Papers

We give a new upper bound on the piercing number $\tau(\mathcal{F})$ of a finite family $\mathcal{F}$ of axis-parallel rectangles in the plane when every $p$ members contain an intersecting pair. Namely, we prove $\tau(\mathcal{F}) = O(p(\log\log p)(\log\log\log p))$, improving the factor $(\log\log p)^2$ in the best previously known bound to $(\log\log p)(\log\log\log p)$. For each $d>2$, when $\mathcal{F}$ is a finite family of axis-parallel boxes in $\mathbb{R}^d$, this yields $\tau(\mathcal{F}) = O(p(\log p)^{d-2}(\log\log p)(\log\log\log p))$. We also study an analogous problem replacing $\mathbb{R}^d=\mathbb{R}\times\cdots\times\mathbb{R}$ with the Cartesian product $T_1\square\cdots\square T_d$ of trees $T_1,\ldots,T_d$, and boxes with objects of the form $S_1\square\cdots\square S_d$ where $S_i$ is a subtree of $T_i$. For a finite family of such objects under the same intersection assumption as above, we establish an upper bound $\tau(\mathcal{F})=O(p(\log p)^{d-1})$ for each $d$. We also construct examples showing that this upper bound cannot be improved in order of magnitude.

Let $\Phi(G)$ denote the number of perfect matchings of a graph $G$. Goedgebeur et al. posted two questions about the perfect matchings of planar 3-connected graphs: does there exist an infinite family of matchable planar 3-connected graphs of girth 5 with a constant number of perfect matchings? Whether there exist infinitely many planar 3-connected graphs with minimum degree 5 which have a constant number of perfect matchings. We present answers to those questions. First, we construct, for every even $n\ge196$, a simple planar 3-connected graph of order $n$ and girth five with exactly $c$ perfect matchings, where $c$ is a fixed positive integer. Second, for a simple planar 3-connected graph $G$ with a perfect matching and minimum degree at least five, we show that $\Phi(G)\ge2^{\lceil(|V(G)|+20)/22\rceil}$. This implies that, for each positive integer $t$, only finitely many such graphs, up to isomorphism, have at most $t$ perfect matchings.

A set $S$ of positive integers is $2$-large if every two-coloring contains arbitrarily long monochromatic arithmetic progressions with common difference in $S$. We prove that every such set meets every nil-Bohr neighborhood of $0$. This proves a conjecture of the first author and gives a sharp answer to a question of Host, Kra, and Maass. We also show that, for every fixed nil-Bohr neighborhood \(B\) of \(0\) and every \(\eps>0\), restricting the common difference to \(B\) increases the extremal number for \(k\)-term arithmetic progressions by at most a factor of \(1+\eps\), provided \(k\) is sufficiently large.

We introduce hybrid expanders, sparse graphs which combine sublinear expansion across all scales with strong constant-factor expansion on smaller, polylogarithmic sets. We show that every $n$-vertex graph of average degree at least $2^{\omega(\log^*n)}$ contains such a subgraph, and use this local expansion to bring into the sparse setting tree-embedding and rollback tools more commonly associated with random or pseudorandom graphs. As an application, for fixed $k\in\mathbb N$, we prove that every graph satisfying this average-degree condition contains $k$ pairwise edge-disjoint cycles with nested vertex sets and compatible cyclic orders, resolving up to a quasi-constant factor a problem of Thomassen, originating in work of Erdős.

In this paper, we study the maximal length of a minimal zero-sum sequence over $\left[\!\left[ -m,m \right]\!\right]^d$ having a support of size $d+1$. When $d=2$, we find the exact value of this maximal length for all $m \ge 2$, as well as the exact structure of all extremal sequences for this problem. When $d=3$, we obtain lower and upper bounds that asymptotically match each other as $m$ tends to infinity. This allows us to improve on the best known lower bounds for the Davenport constant of low-dimensional boxes.

The central object in this work is the Boolean network, a function $f:\{0,1\}^V \to \{0,1\}^V$. Often identified with $f$ is its dynamics $D_f$, the digraph with exactly $\{x \to y : f(x)=y\}$ as its arcs. The interaction graph of $f$ is the digraph $G$ on $V$ which can be thought of as representing the dependencies of the components of $f$ on one another. On the other hand, model-checking is a major field of study in graph algorithms and complexity. We consider problems at the intersection of these paradigms: for a given property $\phi$, we say that a digraph $G$ is $\phi$-dynamic if $G$ is the interaction graph of some Boolean network $f$ such that $D_f \models \phi$. The corresponding decision problem $\phi$-DYNAMIC asks whether $G$ is $\phi$-dynamic. Problems of this form are well-studied in the Boolean networks community. For example, it is a well-known result that, letting $\phi$ be 'No fixed points' (i.e. $\nexists x:x\to x$), $\phi$-DYNAMIC is equivalent to the structural digraph problem NON-ACYCLIC, which is solvable in linear time. We establish, for the first time, properties $\phi$ for which $\phi$-DYNAMIC is computationally hard. These include the 'Unique fixed point' property $\exists!x:x\to x$, and a range of properties concerning the periodic points of $f$ (i.e. vertices of $D_f$ which lie on a cycle). More widely, we exhibit an infinite family of natural properties $\phi$ such that $\phi$-DYNAMIC is $\mathrm{NEXPTIME}$-complete, which we call PERIOD PIECE. As a special case, these include all properties of the form 'Unique cycle of length $k$' and 'Exactly $k$ fixed points' for any $k \ge 1$. On the other hand, we show that if the input digraph has bounded in-degree, then for those same problems $\phi$-DYNAMIC collapses down to $\mathrm{NP}^{\mathrm{NP}}$, the second level of the polynomial hierarchy, though it remains $\mathrm{NP}^{\mathrm{NP}}$-complete.

We introduce $qt$Bench, a benchmark of 29 open problems in algebraic combinatorics. These problems ask for a simple combinatorial algorithm that explains given polynomial coefficients or identities. To help AI agents search for such algorithms, we provide a scaffold for playing with combinatorial objects, countering cheating attempts, and testing candidate solutions. To explore how far frontier AI systems can go on these problems, we collaborated with OpenAI, whose researchers obtained proposed solutions to several of them. This highlights bigger meta-problems: is AI discovery at scale ethical? How should these results be released? How should proof digestion be organized? We discuss these questions and give our views.

Recently, two exponentiations of ultrafilters have been introduced in [7]. We rephrase them in nonstandard terms, and use the nonstandard formulations to characterize some of their properties. We also study their relations with divisibility by introducing factor-picking functions and highly exponential ultrafilters.

Let $\mathbf{G}=\{G_1,\dots,G_{2n}\}$ be a collection of not necessarily distinct balanced bipartite graphs of order $2n$ with a common bipartition $V=(X,Y)$. A cycle $C$ with $V(C)\subseteq V$ and $|E(C)|\leq 2n$ is called rainbow in $\mathbf{G}$, if there exists an injection $\phi\colon E(C)\to [2n]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(C)$. The graph collection $\mathbf{G}$ is said to be rainbow bipancyclic if it contains a rainbow cycle of every even length from 4 to $2n$. In this paper, we show that if $e(G_i)\ge n^2-n+1$ for each $i\in[2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\dots=G_{2n}\cong B_{n}^1$, where $B_{n}^1$ is the graph obtained from $K_{n,n}$ by deleting all edges in its one subgraph $K_{n-1,1}$. As a corollary, we obtain that if $\rho(G_i)\ge \rho(B_{n}^1)>\sqrt{n(n-1)}$ for each $i\in[2n]$, then $\mathbf{G}$ is rainbow bipancyclic unless $G_1=G_2=\dots=G_{2n}\cong B_{n}^1$, which strengthens Theorem $5$ of [Electron. J. Combin., 33(3)(2026), \#P3.29] and Theorem $1.6$ of [Linear Algebra Appl., 743(2026), 250–270] from rainbow Hamiltonicity to rainbow bipancyclicity. We also characterize all bipartite graph collections that are not rainbow bipancyclic under the condition $\rho(G_i)\ge \sqrt{n(n-1)}$ for each $i\in[2n]$

An edge of a multigraph is rich with respect to a proper edge-coloring if all the edges adjacent to it receive mutually distinct colors, and a proper edge-coloring is a rich-neighbor coloring if every non-isolated edge has a rich edge adjacent to it. The least number of colors in such a coloring is the rich-neighbor chromatic index chi'_rn(G). So far it was studied for subcubic graphs only, and multigraphs of larger maximum degree are considered here for the first time. The only upper bound valid in general was the quadratic one inherited from the strong chromatic index, while a single rich edge already requires 2D-1 colors in a D-regular graph, where D denotes the maximum degree. We give an upper linear bound chi'_rn(G) <= 6D-8 which holds for every loopless multigraph G with D >= 3, and we show that a coloring realizing this bound can be found in polynomial time. Consequently chi'_rn(G) = Theta(D). For D-regular graphs with D >= 3 the lower bound is 2D-1, and the upper bound is 5D-8. The bound 5D-8 holds more generally for every multigraph with D >= 3 which has a dominating P_{>=3}-system, and we characterize these multigraphs: such a system exists if and only if k_2(G-T) <= 2|T| holds for every set T of vertices, where k_2 counts the components on two vertices. In particular the bound holds for every graph with 3 <= D <= 4d-4, where d denotes the minimum degree. The paper is concluded with a conjecture on the tight upper bound on chi'_rn(G).

For a fixed graph $T$ and a fixed nonempty family $\mathcal{F}$ of graphs, let $\operatorname{ex}(G, T, \mathcal{F})$ denote the maximum number of copies of $T$ in an $\mathcal{F}$-free subgraph of a graph $G$. Alon and Shikhelman~[Algorithmica, 2022] conjectured that if $e(T)> 0$ and no member of $\mathcal{F}$ is a subgraph of a blow-up of $T$, then for every $\varepsilon>0$, approximating $\operatorname{ex}(G, T, \mathcal{F})$ within an additive error $n^{v(T)-\varepsilon}$ is NP-hard for an input graph $G$ on $n$ vertices. The previously established general cases for this conjecture include $\operatorname{ex}(G, K_2, \mathcal{F})$ by Alon, Shapira and Sudakov~[Ann. of Math., 2009], and $\operatorname{ex}(G, K_m, K_k)$ by Alon and Shikhelman. In this paper, we confirm this conjecture for every bipartite graph $T$ and every clique $T$. Our reductions use iterative leaf additions, symmetric rooted extensions and random transversals to encode the maximum number of edges in a $(k-1)$-colorable subgraph of $G$, where $k=\min_{F\in \mathcal{F}}\chi(F)$.

For positive integers $k,m,n$ with $\gcd(m,n)=1$, we prove a $(km,kn)$-square paths theorem: the symmetric function $(-1)^{k-1}\frac{[km]_q}{[k]_q}p_k[-MX^{m,n}]\cdot1$ is a weighted sum over $(km,kn)$-preference functions. This proves a corrected form of the conjecture of Iraci, Pagaria, Paolini, and Vanden Wyngaerd and generalizes the square paths theorem of Sergel. We encode preference functions by row strict tableaux and closed walks on a set of states, compare them with parking functions for each value of the touch statistic, and apply Mellit's compositional $(km,kn)$-shuffle theorem. Finally, we establish a fall Delta square theorem conjectured by Iraci, Pagaria, and Paolini.

Let $G$ be a split real semisimple group of adjoint type. We prove that the cellular decomposition of the totally nonnegative part of its wonderful compactification is a regular CW complex, confirming a conjecture of Galashin, Karp, and Lam. In particular, the closure of every cell is homeomorphic to a closed ball. The proof combines a Bruhat atlas compatible with total positivity and a CL-labeling of the augmented face poset. The atlas models the strata locally with open Richardson varieties in the flag variety of a larger Kac-Moody group and identifies their totally nonnegative cells.

Conway's $99$-graph is a strongly regular graph with parameters $(v,k,\lambda,\mu) = (99,14,1,2)$ and whose existence has long been an open problem. We use both a sum-of-squares program and the fact that a strongly regular graph with $\lambda = 1, \mu = 2$ has no diamond or $K_{2,3}$ subgraph to derive a loose upper bound for the number of triangles that such a graph could contain. This bound is less than the number of triangles that the graph's spectrum asserts, implying that it does not exist.

A graph is elementary if its edges can be coloured with two colours so that the two edges of every induced path on three vertices receive different colours. Elementary graphs are one of the two basic classes in the Chvátal–Sbihi decomposition of claw-free perfect graphs; they include all line graphs of bipartite multigraphs and all complements of bipartite graphs. We prove that every elementary graph $G$ satisfies $\ch(G)=\chi(G)=\omega(G)$, confirming for this class the conjecture of Gravier and Maffray that claw-free graphs are chromatic-choosable. The proof combines the Maffray–Reed description of elementary graphs as augmentations of line graphs of bipartite multigraphs, Galvin's theorem, an extremal (edge-maximal) choice of counterexample, and a Hall-type lemma describing how list colourings of a complete graph minus a matching can fail.

The object of interest is the maximum number, $h(n)$, of Hamilton cycles in an $n$-tournament. By considering the expected number of Hamilton cycles in various classes of random tournaments, we obtain new asymptotic lower bounds on $h(n)$. The best result so far is approximately $ 2.85584...$ times the expected number $g(n)$ of Hamilton cycles in a random $n$-tournament, and it is conjectured that $h(n)\sim cg(n)$ where $c\approx 2.855958$. The same statements hold for Hamilton paths.

In this paper, we study perfect codes in bi-Cayley graphs, focusing on when two subgroups $H$ and $K$ of a group G can form a perfect code of the form $H_0 \cup K_1$. We first consider the problem for a general group G and then examine what happens for special classes of groups, particularly abelian groups. Along the way, we obtain several results, including a necessary and sufficient condition for normal subgroups to form a perfect code, as well as an equivalent condition for abelian groups in terms of their Sylow 2-subgroups. Finally, we study the same problem for dihedral groups, providing results for this important class of non-abelian groups.

We resolve a conjecture of Łuczak, Mieczkowska and Šileikis on the maximal tail probability of sums of independent and identically distributed nonnegative random variables. Specifically, for every integer $k\ge1$ and every $0\le x\le1/k$, we prove \[ \sup_{\mathbb E X_1\le x}\mathbb{P}(X_1+\cdots+X_k\ge1) =\max\{1-(1-x)^k,(kx)^k\}, \] where the supremum is over nonnegative i.i.d.\ random variables $X_1,\ldots,X_k$. We also characterize all maximizing distributions. Using this result, we prove the Erdős Matching Conjecture for every fixed $k\ge3$, all sufficiently large $s$, and every $n\ge k(s+1)$. More specifically, we show that in this range, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $s+1$ pairwise disjoint edges is \[ \max\left\{ \binom{k(s+1)-1}{k}, \binom nk-\binom{n-s}{k} \right\}. \] Moreover, every extremal family consists either of all $k$-sets meeting a fixed $s$-set or of all $k$-sets contained in a fixed $(k(s+1)-1)$-set.

Let \(A,B\subset\R^2\) be nonempty Borel sets. We prove that \[ \dimH(gA-B)\ge\min\{\dimH A+\dimH B,2\} \] for almost all \(g\in\mathrm O(2)\). This bound is sharp in general, and the proof gives a corresponding statement for the lower Hausdorff dimension of convolutions of Frostman measures. We also prove that \(\LL^2(gA-B)>0\) for almost all \(g\) if both dimensions exceed \((\sqrt{13}-1)/2\). The second result follows from an angular decomposition, refined decoupling and a two-ends Furstenberg estimate. Some applications on intersection patterns are also addressed.

A hereditary graph class has strongly edge-sublinear separators if, for some constant $0 \leqslant b < 1$, every graph in the class with $m$ edges admits a balanced separator of size $\mathcal O(m^b)$. We show that Max Weight Independent Set (MWIS) admits a deterministic quasipolynomial-time approximation scheme (QPTAS) on every such class. In particular, this applies to graphs excluding any fixed graph as an induced minor, by a separator theorem of Korhonen and Lokshtanov [SODA 2024]. Our theorem generalizes the QPTASes for axis-parallel rectangles [Adamaszek and Wiese, FOCS 2013] and polygons [Adamaszek, Har-Peled, and Wiese, JACM 2019], or equivalently for packing connected subgraphs of a planar graph [Pilipczuk, van Leeuwen, and Wiese, Algorithmica 2020]. It also generalizes the QPTAS for MWIS on hereditary classes with strongly sublinear separators measured in vertices [Dvořák, EJC 2018; Dvořák and Lahiri, ESA 2021], as well as the QPTASes for graphs excluding particular induced minors [Chudnovsky et al., arXiv 2025; Bonnet et al., SWAT 2026; Czyżewska et al., arXiv 2026].

Let $\NT(m,n)$ be the total number of parts in the partitions of $n$ with rank $m$, and let $\MW(m,n)$ be the total number of ones in the partitions of $n$ with crank $m$. Using the uniform generalized-rank estimate of Liu and Zhou, we establish asymptotic formulas for $NT(m,n)$ and $\MW(m,n)$. Put $\beta_n=\pi/\sqrt{6n}$ and $L_n=\log(1/\beta_n)$. For every fixed $A>0$, uniformly for $|m|\le A\sqrt n$, we prove \begin{align*} \NT(m,n)&=\frac{p(n)L_n}{4}\sech^2\!\left(\frac{\beta_nm}{2}\right) +O_A\!\bigl(p(n)\log L_n\bigr) \intertext{ and} \MW(m,n)&=\frac{p(n)}4\log(1+e^{-\beta_nm}) \sech^2\!\left(\frac{\beta_nm}{2}\right) +O_A\!\bigl(p(n)\sqrt{\beta_nL_n}\bigr). \end{align*}

Splines on Weyl groups model the equivariant cohomology of regular semisimple Hessenberg varieties, and the natural group action on them encodes the monodromy action. The resulting representations are of great interest in algebraic combinatorics for their connections to Hecke algebra characters, LLT polynomials, and chromatic symmetric functions. We study the analogous construction for dihedral groups, which in general have no underlying geometry. Using purely algebraic methods, we show that these spline modules are free, construct explicit bases, and compute their Hilbert series, recovering the familiar properties of the geometric setting. We also determine the associated dihedral group representations completely. Unlike what is conjectured for the symmetric group case, these need not be permutation representations.

A proper coloring of Euclidean space is a coloring that assigns different colors to points at distance one. Using the planar character-law rigidity theorem of OpenAI, we prove that every finite proper coloring of $\R^3$ can be replaced by a proper Lebesgue-measurable coloring with the same number of colors. As a corollary we obtain that the ordinary and measurable chromatic numbers of $\R^3$ are equal, and both lie between $7$ and $15$.

Entanglement entropies of multipartite pure states satisfy linear inequalities; for stabilizer states, the backbone of quantum error correction, the achievable entropy vectors generate a convex cone. We show that, after a linear lift, the closure of this cone is exactly the self-dual slice of a classical object on one more element than there are parties: the closed cone generated by the rank functions of arrangements of subspaces over $\mathrm{GF}(d)$ for a prime local dimension $d$, and of subgroups of abelian groups of exponent dividing $d$ in general. This correspondence brings two features of rank inequalities into play. First, the purifier acts as an extra variable: already for five parties the stabilizer cone is strictly smaller than the quantum linear rank cone, with which earlier evidence suggested it might coincide; an explicit six-variable inequality witnesses the gap. Second, the characteristic of the field enters, so the cone depends on the local dimension. It does not for up to four parties. With enough parties, the cone of $\mathbb{Z}_d$ qudits is contained in that of $\mathbb{Z}_{d'}$ qudits if and only if $d$ divides $d'$, and already for six parties the logical zero state of Steane's code lies outside the stabilizer cone of every odd local dimension. For this last result we show that the seven-variable characteristic-dependent inequalities we checked all follow from Shannon inequalities on the self-dual slice, and instead transfer one of them through a family of subgroups derived from the Fano plane.

We construct a fixed triangular family of noncommutative polynomials that spans the joint cyclic module of a real symmetric pair from any prescribed vector. In dimension at most $N$, when the second matrix has spectral diameter at most $N-1$, the family has $N(N+1)/2$ members, degree at most $N-1$, and a shared arithmetic circuit of size $O(N^2)$. No rank-dependent choice of coefficients is made. The proof uses a rational differential connection with a small self-adjoint residue: a primitive polynomial exterior vector for an invariant subspace must have an integer degree smaller than one. For connected trees, the probes and the pointed adjacency–degree reconstruction theorem give one explicit self-adjoint polynomial $p_N\in\mathbb{Z}\langle a,d\rangle$ whose trace separates all trees of order at most $N$. Its circuit has $O(N^2)$ gates, formal degree at most $3N-2$, and polynomial bit complexity. Only the original adjacency and degree matrices, the identity, and ordinary matrix arithmetic are used. A separate integer sum-of-squares construction gives a fixed positive semidefinite representative on which the all-one vector is cyclic; its degree is $O(N^2)$. A linear trace-degree lower bound shows that the first construction has the correct asymptotic degree scale.

It is shown that certain finite partially ordered sets arising from unit interval determinantal facet ideals satisfy the $\lambda$-chain condition. In addition, their applications to commutative algebra are briefly discussed.

The extension complexity $\operatorname{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that projects onto it. Motivated by the complexity of neural networks, Hertrich and Loho introduced the virtual extension complexity $\operatorname{vxc}(P)$ of a polytope $P$, which is the minimum of $\operatorname{xc}(Q) + \operatorname{xc}(R)$ among all polytopes $Q$ and $R$ such that $P+Q=R$. We show the following results pertaining to virtual extension complexity. First, we resolve the core open problem on the topic by showing the virtual extension complexity can be exponentially smaller than extension complexity. Second, we show there are matroid base polytopes with exponential virtual extension complexity by adapting a semi-algebraic counting argument by Hrubeš that extends the counting arguments of Rothvoss. Finally, we show that the maximum virtual extension complexity of a polygon with $n$ vertices is $\Theta(\sqrt{n})$. This resolves an analogue of a question on extension complexity of Fiorini, Rothvoss, and Tiwary in 2011 in the case of virtual extension complexity.

We study the behavior of the $k$-th-order nonhomomorphicity of Boolean functions under restriction to affine subspaces. For odd $k\ge3$ and $0\le t<n$, we introduce the $t$-restriction $k$-th-order nonhomomorphicity $P_{k,t}(f)$, defined as the minimum of $P_k(f_{\mid A})$ over affine spaces of codimension at most $t$. For a direct sum of $m$ monomials of degrees at least two, we determine $P_{k,t}$ exactly. When $t<m$, a minimum is obtained by setting one variable equal to zero in each of the $t$ monomials of lowest degree. When $m\le t<n$, the minimum is zero. The proof combines multiplicativity of even Walsh moments with a contraction argument for affine restrictions. We establish a general formula for the Walsh transform of a Boolean function restricted to an affine subspace of arbitrary codimension, yielding an exact spectral characterization of the $t$-restriction normalized nonlinearity $\rnnl_t(f)$. As odd $k$ tends to infinity, $(1-2P_{k,t}(f))^{1/(k+1)}$ converges to $1-2\rnnl_t(f)$. We further prove that every non-affine function has a hyperplane restriction with strictly smaller odd-order nonhomomorphicity. Finally, we determine $\rnnl_1(T_{d,n})$ exactly for all threshold functions with $n\ge3$. This proves the two threshold-function conjectures left open in~\cite{SergeAnaPante2025}, including the previously unresolved odd-majority case, where a minimizing hyperplane is $x_1+x_2=0$.

We study the greedy sequence of positive integers introduced by Slater and Vélez. They proved that it is a permutation of the positive integers but left open whether its sequence of consecutive absolute differences is also a permutation. We prove that it is.

The involution numbers $1,1,2,4,10,26,\ldots$ repeat modulo $3$, $5$ and $7$. We explain this, and many similar patterns, by one rule. Many sequences count permutations in which every cycle of length $m$ carries a weight $w_m$ (involutions, derangements, "sets of lists", and many others in the OEIS). Writing $A(n)$ for such a count, the rule is $A(n+k)\equiv A(n)A(k)$ modulo $\mathrm{lcm}(n,k)$. It generalizes our earlier proof of the case modulo $k$ for one such sequence, but now it holds for all weights at once, as a coefficientwise congruence between universal polynomials, and we find the largest modulus that works for the universal polynomials, $nk/\mathrm{rad}(\gcd(n,k))$, and show that nothing larger does. We then show that passing from $k$ to $pk$ behaves like raising to the $p$-th power modulo a high power of $p$, with a sharp exponent for the polynomials, which gives $A(p^r)\equiv(w_1+(p-1)w_p)^{p^{r-1}}$ modulo $p^r$ for odd $p$, and explains why the prime $2$ is special. A family of sequences defined by Lagrange inversion fits the same framework, which proves Bala's conjecture for the whole family (the single sequence A278070 had been proved earlier by an automated system) and the divisibility observed in the entries A251568 and A251569. As further applications we prove OEIS conjectures of Richman, Hanna, van Hoeij and Bala (A227937), and the theorems apply to 101 OEIS sequences, whose congruences are recorded in a supplementary catalogue. Higher-order versions of the content formula are treated in a separate paper.