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Abstract: Abstract We introduce a symmetry class for higher dimensional partitions—fully complementary higher dimensional partitions (FCPs)—and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product, we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely quasi-transpose-complementary plane partitions, are equinumerous to symmetric plane partitions. PubDate: 2024-04-06

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Abstract: Abstract We define and construct the “canonical reduced word” of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the “run” statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean. PubDate: 2024-04-04

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Abstract: Abstract For an arbitrary set or multiset A of positive integers, we associate the A-partition function \(p_A(n)\) (that is the number of partitions of n whose parts belong to A). We also consider the analogue of the k-colored partition function, namely, \(p_{A,-k}(n)\) . Further, we define a family of polynomials \(f_{A,n}(x)\) which satisfy the equality \(f_{A,n}(k)=p_{A,-k}(n)\) for all \(n\in \mathbb {Z}_{\ge 0}\) and \(k\in \mathbb {N}\) . This paper concerns a polynomialization of the Bessenrodt–Ono inequality, namely $$\begin{aligned} f_{A,a}(x)f_{A,b}(x)>f_{A,a+b}(x), \end{aligned}$$ where a, b are positive integers. We determine efficient criteria for the solutions of this inequality. Moreover, we also investigate a few basic properties related to both functions \(f_{A,n}(x)\) and \(f_{A,n}'(x)\) . PubDate: 2024-04-03

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Abstract: Abstract A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs. PubDate: 2024-03-25

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Abstract: Abstract We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination \(a^3b\) ). Parallel to a complete classification by Cheung, Luk, and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings. PubDate: 2024-03-18

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Abstract: Abstract Inspired by a recent work of Kim, Kim and Lovejoy on two overpartition difference functions, we study some bipartition difference functions, four of which are related to Ramanujan’s identities recorded in his lost notebook. We show that they are always positive by elementary q-series transformations. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract The polytopes \(\mathcal {U}_{I,\overline{J}}\) were introduced by Ceballos, Padrol, and Sarmiento to provide a geometric approach to the study of \((I,\overline{J})\) -Tamari lattices. They observed a connection between certain \(\mathcal {U}_{I,\overline{J}}\) and acyclic root polytopes, and wondered if Mészáros’ subdivision algebra can be used to subdivide all \(\mathcal {U}_{I,\overline{J}}\) . We answer this in the affirmative from two perspectives, one using flow polytopes and the other using root polytopes. We show that \(\mathcal {U}_{I,\overline{J}}\) is integrally equivalent to a flow polytope that can be subdivided using the subdivision algebra. Alternatively, we find a suitable projection of \(\mathcal {U}_{I,\overline{J}}\) to an acyclic root polytope which allows subdivisions of the root polytope to be lifted back to \(\mathcal {U}_{I,\overline{J}}\) . As a consequence, this implies that subdivisions of \(\mathcal {U}_{I,\overline{J}}\) can be obtained with the algebraic interpretation of using reduced forms of monomials in the subdivision algebra. In addition, we show that the \((I,\overline{J})\) -Tamari complex can be obtained as a triangulated flow polytope. PubDate: 2024-03-01

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Abstract: Abstract A connected graph G of diameter \(\textrm{diam}(G) \ge \ell \) is \(\ell \) -distance-balanced if \( W_{xy} = W_{yx} \) for every \(x,y\in V(G)\) 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. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract A k-plane tree is a plane tree whose vertices are assigned labels between 1 and k in such a way that the sum of the labels along any edge is no greater than \(k+1\) . These trees are known to be related to \((k+1)\) -ary trees, and they are counted by a generalised version of the Catalan numbers. We prove a surprisingly simple refined counting formula, where we count trees with a prescribed number of labels of each kind. Several corollaries are derived from this formula, and an analogous theorem is proven for k-noncrossing trees, a similarly defined family of labelled noncrossing trees that are related to \((2k+1)\) -ary trees. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract 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. PubDate: 2024-03-01

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Abstract: Abstract 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”. PubDate: 2024-03-01

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Abstract: Abstract A classical Turán problem asks for the maximum possible number of edges in a graph of a given order that does not contain a particular graph H as a subgraph. It is well-known that the chromatic number of H is the graph parameter which describes the asymptotic behavior of this maximum. Here, we consider an analogous problem for oriented graphs, where compressibility plays the role of the chromatic number. Since any oriented graph having a directed cycle is not contained in any transitive tournament, it makes sense to consider only acyclic oriented graphs as forbidden subgraphs. We provide basic properties of the compressibility, show that the compressibility of acyclic oriented graphs with out-degree at most 2 is polynomial with respect to the maximum length of a directed path, and that the same holds for a larger out-degree bound if the Erdős–Hajnal conjecture is true. Additionally, generalizing previous results on powers of paths and arbitrary orientations of cycles, we determine the compressibility of acyclic oriented graphs with restricted distances of vertices to sinks and sources. PubDate: 2024-02-29

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Abstract: Abstract We introduce a class of polytopes that we call chainlink polytopes and show that they allow us to construct infinite families of pairs of non-isomorphic rational polytopes with the same Ehrhart quasipolynomial. Our construction is based on circular fence posets, a recently introduced class of posets, which admit a non-obvious and nontrivial symmetry in their rank sequences. We show that this symmetry can be lifted to the level of polyhedral models (which we call chainlink polytopes) for these posets. Along the way, we introduce the related class of chainlink posets and show that they exhibit analogous nontrivial symmetry properties. We further prove an outstanding conjecture on the unimodality of rank polynomials of circular fence posets. PubDate: 2024-02-06

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Abstract: Abstract Stanley introduced and studied two lattice polytopes, the order polytope and chain polytope, associated with a finite poset. Recently, Ohsugi and Tsuchiya introduce an enriched version of them, called the enriched order polytope and enriched chain polytope. In this paper, we give a piecewise-linear bijection between these enriched poset polytopes, which is an enriched analogue of Stanley’s transfer map and bijectively proves that they have the same Ehrhart polynomials. Also, we construct explicitly unimodular triangulations of two enriched poset polytopes, which are the order complexes of graded posets. PubDate: 2023-12-22 DOI: 10.1007/s00026-023-00679-7

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Abstract: The approximate uniform sampling of graph realizations with a given degree sequence is an everyday task in several social science, computer science, engineering etc. projects. One approach is using Markov chains. The best available current result about the well-studied switch Markov chain is that it is rapidly mixing on P-stable degree sequences (see DOI :10.1016/j.ejc.2021.103421). The switch Markov chain does not change any degree sequence. However, there are cases where degree intervals are specified rather than a single degree sequence. (A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed.) Rechner, Strowick, and Müller–Hannemann introduced in 2018 the notion of degree interval Markov chain which uses three (separately well studied) local operations (switch, hinge-flip and toggle), and employing on degree sequence realizations where any two sequences under scrutiny have very small coordinate-wise distance. Recently, Amanatidis and Kleer published a beautiful paper ( DOI :10.4230/LIPIcs.STACS.2023.7), showing that the degree interval Markov chain is rapidly mixing if the sequences are coming from a system of very thin intervals which are centered not far from a regular degree sequence. In this paper, we substantially extend their result, showing that the degree interval Markov chain is rapidly mixing if the intervals are centered at P-stable degree sequences. PubDate: 2023-12-21