Annals of Combinatorics最新文献

A connected graph G of diameter (textrm{diam}(G) ge ell ) is (ell )-distance-balanced if (|W_{xy}|=|W_{yx}|) for every (x,yin V(G)) with (d_{G}(x,y)=ell ), where (W_{xy}) is the set of vertices of G that are closer to x than to y. We prove that the generalized Petersen graph GP(n, k) is (textrm{diam}(GP(n,k)))-distance-balanced provided that n is large enough relative to k. This partially solves a conjecture posed by Miklavič and Šparl (Discrete Appl Math 244:143–154, 2018). We also determine (textrm{diam}(GP(n,k))) when n is large enough relative to k.

We consider the Grothendieck polynomials appearing in the K-theory of Grassmannians, which are analogs of Schur polynomials. This paper aims to establish a version of the Murnaghan–Nakayama rule for Grothendieck polynomials of the Grassmannian type. This rule allows us to express the product of a Grothendieck polynomial with a power-sum symmetric polynomial into a linear combination of other Grothendieck polynomials.

In this paper, we focus on the truncations of three classical theta series of Euler and Gauss, and analyze their combinatorial properties which play a key role in proving these truncated identities. Several interesting partition identities are established bijectively.

Kochol [6] gave a new expansion formula for the Tutte polynomial of a matroid using the notion of compatible sets, and asked how this expansion relates to the internal-external activities formula. Here, we provide an answer, which is obtained as a special case of a generalized version of the expansion formula to Las Vergnas’s trivariate Tutte polynomials of matroid perspectives [10]. The same generalization to matroid perspectives and bijection with activities have been independently proven by Kochol in [5] and [7] in parallel with this work, but using different methods. Kochol proves both results recursively using the contraction-deletion relations, whereas we give a more direct proof of the bijection and use that to deduce the compatible sets expansion formula from Las Vergnas’s activities expansion.

Let (F(z_1,dots ,z_d)) be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate residues and saddle-point approximations. Because the singular set of F is the union of hyperplanes, we are able to make explicit the topological decompositions which arise in the multivariate singularity analysis. In addition to effective and explicit asymptotic results, we provide the first results on transitions between different asymptotic regimes, and provide the first software package to verify and compute asymptotics in non-smooth cases of analytic combinatorics in several variables. It is also our hope that this paper will serve as an entry to the more advanced corners of analytic combinatorics in several variables for combinatorialists.

An equidistant X-cactus is a type of rooted, arc-weighted, directed acyclic graph with leaf set X, that is used in biology to represent the evolutionary history of a set (X) of species. In this paper, we introduce and investigate the space of equidistant X-cactuses. This space contains, as a subset, the space of ultrametric trees on X that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning ranked rooted X-cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of (X) that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted X-trees in terms of partitions of X, which provides an alternative proof that the space of ultrametric trees on X is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant X-cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.

Discrete Morse theory, a cell complex-analog to smooth Morse theory allowing homotopic tools in the discrete realm, has been developed over the past few decades since its original formulation by Robin Forman in 1998. In particular, discrete gradient vector fields on simplicial complexes capture important topological features of the structure. We prove that the characteristic polynomials of the Laplacian matrices of a simplicial complex are generating functions for discrete gradient vector fields if the complex is a triangulation of an orientable manifold. Furthermore, we provide a full characterization of the correspondence between rooted forests in higher dimensions and discrete gradient vector fields.

This work highlights the existence of partial symmetries in large families of iterated plethystic coefficients. The plethystic coefficients involved come from the expansion in the Schur basis of iterated plethysms of Schur functions indexed by one-row partitions.The partial symmetries are described in terms of an involution on partitions, the flip involution, that generalizes the ubiquitous (omega ) involution. Schur-positive symmetric functions possessing this partial symmetry are termed flip-symmetric. The operation of taking plethysm with (s_lambda ) preserves flip-symmetry, provided that (lambda ) is a partition of two. Explicit formulas for the iterated plethysms (s_2circ s_bcirc s_a) and (s_ccirc s_2circ s_a), with a,  b,  and c (ge ) 2 allow us to show that these two families of iterated plethysms are flip-symmetric. The article concludes with some observations, remarks, and open questions on the unimodality and asymptotic normality of certain flip-symmetric sequences of iterated plethystic coefficients.

Rectangular standard Young tableaux with 2 or 3 rows are in bijection with (U_q(mathfrak {sl}_2))-webs and (U_q(mathfrak {sl}_3))-webs, respectively. When (mathcal {W}) is a web with a reflection symmetry, the corresponding tableau (T_mathcal {W}) has a rotational symmetry. Folding (T_mathcal {W}) transforms it into a domino tableau (D_mathcal {W}). We study the relationships between these correspondences. For 2-row tableaux, folding a rotationally symmetric tableau corresponds to “literally folding” the web along its axis of symmetry. For 3-row tableaux, we give simple algorithms, which provide direct bijective maps between symmetrical webs and domino tableaux (in both directions). These details of these algorithms reflect the intuitive idea that (D_mathcal {W}) corresponds to “(mathcal {W}) modulo symmetry”.