Annals of Combinatorics最新文献

The q-binomial coefficients are q-analogues of the binomial coefficients, counting the number of k-dimensional subspaces in the n-dimensional vector space ({mathbb {F}}^n_q) over ({mathbb {F}}_{q}.) In this paper, we define a Euclidean analogue of q-binomial coefficients as the number of k-dimensional subspaces which have an orthonormal basis in the quadratic space (({mathbb {F}}_{q}^{n},x_{1}^{2}+x_{2}^{2}+cdots +x_{n}^{2}).) We prove its various combinatorial properties compared with those of q-binomial coefficients. In addition, we formulate the number of subspaces of other quadratic types and study some related properties.

Let (mathcal{S}mathcal{T}_{lambda }(tau )) denote the set of symmetric transversals of a self-conjugate Young diagram (lambda ) which avoid the permutation pattern (tau ). Given two permutations (tau = tau _1tau _2ldots tau _n ) of ({1,2,ldots ,n}) and (sigma =sigma _1sigma _2ldots sigma _m ) of ({1,2,ldots ,m}), the direct sum of (tau ) and (sigma ), denoted by (tau oplus sigma ), is the permutation (tau _1tau _2ldots tau _n (sigma _1+n)(sigma _2+n)ldots (sigma _m+n)). We establish an exterior peak set preserving bijection between (mathcal{S}mathcal{T}_{lambda }(321oplus tau )) and (mathcal{S}mathcal{T}_{lambda }(213oplus tau )) for any pattern (tau ) and any self-conjugate Young diagram (lambda ). Our result is a refinement of part of a result of Bousquet-Mélou–Steingrímsson for pattern-avoiding symmetric transversals. As applications, we derive several enumerative results concerning pattern-avoiding reverse alternating involutions, including two conjectured equalities posed by Barnabei–Bonetti–Castronuovo–Silimbani.

We conjecture a formula for the rational q, t-Catalan polynomial ({mathcal {C}}_{r/s}) that is symmetric in q and t by definition. The conjecture posits that ({mathcal {C}}_{r/s}) can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite (d^*), giving a combinatorial proof of our conjecture on the infinite set of functions ({ {mathcal {C}}_{r/s}^d: requiv 1 mod s, ,,, d le d^*}) is equivalent to a finite counting problem.

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.