steps only depends on the conics and not on the initial point of the sequence. In other words: Poncelet polygons are movable.

We transfer this motion into a flexibility statement about a large class of $(n_4)$ configurations, in which four (straight) lines pass through each point and four points lie on each line. An instance of such configurations in real geometry had been given by Branko Grünbaum and John Rigby in their classical 1990 paper where they constructed a real geometric realisation of a well known combinatorial $(21_4)$ configuration (which had been studied by Felix Klein), now commonly called the Grünbaum–Rigby configuration.

Since then, there has been an intensive search for movable $(n_4)$ configurations, but it is very surprising that the Grünbaum–Rigby $(21_4)$ configuration admits nontrivial motions. It is well-known that the Grünbaum–Rigby configuration is the smallest example of an infinite class of $(n_4)$ configurations, the trivial celestial configurations. A major result of this paper is that all trivial celestial configurations are movable via Poncelet’s Porism and results about properties of Poncelet grids. Alternative approaches via geometry of billiards, in-circle nets, and pentagram maps that relate the subject to discrete integrable systems are given as well.

We develop the geometric theory of equivariant quasisymmetry via a new “quasisymmetric flag variety.” This is a toric complex in the flag variety whose fixed point set is the set of (algebraic) noncrossing partitions, and whose cohomology ring is the ring of quasisymmetric coinvariants.

In this paper, we study the appearance of a spanning subdivision of a clique in graphs satisfying certain pseudorandom conditions. Specifically, we show the following results.

1. (i) There are constants $C\gt 0$ and $c\in (0,1]$ such that, whenever $d/\lambda \ge C$, every $(n,d,\lambda )$-graph contains a spanning subdivision of $K_t$ for all $2\le t \le \min \{cd,c\sqrt {\frac {n}{\log n}}\}$.

2. (ii) There are constants $C\gt 0$ and $c\in (0,1]$ such that, whenever $d/\lambda \ge C\log ^3n$, every $(n,d,\lambda )$-graph contains a spanning nearly balanced subdivision of $K_t$ for all $2\le t \le \min \{cd,c\sqrt {\frac {n}{\log ^3n}}\}$.

3. (iii) For every $\mu \gt 0$, there are constants $c,\varepsilon \in (0,1]$ and $n_0\in \mathbb N$ such that, whenever $n\ge n_0$, every $n$-vertex graph with minimum degree at least $\mu n$ and no bipartite holes of size $\varepsilon n$ contains a spanning nearly balanced subdivision of $K_t$ for all $2\le t \le c\sqrt {n}$.

Reverse mathematics is primarily interested in what set existence axioms are necessary and sufficient in a proof of a theorem. Much work has been done in classifying graph coloring theorems, studying k-regular graphs, k-chromatic graphs, and forests. This article takes inspiration from an old paper by Bean and studies graph coloring theorems restricted to planar graphs. Schmerl showed that these coloring theorems are all equivalent to ${{\mathtt {WKL}}}$ over ${\mathtt {RCA}_{0}}$. In this article, we utilise Weihrauch reducibility to provide a deeper analysis on the uniformity of these implications. Whilst the proofs provided by Schmerl are not obviously non-uniform, we show that in many instances, non-uniformity is indeed necessary.

We explore the combinatorial properties of the set $\mathfrak {S}(k,\ell; n)$ consisting of standard Young tableaux with specific shape constraints. Using bijections with Motzkin and free Motzkin paths, we enumerate $\mathrm {S}_m(3,0;n)$ and $\mathrm {S}_m(2,1;n)$ for all m, and $\mathrm {S}_m(0,3;n)$ and $\mathrm {S}_m(1,2;n)$ for small values of m. We derive explicit formulae such as $\mathrm {S}_2(3,0;n)=M_n - M_{n-1}$ and $\mathrm {S}_4(3,0;n)=M_{n-3}$, where $M_n$ denotes the number of Motzkin paths of length n. Furthermore, we offer an alternative proof of Regev’s identity $\mathrm {S}(2,1;n)=1+\mathrm { S}_{\mathrm {odd}}(2,1;n+1)$, thereby deepening the connection between standard Young tableaux and lattice path combinatorics.

We show that there exist constants $\delta _1,\delta _2\gt 0$ such that if $G$ is an $(n,d,\lambda )$-graph with $\lambda /d\le \delta _1$, then $G$ contains an induced cycle of length at least $\delta _2n/d$. We further demonstrate that, up to a constant factor, this is best possible. Utilising our techniques, we derive that the number of non-isomorphic induced subgraphs of such $G$ is at least exponential in $n\log d/d$, and further demonstrate that this is tight up to a constant factor in the exponent.

In this paper, we introduce the symmetric normaliser graph of a group G. The vertex set of this graph consists of the elements of the group. Vertices x and y are adjacent if x lies in the normaliser of $\langle y \rangle $ and y lies in the normaliser of $\langle x \rangle $. We investigate the position this graph occupies in the hierarchy of graphs defined on groups. We show that the existing hierarchy is further refined by this graph and that the edges of this graph lie between the edges of the commuting graph and the nilpotent graph. For finite groups, we prove a necessary and sufficient condition for the symmetric normaliser graph to be equal to the commuting graph, and similarly, for equality with the nilpotent graph. The edge set of the symmetric normaliser graph is also a subset of the edge set of the Engel graph of a group and has connections to the nongenerating graph of a group.

We prove that if $p \geq n^{-(q-6)/2}$, then asymptotically almost surely the binomial random q-uniform hypergraph $\mathcal{G}^{(q)}(n,p)$ contains an (n, q, 2)-Steiner system, provided n satisfies the necessary divisibility conditions.

In this paper, we formulate a rational analog of the fall Delta theorem and the Delta square conjecture. We find a new dinv statistic on fall-decorated paths on a $(m+k) \times (n+k)$ rectangle that simultaneously extends the previously known dinv statistics on decorated square objects and nondecorated rectangular objects. We prove a symmetric function formula for the $q,t$-generating function of fall-decorated rectangular Dyck paths as a skewing operator applied to $e_{m,n+km}$ and, conditionally on the rectangular paths conjecture, an analog formula for fall-decorated rectangular paths.

In this article, we study a degenerate version of Ramsey’s theorem for pairs and two colors (${\mathsf {RT}}^2_2$), in which the homogeneous sets for color 1 are of bounded size. By ${\mathsf {RT}}^2_2$, it follows that every such coloring admits an infinite homogeneous set for color 0. This statement, called ${\mathsf {BRT}}^2_2$, is known to be computably true, that is, every computable instance admits a computable solution, but the known proofs use $\Sigma ^0_2$-induction ($\mathsf {I}\Sigma ^0_2$). We prove that ${\mathsf {BRT}}^2_2$ follows from the Erdős–Moser theorem but not from the ascending descending sequence principle, and that its computably true version is equivalent to $\mathsf {I}\Sigma ^0_2$ over ${\mathsf {RCA}}_0$.

We give new analogues of Andrews’ celebrated spt-function identity using basic hypergeometric transformations due to Verma and Jain [‘Transformations between basic hypergeometric series on different bases and identities of Rogers–Ramanujan type’, J. Math. Anal. Appl. 76(1) (1980), 230–269].

We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at a rate equal to the product of the sizes of the vertices. We provide an explicit expression for the fluid limit of the number of connected components normalized by its initial value, when the time is smaller than the inverse of the sum of the square of the initial vertex sizes. We also identify the diffusion limit of the rescaled fluctuations around the fluid limit. This is applied to several examples. In the particular setting of the Erdős–Rényi graph process, we give the fluid limit of the normalized number of connected components, and the diffusion limit of the scaled fluctuations in the sub-critical regime, where the mean degree is between zero and one.

We study intersecting families of words from the Erdős–Ko–Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least $3$: maximum intersecting families of words are exactly the stars.

We compute the rate of exponential growth of the free inverse monoid of rank r (and hence an upper bound on the corresponding rate for all r-generated inverse monoids and semigroups). This turns out to be an algebraic number strictly between the obvious bounds of $2r-1$ and $2r$, tending to $2r$ as the rank tends to infinity. We also find an explicit expression for the exponential growth rate of the number of idempotents, and prove that this tends to $\sqrt {e(2r-1)}$ as $r \to \infty $.

In this article we introduce a gluing operation on dimer models. This allows us to construct dimer quivers on arbitrary surfaces. We study how the associated dimer and boundary algebras behave under the gluing and how to determine them from the gluing components. We also use this operation to construct homogeneous dimer quivers on annuli.

We study a random graph model with preferential edge attachment and detachment through the embedding into a generalized Yule model. We show that the in-degree distribution of a vertex chosen uniformly at random follows a power law in the supercritical regime but has an exponential decay in the subcritical. We provide the corresponding asymptotics. In the critical regime we observe an intermediate decay. The regimes are clearly defined in terms of parameter ranges.