Abstract: Abstract 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 methods also give short proofs of Pieri’s rule and Young’s rule. PubDate: 2020-02-20

Abstract: Abstract 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 closely related to the geometric study of affine Deligne–Lusztig varieties, MV polytopes, Hall–Littlewood polynomials, and many more algebraic structures. In this paper, we will introduce various notions of orientations and hence shadows and study some of their algorithmic properties. PubDate: 2020-02-20

Abstract: Abstract 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 Math 312:811–818, 2012). PubDate: 2020-02-20

Abstract: Abstract 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 of their result involving the novel notion of mixed degree that generalizes the degree similar to how the mixed volume generalizes the volume. We discuss and motivate this terminology, also from an algebro-geometric viewpoint, and explain why it extends a previous definition of Soprunov. We also remark how a recent combinatorial result due to Bihan solves a related problem posed by Soprunov. PubDate: 2020-02-19

Abstract: Abstract 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 from certain classes of Motzkin paths to SYT. As a natural framework for some of our bijections, we introduce a class of set partitions which in some sense is dual to the known class of noncrossing partitions. PubDate: 2020-01-18

Abstract: Abstract 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 trees with \( T_{n} = n\) and maximal down degree \(D_{n} = O(n^{0.5-\epsilon }),\) then the number of descents in a labeling of \(T_{n}\) is asymptotically normal. PubDate: 2020-01-16

Abstract: Abstract 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 for this number. PubDate: 2020-01-14

Abstract: Abstract 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. PubDate: 2020-01-14

Abstract: Abstract 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 rule for queer crystals, this provides a new proof that products of Schur P-polynomials are Schur P-positive. Finally, to facilitate applications of queer crystals in the context of Schur P-positivity, we give local axioms for queer regular graphs, generalizing Stembridge’s axioms, that partially characterize queer crystals. PubDate: 2020-01-14

Abstract: Abstract 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 \(\varepsilon \). This is progress toward a conjecture of Lund, Sheffer, and de Zeeuw, that either a single line or circle contains n/2 points of \({\mathcal {P}}\), or the number of distinct perpendicular bisectors is \(\Omega (n^2)\). The proof relies bounding the size of a carefully selected subset of the quadruples \((a,b,c,d) \in {\mathcal {P}}^4\) such that the perpendicular bisector of a and b is the same as the perpendicular bisector of c and d. PubDate: 2020-01-14

Abstract: Abstract 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 of graph with two types of edges which we call a dicot. The partition function of the latter can be written as a determinant, this time of a complex adjacency matrix. This formulation generalises Wu’s assignment of imaginary orientation for the grid graph to planar dicots. As an application, we compute the partition function for a family of non-planar dicots with positive weights. PubDate: 2020-01-14

Abstract: Abstract 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 made by Paul Hanna in 2012. PubDate: 2020-01-14

Abstract: Abstract 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 show that \(U_\lambda \) is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, moreover, the first finite expression for \(U_\lambda \). Combined with work of A. Garsia and J. Remmel, this yields a compact combinatorial (but necessarily nonpositive) formula for the Schur expansion of \(U_\lambda \). PubDate: 2020-01-14

Abstract: Abstract We establish an algorithm for producing formulas for p(n, m, N), the function enumerating partitions of n into at most m parts with no part larger than N. Recent combinatorial results of H. Hahn et al. on a collection of partition identities for p(n, 3, N) are considered. We offer direct proofs of these identities and then place them in a larger context of the unimodality of Gaussian polynomials \(N+m\brack m\) whose coefficients are precisely p(n, m, N). We give complete characterizations of the maximal coefficients of \({M\brack 3}\) and \({M\brack 4}\). Furthermore, we prove a general theorem on the period of quasipolynomials for central/maximal coefficients of Gaussian polynomials. We place some of Hahn’s identities into the context of some known results on differences of partitions into at most m parts, p(n, m), which we then extend to p(n, m, N). PubDate: 2019-11-15

Abstract: Abstract 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. PubDate: 2019-11-11

Abstract: Abstract 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 and 2.2 below. PubDate: 2019-11-01

Abstract: Abstract In this note, we give three identities for partitions with parts separated by parity, which were recently introduced by Andrews. PubDate: 2019-11-01

Abstract: Abstract Refined versions, analytic and combinatorial, are given for classical integer partition theorems. The examples include the Rogers–Ramanujan identities, the Göllnitz–Gordon identities, Euler’s odd = distinct theorem, and the Andrews–Gordon identities. Generalizations of each of these theorems are given where a single part is “marked” or weighted. This allows a single part to be replaced by a new larger part, “shifting” a part, and analogous combinatorial results are given in each case. Versions are also given for marking a sum of parts. PubDate: 2019-11-01

Abstract: Abstract We show that there is a concept of q-translation behind the approach used by Liu to prove summation and transformation identities for q-series. We revisit the q-translation associated with the Askey–Wilson operator introduced in Ismail (Ann Comb 5(3–4):347–362, 2001), simplify its formalism and point out new properties of this translation operator. PubDate: 2019-11-01