We compute the Galois groups of the reductions modulo a prime number p of the generating series of Apéry numbers, Domb numbers and Almkvist–Zudilin numbers. We observe in particular that their behaviour is governed by congruence conditions on p.

Balister, the second author, Groenland, Johnston, and Scott recently showed that there are asymptotically $C4^n/n^{3/4}$ many unordered sequences that occur as degree sequences of graphs with $n$ vertices. Combining limit theory for infinitely divisible distributions with a new connection between a class of random walk trajectories and a subset counting formula from additive number theory, we describe $C$ in terms of Walkup’s number of rooted plane trees. The bijection is related to an instance of the Lévy–Khintchine formula. Our main result complements a result of Stanley, that ordered graphical sequences are related to quasi-forests.

Inspired by the theory of hyperbolic polynomials and Hodge theory, we develop the theory of Lorentzian polynomials on cones. This notion captures the Hodge-Riemann relations of degree zero and one. Motivated by fundamental properties of volume polynomials of Chow rings of simplicial fans, we define a class of multivariate polynomials which we call hereditary polynomials. We give a complete and easily checkable characterization of hereditary Lorentzian polynomials. This characterization is used to give elementary and simple proofs of the Heron-Rota-Welsh conjecture for the characteristic polynomial of a matroid, and the Alexandrov-Fenchel inequalities for convex bodies.

We then characterize Chow rings of simplicial fans which satisfy the Hodge-Riemann relations of degree zero and one, and we prove that this property only depends on the support of the fan.

Several different characterizations of Lorentzian polynomials on cones are provided.

Andrews and El Bachraoui [‘On two-color partitions with odd smallest part’, Preprint (2024), arXiv:2410. 14190] recently investigated identities involving two-colour partitions, with particular emphasis on their connection to overpartitions, and posed questions regarding possible companion results. Subsequently, Chen and Zou [‘Combinatorial proofs for two-colour partitions’, Bull. Aust. Math. Soc. 113(1) (2025), to appear] obtained some companion results by employing q-series identities and generating functions. In addition, they presented a combinatorial proof for one of their own results and one of the results of Andrews and El Bachraoui. They posed questions regarding combinatorial proofs of the remaining companion results. In this paper, we provide such proofs.

This paper is concerned with a duality between $r$-regular permutations and $r$-cycle permutations, and a monotone property due to Bóna-McLennan-White on the probability $p_r(n)$ for a random permutation of $\{1,2,\ldots, n\}$ to have an $r$th root, where $r$ is a prime. For $r=2$, the duality relates permutations with odd cycles to permutations with even cycles. For the general case where $r\geq 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles coloured by one of the colours $1, 2, \ldots, r-1 $. In this setup, we discover a bijection between $r$-regular permutations and enriched $r$-cycle permutations, which in turn yields a stronger version of an inequality of Bóna-McLennan-White. This leads to a fully combinatorial understanding of the monotone property, thereby answering their question. When $r$ is a prime power $q^l$, we further show that $p_r(n)$ is monotone. In the case that $n+1 \not\equiv 0 \pmod q$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.

In this article, we generalize Andrews’ partitions separated by parity to overpartitions in two ways. We investigate the generating functions for $16$ overpartition families whose parts are separated by parity, and we prove various q-series identities for these functions. These identities include relations to modular forms, q-hypergeometric series, and mock modular forms.

In his seminal work on partitions and divisor functions, MacMahon introduced the following two q-series:

\begin{align*}A_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k} \frac{q^{s_1+s_2+\cdots+s_k}}{(1-q^{s_1})^2(1-q^{s_2})^2\cdots(1-q^{s_k})^2},\qquad\quad\quad\quad\\[6pt]C_k(q):= \sum_{0 \lt s_1 \lt s_2 \lt \cdots \lt s_k}\frac{q^{2(s_1+s_2+\cdots+s_k)-k}}{(1-q^{2s_1-1})^2(1-q^{2s_2-1})^2\cdots(1-q^{2s_k-1})^2}.\end{align*}

In 2013, Andrews and Rose proved that $A_k(q)$ and $C_k(q)$ are quasimodular forms of weight $\leq 2k$. Recently, Ono and Singh proved two interesting identities involving $A_k(q)$ and $C_k(q)$ and showed that the generating functions for the three-coloured partition function $p_3(n)$ and the overpartition function $\overline{p}(n)$ have infinitely many closed formulas in terms of MacMahon’s quasimodular forms $A_k(q)$ and $C_k(q)$. In this paper, we introduce the finite forms $A_{k,n}(q)$ and $C_{k,n}(q)$ of MacMahon’s q-series $A_k(q)$ and $C_k(q)$ and prove two identities which generalize Ono–Singh’s identities. We also prove some new identities involving $A_{k,n}(q)$, $C_{k,n}(q)$ and certain infinite products based on two Bailey pairs. Those identities are analogous to Ono–Singh’s identities.

We introduce a natural weighted enumeration of lattice points in a polytope, and give a Brion-type formula for the corresponding generating function. The weighting has combinatorial significance, and its generating function may be viewed as a generalization of the Rogers–Szegő polynomials. It also arises from the geometry of the toric arc scheme associated to the normal fan of the polytope. We show that the asymptotic behaviour of thecoefficients at $q=1$ is Gaussian.

For each of the four particle processes given by Dieker and Warren, we show the n-step transition kernels are given by the (dual) (weak) refined symmetric Grothendieck functions up to a simple overall factor. We do so by encoding the particle dynamics as the basis of free fermions first introduced by the first author, which we translate into deformed Schur operators acting on partitions. We provide a direct combinatorial proof of this relationship in each case, where the defining tableaux naturally describe the particle motions.

We obtain explicit expressions for the class in the Grothendieck group of varieties of the moduli space $\overline {\mathcal{M}}_{0,n}$. This information is equivalent to the Poincaré polynomial and yields explicit expressions for the Betti numbers of $\overline {\mathcal{M}}_{0,n}$ in terms of Stirling or Bernoulli numbers. The expressions are obtained by solving a differential equation characterizing the generating function for the Poincaré polynomials, determined by Manin in the 1990s and equivalent to Keel’s recursion for the Betti numbers of $\overline {\mathcal{M}}_{0,n}$. Our proof reduces the solution to two combinatorial identities, verified by applying Lagrange series. We also study generating functions for the individual Betti numbers. These functions are determined by a set of polynomials $p^{(k)}_m(z)$, $k\geqslant m$. These polynomials are conjecturally log-concave; we verify this conjecture for several infinite families, corresponding to generating functions for $2k$-Betti numbers of $\overline {\mathcal{M}}_{0,n}$ for all $k\leqslant 100$. Further, studying the polynomials $p^{(k)}_m(z)$, we prove that the generating function for the Grothendieck class can be written in terms of a series of rational functions in the principal branch of the Lambert W-function. We include an interpretation of the main result in terms of Stirling matrices and a discussion of the Euler characteristic of $\overline {\mathcal{M}}_{0,n}$.

We consider a class of sequence $c(n)$ with the following asymptotic form:

$$ \begin{align*}c(n) \sim \frac C{n^\kappa} \exp\left(\sum_{\lambda\in\mathcal S}A_\lambda n^\lambda \right) \sum_{\mu\in\mathcal T} \frac{\beta_\mu}{n^\mu} \qquad (n \to \infty).\end{align*} $$

$c(n)$

$c(n)$

$\ell $

$S_n$

$\mathfrak {su}(3)$

$\mathfrak {so}(5),$

We prove that on any transitive graph G with infinitely many ends, a self-avoiding walk of length n is ballistic with extremely high probability, in the sense that there exist constants $c,t>0$ such that $\mathbb {P}_n(d_G(w_0,w_n)\geq cn)\geq 1-e^{-tn}$ for every $n\geq 1$. Furthermore, we show that the number of self-avoiding walks of length n grows asymptotically like $\mu _w^n$, in the sense that there exists $C>0$ such that $\mu _w^n\leq c_n\leq C\mu _w^n$ for every $n\geq 1$. These results generalise earlier work by Li (J. Comb. Theory Ser. A, 2020). The key to this greater generality is that in contrast to Li’s approach, our proof does not require the existence of a special structure that enables the construction of separating patterns. Our results also extend more generally to quasi-transitive graphs with infinitely many ends, satisfying the additional technical property that there is a quasi-transitive group of automorphisms of G which does not fix an end of G.

In this paper, we will investigate the following inequality

\begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*}

which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.

Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

For any integer $t \geq 2$, we prove a local limit theorem (LLT) with an explicit convergence rate for the number of parts in a uniformly chosen t-regular partition. When $t = 2$, this recovers the LLT for partitions into distinct parts, as previously established in the work of Szekeres [‘Asymptotic distributions of the number and size of parts in unequal partitions’, Bull. Aust. Math. Soc. 36 (1987), 89–97].

We establish a polynomial ergodic theorem for actions of the affine group of a countable field K. As an application, we deduce—via a variant of Furstenberg’s correspondence principle—that for fields of characteristic zero, any ‘large’ set $E\subset K$ contains ‘many’ patterns of the form $\{p(u)+v,uv\}$, for every non-constant polynomial $p(x)\in K[x]$. Our methods are flexible enough that they allow us to recover analogous density results in the setting of finite fields and, with the aid of a finitistic variant of Bergelson’s ‘colouring trick’, show that for $r\in \mathbb N$ fixed, any r-colouring of a large enough finite field will contain monochromatic patterns of the form $\{u,p(u)+v,uv\}$. In a different direction, we obtain a double ergodic theorem for actions of the affine group of a countable field. An adaptation of the argument for affine actions of finite fields leads to a generalization of a theorem of Shkredov. Finally, to highlight the utility of the aforementioned finitistic ‘colouring trick’, we provide a conditional, elementary generalization of Green and Sanders’ $\{u,v,u+v,uv\}$ theorem.

Vertically symmetric alternating sign matrices (VSASMs) of order $2n+1$ are known to be equinumerous with lozenge tilings of a hexagon with side lengths $2n+2,2n,2n+2,2n,2n+2,2n$ and a central triangular hole of size $2$ that exhibit a cyclical as well as a vertical symmetry, but finding an explicit bijection proving this belongs to the most difficult problems in bijective combinatorics. Towards constructing such a bijection, we generalize the result by introducing certain natural extensions for both objects along with $n+3$ parameters and show that the multivariate generating functions with respect to these parameters coincide. This is a significant step from a constant number of equidistributed statistics to a linear number of statistics in n. The equinumeracy of VSASMs and the lozenge tilings is then an easy consequence of this result, which is obtained by specializing the generating functions to signed enumerations for both types of objects and then applying certain sign-reversing involutions. Another main result concerns the expansion of the multivariate generating function into symplectic characters as a sum over totally symmetric self-complementary plane partitions, which is in perfect analogy to the situation for ordinary ASMs where the Schur expansion can be written as a sum over totally symmetric plane partitions. This is exciting as it is reminiscent of the well-known Cauchy identity, and the Cauchy identity does have a bijective proof using the Robinson-Schensted-Knuth correspondence, and thus the result raises the question of whether there is a variation of the Robinson–Schensted–Knuth correspondence that does eventually lead to a bijective proof.

The Chan–Robbins–Yuen polytope ($CRY_n$) of order n is a face of the Birkhoff polytope of doubly stochastic matrices that is also a flow polytope of the directed complete graph $K_{n+1}$ with netflow $(1,0,0, \ldots , 0, -1)$. The volume and lattice points of this polytope have been actively studied; however, its face structure has received less attention. We give generating functions and explicit formulas for computing the f-vector by using Hille’s (2003) result bijecting faces of a flow polytope to certain graphs, as well as Andresen–Kjeldsen’s (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes of the complete graph having arbitrary (non-negative) netflow vectors and recover the f-vector of the Tesler polytope of Mészáros–Morales–Rhoades (2017).

We give combinatorially controlled series solutions to Dyson–Schwinger equations with multiple insertion places using tubings of rooted trees and investigate the algebraic relation between such solutions and the renormalization group equation.

Modified ascent sequences, initially defined as the bijective images of ascent sequences under a certain hat map, have also been characterized as Cayley permutations where each entry is a leftmost copy if and only if it is an ascent top. These sequences play a significant role in the study of Fishburn structures. In this paper, we investigate (primitive) modified ascent sequences avoiding a pattern of length 4 by combining combinatorial and algebraic techniques, including the application of the kernel method. Our results confirm several conjectures posed by Cerbai.