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.

The lattice walks in the plane starting at the origin $\mathbf {0}$ with steps in $\{-1,0,1\}^{2}\setminus \{\mathbf {0}\}$ are called king walks. We investigate enumeration and divisibility for higher dimensional king walks confined to certain regions. Specifically, we establish an explicit formula for the number of $(r+s)$-dimensional king walks of length n ending at $(a_1,\ldots ,a_r,b_1,\ldots ,b_s)$ which never dip below $x_i=0$ for $i=1,\ldots ,r$. We also derive divisibility properties for the number of $(r+s)$-dimensional king walks of length p (an odd prime) through group actions.

Let $t\geq 2$ and $k\geq 1$ be integers. A t-regular partition of a positive integer n is a partition of n such that none of its parts is divisible by t. Let $b_{t,k}(n)$ denote the number of hooks of length k in all the t-regular partitions of n. In this article, we prove some inequalities for $b_{t,k}(n)$ for fixed values of k. We prove that for any $t\geq 2$, $b_{t+1,1}(n)\geq b_{t,1}(n)$, for all $n\geq 0$. We also prove that $b_{3,2}(n)\geq b_{2,2}(n)$ for all $n>3$, and $b_{3,3}(n)\geq b_{2,3}(n)$ for all $n\geq 0$. Finally, we state some problems for future works.

We prove a criterion of when the dual character $\chi _{D}(x)$ of the flagged Weyl module associated a diagram D in the grid $[n]\times [n]$ is zero-one, that is, the coefficients of monomials in $\chi _{D}(x)$ are either 0 or 1. This settles a conjecture proposed by Mészáros–St. Dizier–Tanjaya. Since Schubert polynomials and key polynomials occur as special cases of dual flagged Weyl characters, our approach provides a new and unified proof of known criteria for zero-one Schubert/key polynomials due to Fink–Mészáros–St. Dizier and Hodges–Yong, respectively.

This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under which these semisimple Hessenberg varieties are irreducible. We determine the dimension of each irreducible Hessenberg variety under consideration and show that the number of such varieties is a Catalan number. We then apply a theorem of Brion to compute a polynomial representative for the cohomology class of each such variety. Additionally, we calculate the intersections of a standard (Schubert) hyperplane section of the flag variety with each of our Hessenberg varieties and prove that this intersection possesses a cohomological multiplicity-free property.

We introduce a family of polynomials, which arise in three distinct ways: in the large N expansion of a matrix integral, as a weighted enumeration of factorizations of permutations, and via the topological recursion. More explicitly, we interpret the complex Grassmannian $\mathrm {Gr}(M,N)$ as the space of $N \times N$ idempotent Hermitian matrices of rank M and develop a Weingarten calculus to integrate products of matrix elements over it. In the regime of large N and fixed ratio $\frac {M}{N}$, such integrals have expansions whose coefficients count factorizations of permutations into monotone sequences of transpositions, with each sequence weighted by a monomial in $t = 1 - \frac {N}{M}$. This gives rise to the desired polynomials, which specialise to the monotone Hurwitz numbers when $t = 1$. These so-called deformed monotone Hurwitz numbers satisfy a cut-and-join recursion, a one-point recursion, and the topological recursion. Furthermore, we conjecture on the basis of overwhelming empirical evidence that the deformed monotone Hurwitz numbers are real-rooted polynomials whose roots satisfy remarkable interlacing phenomena. An outcome of our work is the viewpoint that the topological recursion can be used to “topologise” sequences of polynomials, and we claim that the resulting families of polynomials may possess interesting properties.

A partition is called a t-core if none of its hook lengths is a multiple of t. Let $a_t(n)$ denote the number of t-core partitions of n. Garvan, Kim and Stanton proved that for any $n\geq1$ and $m\geq1$, $a_t\big(t^mn-(t^2-1)/24\big)\equiv0\pmod{t^m}$, where $t\in\{5,7,11\}$. Let $A_{t,k}(n)$ denote the number of partition k-tuples of n with t-cores. Several scholars have been subsequently investigated congruence properties modulo high powers of 5 for $A_{5,k}(n)$ with $k\in\{2,3,4\}$. In this paper, by utilizing a recurrence related to the modular equation of fifth order, we establish dozens of congruence families modulo high powers of 5 satisfied by $A_{5,k}(n)$, where $4\leq k\leq25$. Moreover, we deduce an infinite family of internal congruences modulo high powers of 5 for $A_{5,4}(n)$. In particular, we generalize greatly a recent result on a congruence family modulo high powers of 5 enjoyed by $A_{5,4}(n)$, which was proved by Saikia, Sarma and Talukdar (Indian J. Pure Appl. Math., 2024). Finally, we conjecture that there exists a similar phenomenon for $A_{5,k}(n)$ with $k\geq26$.

The payoff in the Chow–Robbins coin-tossing game is the proportion of heads when you stop. Stopping to maximize expectation was addressed by Chow and Robbins (1965), who proved there exist integers ${k_n}$ such that it is optimal to stop at n tosses when heads minus tails is ${k_n}$. Finding ${k_n}$ was unsolved except for finitely many cases by computer. We prove an $o(n^{-1/4})$ estimate of the stopping boundary of Dvoretsky (1967), which then proves ${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta (\! -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil $ except for n in a set of density asymptotic to 0, at a power law rate. Here, $\alpha$ is the Shepp–Walker constant from the Brownian motion analog, and $\zeta$ is Riemann’s zeta function. An $n^{ - 1/4}$ dependence was conjectured by Christensen and Fischer (2022). Our proof uses moments involving Catalan and Shapiro Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund (2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod’s embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in another way. We use them first for yet many more examples and a conjecture, then algebraically in the tree, with feedback to get much sharper Value bounds near the border, and analytic results. Also, we give a formula that gives the exact optimal stop rule for all n up to about a third of a billion; it uses the analytic result plus terms arrived at empirically.