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  • Applications of Gaussian Binomials to Coding Theory for Deletion Error Correction
    Ann. Comb. (IF 0.65) Pub Date : 2020-06-03
    Manabu Hagiwara, Justin Kong

    We present new applications of q-binomials, also known as Gaussian binomial coefficients. Our main theorems determine cardinalities of certain error-correcting codes based on Varshamov–Tenengolts codes and prove a curious phenomenon relating to deletion spheres.

    更新日期:2020-06-03
  • General Colored Partition Identities
    Ann. Comb. (IF 0.65) Pub Date : 2020-06-03
    Sun Kim

    Ramanujan’s modular equations of degrees 3, 5, 7, 11 and 23 yield beautiful colored partition identities. Warnaar analytically generalized the modular equations of degrees 3 and 7, and thereafter, the author found bijective proofs of those partitions identities and recently, established an analytic generalization of the modular equations of degrees 5, 11 and 23. The partition identities of degrees

    更新日期:2020-06-03
  • Gorenstein Graphic Matroids from Multigraphs
    Ann. Comb. (IF 0.65) Pub Date : 2020-05-30
    Max Kölbl

    A matroid is Gorenstein if its toric variety is. Hibi, Lasoń, Matsuda, Michałek, and Vodička provided a full graph-theoretic classification of Gorenstein matroids associated to simple graphs. We extend this classification to multigraphs.

    更新日期:2020-05-30
  • Revisiting Pattern Avoidance and Quasisymmetric Functions
    Ann. Comb. (IF 0.65) Pub Date : 2020-05-13
    Jonathan S. Bloom, Bruce E. Sagan

    Let \({{\mathfrak {S}}}_n\) be the nth symmetric group. Given a set of permutations \(\Pi \), we denote by \({{\mathfrak {S}}}_n(\Pi )\) the set of permutations in \({{\mathfrak {S}}}_n\) which avoid \(\Pi \) in the sense of pattern avoidance. Consider the generating function \(Q_n(\Pi )=\sum _\sigma F_{{{\,\mathrm{Des}\,}}\sigma }\) where the sum is over all \(\sigma \in {{\mathfrak {S}}}_n(\Pi )\)

    更新日期:2020-05-13
  • Combinatorics of the Deodhar Decomposition of the Grassmannian
    Ann. Comb. (IF 0.65) Pub Date : 2020-03-11
    Cameron Marcott

    The Deodhar decomposition of the Grassmannian is a refinement of the Schubert, Richardson, and positroid stratifications of the Grassmannian. Go-diagrams are certain fillings of Ferrers diagrams with black stones, white stones, and pluses which index Deodhar components in the Grassmannian. We provide a series of corrective flips on diagrams which may be used to transform arbitrary fillings of Ferrers

    更新日期:2020-03-11
  • Dungeons and Dragons: Combinatorics for the $$\varvec{dP_3}$$dP3 Quiver
    Ann. Comb. (IF 0.65) Pub Date : 2020-02-25
    Tri Lai, Gregg Musiker

    In this paper, we utilize the machinery of cluster algebras, quiver mutations, and brane tilings to study a variety of historical enumerative combinatorics questions all under one roof. Previous work (Zhang in Cluster variables and perfect matchings of subgraphs of the \(dP_{3}\) lattice, http://www.math.umn.edu/~reiner/REU/Zhang2012.pdf. arXiv:1511.0655, 2012; Leoni et al. in J Phys A Math Theor 47:474011

    更新日期:2020-02-25
  • A Proof of the Murnaghan–Nakayama Rule Using Specht Modules and Tableau Combinatorics
    Ann. Comb. (IF 0.65) Pub Date : 2020-02-20
    Jasdeep Kochhar, Mark Wildon

    The Murnaghan–Nakayama rule is a combinatorial rule for the character values of symmetric groups. We give a new combinatorial proof by explicitly finding the trace of the representing matrices in the standard basis of Specht modules. This gives an essentially bijective proof of the rule. A key lemma is an extension of a straightening result proved by the second author to skew tableaux. Our module theoretic

    更新日期:2020-02-20
  • Shadows in Coxeter Groups
    Ann. Comb. (IF 0.65) Pub Date : 2020-02-20
    Marius Graeber, Petra Schwer

    For a given w in a Coxeter group W, the elements u smaller than w in Bruhat order can be seen as the end alcoves of stammering galleries of type w in the Coxeter complex \(\Sigma \). We generalize this notion and consider sets of end alcoves of galleries that are positively folded with respect to certain orientation \(\phi \) of \(\Sigma \). We call these sets shadows. Positively folded galleries are

    更新日期:2020-02-20
  • On the Correlation Measures of Subsets
    Ann. Comb. (IF 0.65) Pub Date : 2020-02-20
    Huaning Liu, Christian Mauduit

    In a series of papers, Dartyge and Sárközy (partly with other coauthors) studied pseudorandom subsets. In this paper, we study the minimal values of correlation measures of pseudorandom subsets by extending the methods introduced in Alon et al. (Comb Probab Comput 15:1–29, 2006), Anantharam (Discrete Math 308:6203–6209, 2008), Gyarmati (Stud Sci Math Hung 42:79–93, 2005) and Gyarmati and Mauduit (Discrete

    更新日期:2020-02-20
  • The Mixed Degree of Families of Lattice Polytopes
    Ann. Comb. (IF 0.65) Pub Date : 2020-02-19
    Benjamin Nill

    The degree of a lattice polytope is a notion in Ehrhart theory that was studied quite intensively over previous years. It is well known that a lattice polytope has normalized volume one if and only if its degree is zero. Recently, Esterov and Gusev gave a complete classification result for families of n lattice polytopes in \({\mathbb {R}}^n\) whose mixed volume equals one. Here, we give a reformulation

    更新日期:2020-02-19
  • From Dyck Paths to Standard Young Tableaux
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-18
    Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner

    We present nine bijections between classes of Dyck paths and classes of standard Young tableaux (SYT). In particular, we consider SYT of flag and rectangular shapes, we give Dyck path descriptions for certain SYT of height at most 3, and we introduce a special class of labeled Dyck paths of semilength n that is shown to be in bijection with the set of all SYT with n boxes. In addition, we present bijections

    更新日期:2020-01-18
  • Tree Descent Polynomials: Unimodality and Central Limit Theorem
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-16
    Amy Grady, Svetlana Poznanović

    For a poset whose Hasse diagram is a rooted plane forest F, we consider the corresponding tree descent polynomial \(A_F(q)\), which is a generating function of the number of descents of the labelings of F. When the forest is a path, \(A_F(q)\) specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of \(A_F(q)\) is unimodal and that if \(\{T_{n}\}\) is a sequence of

    更新日期:2020-01-16
  • On the Number of Even Parts in All Partitions of $$\varvec{n}$$n into Distinct Parts
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    George E. Andrews, Mircea Merca

    A famous theorem of Euler asserts that there are as many partitions of n into distinct parts as there are partitions into odd parts. The even parts in partitions of n into distinct parts play an important role in the Euler–Glaisher bijective proof of this result. In this paper, we investigate the number of even parts in all partitions of n into distinct parts providing new combinatorial interpretations

    更新日期:2020-01-14
  • A Tableau Formula of Double Grothendieck Polynomials for 321-Avoiding Permutations
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    Tomoo Matsumura

    In this article, we prove a tableau formula for the double Grothendieck polynomials associated to 321-avoiding permutations. The proof is based on the compatibility of the formula with the K-theoretic divided difference operators. Our formula specializes to the one obtained by Chen et al. (Eur J Combin 25(8):1181–1196, 2004) for the (double) skew Schur polynomials.

    更新日期:2020-01-14
  • Toward a Local Characterization of Crystals for the Quantum Queer Superalgebra
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    Sami Assaf, Ezgi Kantarci Oguz

    We define operators on semistandard shifted tableaux and use Stembridge’s local characterization for regular graphs to prove they define a crystal structure. This gives a new proof that Schur P-polynomials are Schur positive. We define queer crystal operators (also called odd Kashiwara operators) to construct a connected queer crystal on semistandard shifted tableaux of a given shape. Using the tensor

    更新日期:2020-01-14
  • The Genomic Schur Function is Fundamental-Positive
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    O. Pechenik

    In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood–Richardson coefficients in torus-equivariant K-theory of Grassmannians. We then studied the genomic Schur function \(U_\lambda \), a generating function for such tableaux, showing that it is nontrivially a symmetric function, although generally not Schur-positive. Here, we

    更新日期:2020-01-14
  • A Refined Energy Bound for Distinct Perpendicular Bisectors
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    Ben Lund

    Let \({\mathcal {P}}\) be a set of n points in the Euclidean plane. We prove that, for any \(\varepsilon > 0\), either a single line or circle contains n/2 points of \({\mathcal {P}}\), or the number of distinct perpendicular bisectors determined by pairs of points in \({\mathcal {P}}\) is \(\Omega (n^{52/35 - \varepsilon })\), where the constant implied by the \(\Omega \) notation depends on \(\varepsilon

    更新日期:2020-01-14
  • Squareness for the Monopole-Dimer Model
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    Arvind Ayyer

    The monopole-dimer model introduced recently is an exactly solvable signed generalisation of the dimer model. We show that the partition function of the monopole-dimer model on a graph invariant under a fixed-point free involution is a perfect square. We give a combinatorial interpretation of the square root of the partition function for such graphs in terms of a monopole-dimer model on a new kind

    更新日期:2020-01-14
  • On a Conjecture of Hanna Connecting Distinct Part and Complete Partitions
    Ann. Comb. (IF 0.65) Pub Date : 2020-01-14
    George E. Andrews, George Beck, Brian Hopkins

    Complete partitions are a generalization of MacMahon’s perfect partitions; we further generalize these by defining k-step partitions. A matrix equation shows an unexpected connection between k-step partitions and distinct part partitions. We provide two proofs of the corresponding theorem, one using generating functions and one combinatorial. The algebraic proof relies on a generalization of a conjecture

    更新日期:2020-01-14
  • A Simple Proof of a Congruence for a Series Involving the Little q -Jacobi Polynomials
    Ann. Comb. (IF 0.65) Pub Date : 2019-11-11
    Atul Dixit

    We give a simple and a more explicit proof of a mod 4 congruence for a series involving the little q-Jacobi polynomials which arose in a recent study of a certain restricted overpartition function.

    更新日期:2019-11-11
  • Interval Graph Limits.
    Ann. Comb. Pub Date : 2013-03-01
    Persi Diaconis,Susan Holmes,Svante Janson

    We work out a graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function W (x, y) on the unit square, with x and y uniform on the interval (0, 1). Instead, we fix a W and change the underlying distribution of the coordinates x and y. We find choices such that our limits are continuous. Connections to random interval

    更新日期:2019-11-01
  • A $$\varvec{q}$$q -Analogue for Euler’s $$\varvec{\zeta (6)=\pi ^6/945}$$ζ(6)=π6/945
    Ann. Comb. (IF 0.65) Pub Date : 2019-10-30
    Ankush Goswami

    Recently, Sun (Two q-analogues of Euler’s formula \(\zeta (2)=\pi ^2/6\). arXiv:1802.01473, 2018) obtained q-analogues of Euler’s formula for \(\zeta (2)\) and \(\zeta (4)\). Sun’s formulas were based on identities satisfied by triangular numbers and properties of Euler’s q-Gamma function. In this paper, we obtain a q-analogue of \(\zeta (6)=\pi ^6/945\). Our main results are stated in Theorems 2.1

    更新日期:2019-10-30
  • A Note on Andrews’ Partitions with Parts Separated by Parity
    Ann. Comb. (IF 0.65) Pub Date : 2019-10-16
    Kathrin Bringmann, Chris Jennings-Shaffer

    In this note, we give three identities for partitions with parts separated by parity, which were recently introduced by Andrews.

    更新日期:2019-10-16
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