Annals of Combinatorics最新文献

We prove NP-hardness results for determining whether ice quivers are mutation equivalent to quivers with given properties, specifically, determining whether an ice quiver is mutation equivalent to an ice quiver with exactly k arrows between any two of its vertices is NP-hard. Also, determining whether an ice quiver is mutation equivalent to a quiver with no edges between frozen vertices is strongly NP-hard. Finally, we present a characterization of mutation classes of ice quivers with two mutable vertices.

Let (S_k times S_{n-k}) be a maximal Young subgroup of the symmetric group (S_n). We introduce a basis ({{mathcal {B}}}_{n,k}) for the coset space (S_n/S_k times S_{n-k}) that is naturally parametrized by the set of standard Young tableaux with n boxes, at most two rows, and at most k boxes in the second row. The basis ({{mathcal {B}}}_{n,k}) has positivity properties that resemble those of a root system, and there is a composition series of the coset space in which each term is spanned by the basis elements that it contains. We prove that the spherical functions of the associated Gelfand pair are nonnegative linear combinations of the ({{mathcal {B}}}_{n,k}).

Petersen and Tenner defined the depth statistic for Coxeter group elements which, in the symmetric group, can be described in terms of a cost function on transpositions. We generalize that cost function to the other classical (finite and affine) Weyl groups, letting the cost of an individual reflection t be the distance between the integers transposed by t in the combinatorial representation of the group (à la Eriksson and Eriksson). Arbitrary group elements then have a well-defined cost, obtained by minimizing the sum of the transposition costs among all factorizations of the element. We show that the cost of arbitrary elements can be computed directly from the elements themselves using a simple, intrinsic formula.

A permutation statistic ({{,textrm{st},}}) is said to be shuffle-compatible if the distribution of ({{,textrm{st},}}) over the set of shuffles of two disjoint permutations (pi ) and (sigma ) depends only on ({{,textrm{st},}}pi ), ({{,textrm{st},}}sigma ), and the lengths of (pi ) and (sigma ). Shuffle-compatibility is implicit in Stanley’s early work on P-partitions, and was first explicitly studied by Gessel and Zhuang, who developed an algebraic framework for shuffle-compatibility centered around their notion of the shuffle algebra of a shuffle-compatible statistic. For a family of statistics called descent statistics, these shuffle algebras are isomorphic to quotients of the algebra of quasisymmetric functions. Recently, Domagalski, Liang, Minnich, Sagan, Schmidt, and Sietsema defined a version of shuffle-compatibility for statistics on cyclic permutations, and studied cyclic shuffle-compatibility through purely combinatorial means. In this paper, we define the cyclic shuffle algebra of a cyclic shuffle-compatible statistic, and develop an algebraic framework for cyclic shuffle-compatibility in which the role of quasisymmetric functions is replaced by the cyclic quasisymmetric functions recently introduced by Adin, Gessel, Reiner, and Roichman. We use our theory to provide explicit descriptions for the cyclic shuffle algebras of various cyclic permutation statistics, which in turn gives algebraic proofs for their cyclic shuffle-compatibility.

A permutation is called Grassmannian if it has at most one descent. The study of pattern avoidance in such permutations was initiated by Gil and Tomasko in 2021. We continue this work by studying Grassmannian permutations that avoid an increasing pattern. In particular, we count the Grassmannian permutations of size m avoiding the identity permutation of size k,  thus solving a conjecture made by Weiner. We also refine our counts to special classes such as odd Grassmannian permutations and Grassmannian involutions. We prove most of our results by relating Grassmannian permutations to Dyck paths and binary words.

To any simple graph (G), the clique graph operator (K) assigns the graph (K(G)), which is the intersection graph of the maximal complete subgraphs of (G). The iterated clique graphs are defined by (K^{0}(G)=G) and (K^{n}(G)=K(K^{n-1}(G))) for (nge 1). We associate topological concepts to graphs by means of the simplicial complex (textrm{Cl}(G)) of complete subgraphs of (G). Hence, we say that the graphs (G_{1}) and (G_{2}) are homotopic whenever (textrm{Cl}(G_{1})) and (textrm{Cl}(G_{2})) are. A graph (G) such that (K^{n}(G)simeq G) for all (nge 1) is called (K)-homotopy permanent. A graph is Helly if the collection of maximal complete subgraphs of (G) has the Helly property. Let (G) be a Helly graph. Escalante (1973) proved that (K(G)) is Helly, and Prisner (1992) proved that (Gsimeq K(G)), and so Helly graphs are (K)-homotopy permanent. We conjecture that if a graph (G) satisfies that (K^{m}(G)) is Helly for some (mge 1), then (G) is (K)-homotopy permanent. If a connected graph has maximum degree at most four and is different from the octahedral graph, we say that it is a low degree graph. It was recently proven that all low-degree graphs (G) satisfy that (K^{2}(G)) is Helly. In this paper, we show that all low-degree graphs have the homotopy type of a wedge or circumferences, and that they are (K)-homotopy permanent.

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.

Haglund et al. (Trans Am Math Soc 370(6):4029–4057, 2018) introduced their Delta conjectures, which give two different combinatorial interpretations of the symmetric function (Delta '_{e_{n-k-1}} e_n) in terms of rise-decorated or valley-decorated labelled Dyck paths. While the rise version has been recently proved (D’Adderio and Mellit in Adv Math 402:108342, 2022; Blasiak et al. in A Proof of the Extended Delta Conjecture, arXiv:2102.08815, 2021), not much is known about the valley version. In this work, we prove the Schröder case of the valley Delta conjecture, the Schröder case of its square version (Iraci and Wyngaerd in Ann Combin 25(1):195–227, 2021), and the Catalan case of its extended version (Qiu and Wilson in J Combin Theory Ser A 175:105271, 2020). Furthermore, assuming the symmetry of (a refinement of) the combinatorial side of the extended valley Delta conjecture, we deduce also the Catalan case of its square version (Iraci and Wyngaerd 2021).

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.