Annals of Combinatorics最新文献

We correct a theorem on caterpillar posets in Lesnevich and McNamara (Ann Comb 26(1):171–204, 2022). In strengthening the hypotheses on the caterpillar posets we consider, we are also able to strengthen the conclusion on the types of positivity that result.

We use Maya diagrams to refine the criterion by Fulton and Woodward for the smallest powers of the quantum parameter q that occur in a product of Schubert classes in the (small) quantum cohomology of partial flags. Our approach using Maya diagrams yields a combinatorial proof that the minimal quantum degrees are unique for partial flags. Furthermore, visual combinatorial rules are given to perform precise calculations.

In this paper, we give a new combinatorial interpretation for the Rogers–Ramanujan–Gordon partitions for (k=3). Our interpretation is given by base partition and moves ideas. We conclude the paper with some research questions related to the generalization of this approach.

We introduce two simplicial complexes, the noncrossing matching complex and the noncrossing bipartite complex. Both complexes are intimately related to the bubble lattice introduced in our earlier article “Bubble Lattices I: Structure” (arXiv:2202.02874). We study these complexes from both an enumerative and a topological point of view. In particular, we prove that these complexes are shellable and give explicit formulas for certain refined face numbers. Lastly, we conjecture an intriguing connection of these refined face numbers to the so-called M-triangle of the shuffle lattice.

Randomly sampling an acyclic orientation on the complete bipartite graph (K_{n,k}) with parts of size n and k, we investigate the length of the longest path. We provide a probability generating function for the distribution of the longest path length, and we use analytic combinatorics to perform asymptotic analysis of the probability distribution in the case of equal part sizes (n=k) tending toward infinity. We show that the distribution is asymptotically Gaussian, and we obtain precise asymptotics for the mean and variance. These results address a question asked by Peter J. Cameron.

For simple graphs X and Y on n vertices, the friends-and-strangers graph (textsf{FS}(X,Y)) is the graph whose vertex set consists of all bijections (sigma : V(X) rightarrow V(Y)), where two bijections (sigma ) and (sigma ') are adjacent if and only if they agree on all but two adjacent vertices (a, b in V(X)) such that (sigma (a), sigma (b) in V(Y)) are adjacent in Y. Resolving a conjecture of Wang, Lu, and Chen, we completely characterize the connectedness of (textsf{FS}(X, Y)) when Y is a complete bipartite graph. We further extend this result to when Y is a complete multipartite graph. We also determine when (textsf{FS}(X, Y)) has exactly two connected components where X is bipartite and Y is a complete bipartite graph.

In recent years, the techniques of analytic combinatorics in several variables (ACSV) have been applied to determine asymptotics for several families of lattice path models restricted to the orthant ({mathbb {N}}^d) and defined by step sets ({mathcal {S}}subset {-1,0,1}^dsetminus {textbf{0}}). Using the theory of ACSV for smooth singular sets, Melczer and Mishna determined asymptotics for the number of walks in any model whose set of steps ({mathcal {S}}) is ‘highly symmetric’ (symmetric over every axis). Building on this work, Melczer and Wilson determined asymptotics for all models where ({mathcal {S}}) is ‘mostly symmetric’ (symmetric over all but one axis) except for models whose set of steps have a vector sum of zero but are not highly symmetric. In this paper, we complete the asymptotic classification of the mostly symmetric case by analyzing a family of saddle-point-like integrals whose amplitudes are singular near their saddle points.

We study the interplay between Sabbah’s mixed Hodge structure for tame regular functions and Ehrhart theory for polytopes. We first analyze the Poincaré polynomial of the Hodge filtration of this mixed Hodge structure (we call this Poincaré polynomial the (theta )-vector). Using the symmetry of the Hodge numbers involved, we show that it shares many properties with the (h^*)-vector of a polytope. For instance, we define from the (theta )-vector the Hodge–Ehrhart polynomial of a general tame function and we show that it satisfies a reciprocity law, analogous to the one satisfied by the Ehrhart polynomial of a polytope. We study the roots of this Hodge–Ehrhart polynomial, in particular their distribution around some critical lines. Using techniques coming from singularity theory, we also show a Thom–Sebastiani type theorem for the (theta )-vector. Finally, we offer some linear inequalities among the coefficients of the (theta )-vectors which could be helpful to test if a polynomial is a (theta )-vector or not. In the very particular case of convenient and nondegenerate Laurent polynomials, we show (using the Brieskorn lattice and the V-filtration) that the previous results agree with the classical ones in combinatorics and we emphasize various combinatorial properties of Sabbah’s Hodge numbers: on the way, this provides an alternative interpretation of prior results about the (limit) Hodge numbers of hypersurfaces in a torus obtained in a different framework by Danilov–Khovanskiĭ and more recently by Katz–Stapledon.

Partition hook lengths have wide-ranging applications in combinatorics, number theory, physics, and representation theory. We study two infinite families of random variables associated with t-hooks. For fixed (tge 1,) if (Y_{t;,n}) counts the number of hooks of length t in a random integer partition of n, we prove a uniform local limit theorem for (Y_{t;,n}) on any bounded set of ({mathbb {R}}.) To achieve this, we establish an asymptotic formula with a power-saving error term for the number of partitions of n with m many t-hooks. In contrast, we define ({widehat{Y}}_{t;,n}) as the count of hooks divisible by t in a randomly chosen partition of n. While ({widehat{Y}}_{t;,n}) converges in distribution, we show that it fails to satisfy the local limit theorem for any (t ge 2). The proofs employ the multivariable saddle-point method, asymptotic formulas for the number of t-core partitions from Anderson and Lulov–Pittel, and estimates of certain exponential sums. Notably, for (t=4,) the analysis involves the asymptotic behavior of class numbers of imaginary quadratic fields.

Recent works have sought to realize certain families of orthogonal, symmetric polynomials as partition functions of well-chosen classes of solvable lattice models. Many of these use Boltzmann weights arising from the trigonometric six-vertex model R-matrix (or generalizations or specializations of these weights). In this paper, we seek new variants of bosonic models on lattices designed for Cartan type C root systems, whose partition functions match the zonal spherical function in type C. Under general assumptions, we find that this is possible for all highest weights in rank two and three, but not for higher rank. In ranks two and three, this may be regarded as a new generating function formula for zonal spherical functions (also known as Hall–Littlewood polynomials) in type C.