Papers

By a simple homogenization process, one may turn a $q$-analog into a symmetric $(q,t)$-analog. Although simple, this bijective correspondence makes many concepts and identities become more natural, and proofs follow readily from classical properties of symmetric functions in two variables. A more intricate non-homogeneous extension is also developed. We illustrate this approach, revisiting recent developments in the study of $\gamma$-positivity, Lucas analogues, and the monoid of Cyclotomic generating functions. This also leads naturally to new constructions and conjectures. We further explore other avenues of investigations, included graded and equivariant $\gamma$-positivity and $\gamma$-anti-positivity (also known as alternatingly $\gamma$-positive).

Fix integers $e\ge2$ and $r\ge1$. In this paper we study the $r$-th generalized covering radius $\rho_r\left(BCH(e,m)\right)$ of the binary primitive $e$-error-correcting BCH code $BCH(e,m)$. By using an algebraic-geometric reformulation of the covering problem together with an explicit Lang-Weil estimate, we prove that \[\rho_r\bigl(\BCH(e,m)\bigr)\le(r+1)e-1\] for all sufficiently large $m$. For $e\ge7$, this improves a recent result of Belinsky–Zabokritskiy. Our proof gives a substantially simpler geometric approach to this upper bound. In particular it implies that \[\rho_2\bigl(BCH(e,m)\bigr)=3e-1\] for all sufficiently large $m$. Previously it was only known that \[\rho_2\bigl(\BCH(e,m)\bigr) \in \left\{3e-1,3e\right\}\] for all sufficiently large $m$.

We give the first polynomial-time algorithm, after half a century, for the celebrated Győri-Lovász theorem, which resolved a conjecture of Frank (1975). The theorem, one of the simplest existential theorems to explain, states that every $k$-connected graph can be partitioned into $k$ disjoint connected subgraphs of arbitrary prescribed positive sizes. This is a fundamental structural result with broad applications, such as flexible allocation of connected subnetworks of prescribed sizes in sufficiently connected cloud infrastructures. While Lovász (1977) gave a highly non-constructive proof for a stronger directed version using algebraic topology, Győri's original constructive proof (1976) requires exponential time. Despite more than 50 years of effort, no polynomial-time algorithm was known even for $k>4$. Determining the computational complexity of the Győri-Lovász theorem—whether it admits even a sub-exponential-time algorithm or is computationally hard (in particular, PLS-complete or PPAD)—has remained one of the central open problems in algorithmic graph theory. In this paper, we finally resolve this long-standing problem by a fundamentally new proof of the existential theorem via introducing the novel concept of flow-essential assignment, which genuinely marries matching and cut structures and yields the first polynomial-time constructive algorithm for the Győri-Lovász theorem. In fact, we obtain a polynomial-time algorithm for Lovász's stronger directed version, whose proof was non-constructive even for DAGs; for DAGs, we further obtain a near-linear-time algorithm. We also develop polynomial-time algorithms for weighted generalizations where the seminal work of Chen, Kleinberg, Lovász, Rajaraman, Sundaram, and Vetta (JACM'07) on confluent flows established only existential non-constructive results.

In this note we count the number of solutions to a non-degenerate quadratic form in eight prime variables of off-diagonal rank $3$. It is a continuation of work by L. Zhao, who counted solutions to forms in at least nine variables, by B. Green, who counted solutions to 'generic' (which implies off-diagonal rank $4$) forms in eight variables, and by J. Dobrowolski, who counted solutions to forms in eight variables of off-diagonal rank at most $2$.

A finite connected graph is rarely a Cayley graph. We measure how far it is from being one: given $G$ with $n$ vertices and $m$ edges, how few edges must be added, or added and deleted, before the result is a Cayley graph of an abelian group of order $n$ on the same vertex set? This defines two invariants, the completion number $\gamma^{+}$ (additions only) and the Cayley edit distance $\gamma_{\triangle}$ (both), each normalized by $m$. We show that deciding the edit version is NP-complete already for a fixed cyclic host, by a reduction from Hamiltonian Cycle in which the edit cost of a labeling is $n+m-2k$ when it realizes a longest path with $k$ edges; the optimal cost is $m-n+2pp(G)$, bounded in polynomial time by the matching number. We prove that irregularity alone forces $\gamma^{+}(G)\ge n\Delta^{*}/(2m)-1$, where $\Delta^{*}$ is the least $d\ge\Delta$ with $nd$ even, computable in linear time from the degree sequence; we characterize equality exactly. It is attained on the star, where $\gamma^{+}(K_{1,q})=(q-1)/2$ and the star maximizes $\gamma^{+}$, while $\gamma_{\triangle}$ stays bounded by an absolute constant. We determine paths and grids exactly, $\gamma^{+}(P_n)=\gamma^{+}(P_n\,\square\,P_n)=1/(n-1)$, and show $\gamma_{\triangle}(K_{1,q})\to 2$, not the $3/2$ suggested by the additive case. We report an exhaustive certified census of all $995$ connected graphs on at most seven vertices. The degree bound is attained on $89.4\%$ and the two invariants separate strictly on $84.7\%$, though both rates vary sharply with order: attainment $100\%,100\%,84.8\%,89.7\%$ and separation $0\%,61.9\%,73.2\%,87.7\%$ for $n=4,5,6,7$, dominated by the $853$ graphs on seven vertices. The star uniquely maximizes both. Edit count and the bi-Lipschitz distortion of the completed host are independent, moving oppositely on stars and paths.Data and certificates at doi:10.5281/zenodo.21852006.

Let $r\geq3$ be fixed, and let $\mathbf{G}_n$ be the set of all simple graphs with vertex set $[n]=\{1,\ldots,n\}$. We consider an exponential random graph model which gives higher probability to $G \in \mathbf{G}_n$ than to $H \in \mathbf{G}_n$ if $G$ has fewer $r$-cliques than $H$. But all graphs in $\mathbf{G}_n$ have positive probability. The degree to which graphs with fewer $r$-cliques are given higher probability is determined by a positive weight $w$. We prove that, asymptotically almost surely as $n \to \infty$, a random graph from $\mathbf{G}_n$ has a vertex partition into $r-1$ parts of roughly equal size, the density of edges between the parts is close to $1/2$, and for every $\varepsilon > 0$ the density of edges within any part is less than $\varepsilon$. The asymptotic structural properties are independent of the weight $w$ as long as it is positive. We also extend the result to the context of several clique sizes, each one with its own weight.

A classical result of Andrásfai, Erdős, and Sós states that every $n$-vertex graph with odd girth at least $2k+1$ and minimum degree larger than $\frac{2n}{2k+1}$ is bipartite. Rather than imposing a minimum-degree condition, in this paper we investigate conditions on algebraic connectivity that force graphs of given odd girth to have a simple structure. The algebraic connectivity of a graph $G$, denoted by $\mu_2(G)$, is the second smallest eigenvalue of its Laplacian matrix. Our main results are as follows. 1. Every $n$-vertex triangle-free graph $G$ with $\mu_2(G)\geq \frac{n}{3}$ is bipartite. Moreover, the constant $\frac{1}{3}$ is asymptotically best possible. 2. For $k\geq 3$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $\mu_2(G)>\frac{4n}{6k-1}$ is bipartite. 3. For $k\geq 22$, every $n$-vertex graph $G$ of odd girth at least $2k+1$ with $\mu_2(G)>\frac{3456n}{k^3}$ is bipartite. Moreover, the term $k^{-3}$ is asymptotically best possible.

In a preceding article, we introduced stable symmetric series encoding simultaneously the normalized conjugacy classes of all symmetric groups. The same rational series remain stable for the Jack-deformed Goulden–Jackson product. We investigate their two-parameter Macdonald analogue. Starting from the $(q,t)$-deformed class product (dual to the coproduct diagonal on the $J$-basis), we construct the unique infinite series whose multiplication realizes any shifted Macdonald eigenvalue. In contrast with the classical and Jack cases, this transform is no longer multiplication by a fixed explicit series. We identify it with a composition of a Cauchy multiplication, the integral nabla operator, and a simple diagonal operator. We then determine the series realizing the Nazarov–Sklyanin operators $A^{(k)}$, derive a single generating series for all column partitions, and compare our construction with the Macdonald characters and Theta operators of Ben Dali and D'Adderio.

Kashiwara–Nakashima tableaux and Gelfand–Tsetlin patterns of orthogonal type are famous as combinatorial models of the finite-dimensional irreducible representations of the special orthogonal Lie algebras $\mathfrak{so}_N$. The former is constructed based on the representation theory of quantum groups, especially the theory of crystals, while the latter based on the branching rule for $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$. In the present paper, we construct a natural bijection between them by means of the representation theory of quantum symmetric pairs corresponding to $(\mathfrak{so}_N,\mathfrak{so}_{N-1})$.

Breakpoint graphs originate in comparative genomics, where their alternating cycles encode relationships between genomes. We study a constrained class of three-colored breakpoint graphs associated with permutations and develop cycle-refined generating functions for two extremal families. These families have a natural topological interpretation: their canonical surfaces are, respectively, the sphere and the projective plane. The spherical family is characterized by noncrossing configurations, while the projective-plane family admits a different decomposition involving a distinguished family of Möbius ladders. The resulting generating-function equations retain the full cycle structure but nevertheless admit substantial reductions. This leads to explicit Catalan-weighted evaluations, polynomiality results for refined cycle statistics, and a connection between a natural diagonal specialization and noncrossing trees. The two topological families exhibit markedly different combinatorial mechanisms, providing complementary examples of how local transformations of breakpoint graphs can control refined permutation enumerations. As a further application, the same Catalan-weighted sums arise in asymptotic unitary-Weingarten expansions for entanglement of random Gaussian states in linear optics. The combinatorial results determine the leading and constant-order moment polynomials entering the Rényi entropy expansion, with the projective-plane contribution giving the finite-size constant correction.

Let $\zeta(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erdős and Gimbel asked whether, for $G_n\sim G(n,1/2)$, the difference $\chi(G_n)-\zeta(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\to\infty$ and prove that $\mathbb P(\chi(G_n)-\zeta(G_n)\ge ((\log 2)^2/4)\log(200/153)\,n/(\log n)^3)\to1$. This gives a lower bound at the conjectured scale $n/(\log n)^3$. We also obtain a phase-resolved refinement: if $\delta_n$ is the fractional part of the standard independence-number center, then the coefficient may be replaced by $(\log 2)^2A_4(\delta_n)/4-o(1)$, where $A_4$ is explicit, continuous, nonconstant, and satisfies $A_4(\delta)>\log(200/153)$ for every $\delta\in[0,1]$. The proof uses signed cocoloring profiles supported on four consecutive class sizes and remains uniform across jumps of the natural class-size cutoff. An exact signed-overlap identity separates local cell rewards from a binary cycle-space factor. A canonical decomposition into high cells and a capped residual matching, together with an endpoint-table comparison and an injective restriction of residual even edge sets, yields the required second-moment bound. A bounded-differences argument then amplifies the resulting rare signed witness to a high-probability cocoloring.

The independence complex and perfect matching complex of a graph are simplicial complexes encoding, respectively, its independent sets and perfect matchings. Determining their homotopy types is generally difficult, with explicit descriptions known mainly for highly structured graph families. In this article, we determine the homotopy types of these complexes for the Möbius ladder graphs $M_{2n}$ and circular ladder graphs $\mathcal{C}_{2n}$. The Möbius ladder graphs $M_{2n}$ are highly symmetric cubic graphs obtained from a $2n$-cycle by joining opposite vertices, while the circular ladder graphs $\mathcal{C}_{2n}$ are the Cartesian products of an $n$-cycle and a path of length one. We show that $\operatorname{Ind}(M_{2n})$ and $\operatorname{Ind}(\mathcal{C}_{2n})$ have the homotopy type of wedges of spheres, with the numbers and dimensions of the spheres exhibiting periodic behavior according to $n$ modulo $4$. For the perfect matching complex $\mathcal{M}_p(M_{2n})$, its homotopy type is a wedge of two copies of $\mathbb{S}^{(n-2)/2}$ when $n$ is even, while for odd $n$ it has the homotopy type of a wedge of spheres whose numbers and dimensions depend periodically on $n$ modulo $6$. The perfect matching complex $\mathcal{M}_p(\mathcal{C}_{2n})$ is contractible for odd $n$, whereas for even $n$ its homotopy type is a wedge of spheres, with the numbers and dimensions determined periodically by $n$ modulo $6$. Thus, we obtain explicit homotopy types for the independence and perfect matching complexes of two highly symmetric families of cubic graphs, which are also relevant in crystallization theory and the combinatorial representation of PL manifolds.

We introduce the second order Zarankiewicz number $z_2(m,n)$ for irreducible doubly simple biquadratic forms with $|E_1|=z(m,n)$, together with the intermediate recursive-line and signed parameters $z_{RL}(m,n)$ and $z_{SL}(m,n)$. They satisfy the unconditional hierarchy \[ \operatorname{BSR}(m,n) \ge z_2(m,n) \ge z_{SL}(m,n) \ge z_{RL}(m,n) \ge z_{wL}(m,n) \ge z(m,n), \] where $z_{RL}$ is defined by the strengthened recursive rectangle criterion $(RW3^+)$ together with the conditions $(S)$ and $C_4$-freeness of $G_1$, and $z_{SL}$ is defined by the signed criterion $(RW3^\pm)$. We show that $(RW3^+)$ is sound and strictly weaker than the literal weak cross-cell test $(W3)$ on the weak-admissible class, and that $(RW3^\pm)$ is likewise sound. At the smallest four-column cases we obtain \[ z_2(5,4)=z_{SL}(5,4)=z_{RL}(5,4)=13>12=z_{wL}(5,4), \] and \[ z_2(6,4)=z_{SL}(6,4)=z_{RL}(6,4)=16>14=z_{wL}(6,4). \] In three columns this yields \[ z_2(m,3)=z_{RL}(m,3)=2m \qquad \text{for all } m\ge 3, \] with strict separation from $z_{wL}(m,3)$ for every $m\ge10$, and exact gap $\lfloor(m-3)/3\rfloor$ for $m\ge16$. Further finite computations give $z_{RL}(5,5)=17$, $z_{RL}(7,4)=19$, and $z_{RL}(7,7)\ge32>28=z_{wL}(7,7)$. Along $N=2p$ with $p$ an odd prime, we obtain the cubic asymptotic separation \[ z_2\!\left(\binom{N}{2},N\right)-z_{wL}\!\left(\binom{N}{2},N\right)\ge \left(\frac{1}{16}-o(1)\right)N^3. \] For the exceptional incidence case $p=3$ we prove $z_{SL}(15,6)=z_2(15,6)=60$. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all $m\ge15$. The conjectural equality $z_2=z_{RL}$ is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.

For a graph $H$, let $\operatorname{ex}(Q_n, H)$ be the largest number of edges in a subgraph of the hypercube $Q_n$ of dimension $n$ that contains no subgraph isomorphic to $H$. We define $\pi_\square(H)=\lim_{n\to\infty}\operatorname{ex}(Q_n,H)/|E(Q_n)|$ as the Turán density of $H$ in the hypercube. Although determining $\pi_\square(H)$ for general $H$ remains widely open, Conlon found a large class of graphs with zero Turán density in a hypercube. In this note, we address the case when $\pi_{\square} (H)>0$. If a graph $H$ cannot be embedded in an edge layer of a hypercube, then $\pi_{\square} (H)\geq 1/2$, as can be seen by taking every other edge layer of $Q_n$. Previously known examples of layered graphs of positive Turán density in a hypercube include graphs containing cycles of length $6$ or $10$, as well as examples of girth $8$. Also, to the best of our knowledge, all previously known examples of graphs with Turán densities arbitrarily close to one are not layered. We show that for every $g \geq 3$, there are layered graphs of girth greater than $g$ whose Turán densities in a hypercube are arbitrarily close to one. Moreover, these graphs have an induced embedding into an edge layer of a hypercube.

We determine the $k$-colour size–Ramsey number of even cycles up to absolute constant factors. For every $k\ge2$ and every even $n\ge100\log k$, \[ \widehat R_k(C_n)=\Theta(k^2\log k)n. \] The lower bound follows from the corresponding result of Beke, Li and Sahasrabudhe for paths, while our upper bound improves the previous best estimate $O(k^{34}n)$ of Javadi, Kohayakawa and Miralaei.

We study the fast-robber version of the Cops and Robbers game on finite strongly connected abelian Cayley digraphs, including undirected Cayley graphs as the symmetric case. For bounded out-degree $D$, we show that $c_{1,\infty}\left(\Gamma\right)=O_D\left(n^{1-\frac{1}{D}}\right)$; in the undirected case with $D\ge2$, this improves to the optimal exponent $1-\frac{1}{\left\lfloor \frac{D}{2}\right\rfloor}$. We also establish the degree-independent bound $c_{1,\infty}\left(\Gamma\right)=O\!\left(\frac{n\left(\log\log n\right)^2}{\log n}\right)$. These estimates follow from an optimized character-theoretic cyclic sweep over subgroup quotients.

Jaramillo-Velez recently introduced edge codes, a new class of toric evaluation codes constructed from the edges of a (hyper)graph $\mathcal{H}$. In the case that $\mathcal{H}$ is a tree, Jaramillo-Velez computed both the minimum distance and the weight distribution of the associated code. In this paper, we study edge codes associated to unicyclic graphs. Our most striking result is that computing the parameters of these codes is subtle in the case that the induced cycle has an even length because these values will depend on certain conditions regarding the length of the cycle and the size of the base field.

We study the forbidden poset analog of the maximal anti-Ramsey problem introduced for graphs by Burr, Erd\H os, Graham, and Sós. For integers $m\le 2^n$ and poset $P=(P,\preceq)$, we introduce $\mathrm{ar_m}(n,m,P)$ (and $\mathrm{ar^*_m}(n,m,P)$) to denote the minimum integer $k$ such that there exists a family $\mathcal{F}\subseteq 2^{[n]}$ with $|\mathcal{F}|=m$ and a coloring $\mathcal{F}\rightarrow [k]$ with all weak (strong) copies of $P$ being rainbow. As long as there exist $P$-free families of size $m$, these parameters equal 1. It is known that the largest size $La(n,P)$ ($La^*(n,P)$) of weak (strong) $P$-free families has order of magnitude $\Theta(\binom{n}{\lfloor n/2\rfloor})$ unless $P$ is the antichain $A_k$ on $k$ elements. In this paper we study $\mathrm{ar_m}(n,m,P)$ and $\mathrm{ar^*_m}(n,m,P)$ in two regimes of $m$. We determine the asymptotics of these parameters for all posets $P$ when $m=2^n$. We also consider the case $m=\Theta(\binom{n}{\lfloor n/2\rfloor})$. It is shown that for any connected poset $P$ and integer $k$, there exist integers $m_{P,k}$ and $m^*_{P,k}$ such that to color the middle $k$ layers of the Boolean lattice with all weak or strong copies of $P$ being rainbow, one needs $\Theta(n^{m_{P,k}})$ or $\Theta(n^{m^*_{P,k}})$ colors. For tree posets $T$, one has $m_{T,k}=m^*_{T,k}$. We conjecture that for any tree poset $T$, and positive real $\varepsilon$, $\mathrm{ar_m}(n,m,T),\mathrm{ar^*_m}(n,m,T)=\Omega(n^{m_{T,k}})$ holds provided $m\ge (k-1+\varepsilon)\binom{n}{\lfloor n/2\rfloor}$. We prove our conjecture on $\mathrm{ar_m}(n,m,T)$ for an infinite class of tree posets.

Smith's conjecture asserts that in every $k$-connected graph with $k\geq 2$, any two longest cycles intersect in at least $k$ vertices. In this work, we establish an $\Omega(k^{8/11})$ bound for this conjecture, improving upon the $\Omega(k^{2/3})$ bound of Ma and Zhao. Our proof combines a Ramsey theoretic refinement of the traditional Turán-type approach with computer search.

Perfect state transfer on graphs has attracted extensive attention due to its application in quantum information and quantum computation. Explicit characterizations of connection sets admitting perfect state transfer in Cayley graphs are rare and, so far, are known only for a few abelian Cayley graphs. In this paper, we characterize the conjugation-closed connection sets of connected Cayley graphs over dihedral groups that admit perfect state transfer. By applying Ramanujan sums, Möbius inversion, and arguments based on the $p$-adic exponential valuation of rational numbers, we convert the eigenvalue constraints imposed by perfect state transfer into explicit structural conditions on the connection set. This yields a complete and practical characterization, which gives an effective criterion for recognizing and constructing such Cayley graphs and also determines the exact minimum perfect state transfer time.

Marton's covering conjecture studies finite sets in high-dimensional binary spaces whose pairwise sums create relatively few new elements. It predicts that every such set can be described efficiently by shifted copies of one linear subspace of comparable size. Gowers, Green, Manners, and Tao [GGMT25] proved the conjecture with exponent $12$. Liao [L24] improved the exponent to $9$. We improve it further to $5.287$.

Let $G$ be a permutation group that acts $2$-transitively on the finite set $V$ and let $\mathcal{T}=(V,T)$ be a two-graph whose automorphism group contains $G$. In this paper, we classify those strongly regular graphs $\Gamma$ with vertex set $V$ whose automorphism group is a transitive maximal subgroup of $G$ and whose associated two-graph is $\mathcal{T}$. In doing so, we obtain a new family of vertex-transitive strongly regular graphs whose associated two-graph arises from $P\Sigma L(2,q)$.

We find explicit ridge unfoldings of the three exceptional 4D polytopes (24-cell, 120-cell, 600-cell) that result in overlaps of their facets. These failures bring an end to the full classification of regular polytopes with the all-net property.

The exact Shannon capacity is unknown for every odd cycle beyond the five-cycle $C_5$, making odd cycles a central open problem in zero-error information theory. Improving the known lower bounds requires constructing large independent sets in strong powers of these graphs. Recent AI-assisted work has produced a rapid sequence of improvements: building on the construction of Itty et al., Gao developed a recursive product construction for combining structured independent sets, and Buys, Polak, and Zuiddam (BPZ) subsequently strengthened this through a richer recursion framework. We continue this line of AI-assisted exploration and introduce a heterogeneous refinement of these constructions. The central observation is that the usefulness of an intermediate construction depends not only on the size of its current main independent set, but also on the auxiliary structure it carries into subsequent recursion. Consequently, different parts of that auxiliary structure need not use the same independent set, and different occurrences in a recursion need not use the same intermediate representation. We formalize this for Gao's binary product and derive explicit propagation rules showing how heterogeneous choices strengthen the resulting gadget while leaving its current code size unchanged, then extend the principle to the more general BPZ framework, tailoring constructions to the distinct roles they play within the recursion. Applying these refinements to the seven-cycle $C_7$, we obtain an independent set in $C_7^{\boxtimes 500}$ yielding $\Theta(C_7)\ge 3.25883262\ldots$, improving the best known lower bound. Beyond the numerical gain, the results illustrate a general principle for recursive zero-error constructions: intermediate structures with the same dimension and current code size can have different downstream value depending on where and how they are used in the recursion.

We give the recurrence relation for counting polynomials for permutations avoiding the $\321$ vincular pattern, with respect to inversion numbers and descent numbers, and consequently give the $q$-exponential generating function for these counting polynomials. Furthermore, we give another explanation for these counting polynomials in terms of restricted-growth words.

We investigate the filtration structure of the skein algebra induced by the (quantum) trace map. Utilizing the structure, we prove that relative $\operatorname{SL}(3, \mathbb{C})$ character varieties of punctured surfaces admit log Calabi-Yau compactifications and prove the (weak) geometric P=W conjecture.

A classification is given of regular, rotary, and birotary Cayley maps of which the vertex number is coprime to the valency. The classification is involved in constructions of new examples of interesting Cayley maps.

Recently Dixit, Kumar, and Srivastava investigated what they called Rascoe and non-Rascoe partitions. These are defined to be the set of distinct partitions where the length of the partition is a part of the partition and is not a part respectively. In this note we provide a $q$-series theoretic proof of two identities regarding the generating function for unrestricted Rascoe and non-Rascoe partitions fulfilling a request of Dixit, Kumar, and Srivastava. We also prove a conjecture of Beck relating non-Rascoe partitions and the rank of a partition.

This paper establishes a comprehensive combinatorial and arithmetic framework for mixed Stirling and mixed Bell numbers, bridging partition structures, Touchard polynomials, and prime-power congruences. Furthermore, we develop to $p$-adic valuation theory, proving prime-power Touchard congruences and higher-order modulo-$p^2$ refinements that generalize classical arithmetic properties of combinatorial sequences.

Moustrou, Riener, and Verdure related the dominance order on partitions to Specht ideal and variety inclusion for $S_n$; we previously extended this to direct products of symmetric groups. Here we take a first step toward the imprimitive complex reflection group $G(r,1,n)=(\mathbb{Z}/r\mathbb{Z})\wr S_n$. The correct geometric invariant of a point $x\in K^n$ is a pair $(\mu,z)$: a classical partition $\mu$ on the $r$-th powers of the nonzero coordinates, together with the number $z$ of zero coordinates. We construct a $G(r,1,n)$-invariant ideal $I_{(\mu,z)}$ for each such pair, and show that a dominance order on $(\mu,z)$, with $z$ taking strict priority over $\mu$, exactly characterizes both ideal inclusion and reverse variety inclusion. We settle a monomial criterion in three regimes, recover the classical $B_n$-Specht ideal theory at $r=2$ when $z=0$, and establish radicality of $I_{(\mu,z)}$ at the two extremes of the $z=0$ level. Four computational examples, checked exhaustively for group invariance, illustrate the construction throughout.