Cyclic Tableaux and Symmetric Functions

We introduce the notion of cyclic tableaux and develop involutions for Waring's formulas expressing the power sum symmetric function p_n in terms of the elementary symmetric function e_n and the homogeneous symmetric function h_n. The coefficients appearing in Waring's formulas are shown to be a cyclic analog of the multinomial coefficients, a fact that seems to have been neglected before. Our involutions also spell out the duality between these two forms of Waring's formulas, which turns out to be exactly the “duality between sets and multisets.” We also present an involution for permutations in cycle notation, leading to probably the simplest combinatorial interpretation of the Möbius function of the partition lattice and a purely combinatorial treatment of the fundamental theorem on symmetric functions. This paper is motivated by Chebyshev polynomials in connection with Waring's formula in two variables.     

Weyl Group $${varvec{q}}$$ q -Kreweras Numbers and Cyclic Sieving

Catalan numbers are known to count noncrossing set partitions, while Narayana and Kreweras numbers refine this count according to the number of blocks in the set partition, and by its collection of block sizes. Motivated by reflection group generalizations of Catalan numbers and their q-analogues, this paper concerns a definition of q-Kreweras numbers for finite Weyl groups W, refining the q-Catalan numbers for W, and arising from work of the second author. We give explicit formulas in all types for the q-Kreweras numbers. In the classical types A, B, C, we also record formulas for the q-Narayana numbers and in the process show that the formulas depend only on the Weyl group (that is, they coincide in types B and C). In addition, we verify that in the classical types A, B, C, D the q-Kreweras numbers obey the expected cyclic sieving phenomena when evaluated at appropriate roots of unity.

     

Cyclic Sieving of Noncrossing Partitions for Complex Reflection Groups

We prove an instance of the cyclic sieving phenomenon, occurring in the context of noncrossing parititions for well-generated complex reflection groups.     

Brauer Diagrams, Updown Tableaux and Nilpotent Matrices

We interpret geometrically a variant of the Robinson-Schensted correspondence which links Brauer diagrams with updown tableaux, in the spirit of Steinberg's result [32] on the original Robinson-Schensted correspondence. Our result uses the variety of all where is a complete flag in is a nondegenerate alternating bilinear form on and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both and instead of Steinberg's variety of where are two complete flags in and N is a nilpotent element of the Lie algebra of the simultaneous stabilizer of both .     

Affine Invariant and Cyclic Codes over p-adic Numbers and Finite Rings

Affine invariant and cyclic codes over p-adic numbers and over integers modulo p ^d are studied. It has been determined what cyclic codes have an extension that is affine invariant.     

Noether Numbers for Subrepresentations of Cyclic Groups of Prime Order

Let W be a finite-dimensional Z/p-module over a field, k, ofcharacteristic p. The maximum degree of an indecomposable elementof the algebra of invariants, k[W]^Z/p, is called the Noethernumber of the representation, and is denoted by rß(W).A lower bound for rß(W) is derived, and it is shownthat if U is a Z/p submodule of W, then rß(U) rß(W).Aset of generators, in fact a SAGBI basis, is constructed fork[V₂ V₃]^Z/p, where V_n is the indecomposable Z/p-module of dimensionn. 2000 Mathematics Subject Classification 13A50, 20J06.     

On the Orthogonal Tableaux of Koike and Terada

Many different definitions have been given for semistandard odd and even orthogonal tableaux which enumerate the corresponding irreducible orthogonal characters. Weightpreserving bijections have been provided between some of these sets of tableaux (see [3], [8]). We give bijections between the (semistandard) odd orthogonal Koike-Terada tableaux and the odd orthogonal Sundaram-tableaux and between the even orthogonal Koike-Terada tableaux and the even orthogonal King-Welsh tableaux. As well, we define an even version of orthogonal Sundaram tableaux which enumerate the irreducible characters of O(2n).     

Closed-form expression for Hankel determinants of the Narayana polynomials

We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana polynomials.     

The Distributions of the Entries of Young Tableaux

Let T be a standard Young tableau of shape λ⊢k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n→∞, equal to f^λ/k!, where f^λ is the number of all tableaux of shape λ. In other words, the probability that a large tableau contains T is equal to the number of tableaux whose shape is that of T, divided by k!. We give several applications, to the probabilities that a set of prescribed entries will appear in a set of prescribed cells of a tableau, and to the probabilities that subtableaux of given shapes will occur. Our argument rests on a notion of quasirandomness of families of permutations, and we give sufficient conditions for this to hold.     

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux – ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.     

On the Numbers of Orbits of Permutations under an Operator Related to Eulerian Numbers

In two previous papers an operator on permutations was introduced and its applications to Eulerian numbers were discussed by means of periods and orbits under the operator. In this paper, observing particular subsequences of permutations, an explicit formula for the number of orbits is given for each period. Several identities concerning the number of orbits and its related numbers are also derived.AMS Subject Classification: 05A05, 05A10.     

On the Damping-Induced Self-Recovery Phenomenon in Mechanical Systems with Several Unactuated Cyclic Variables

The damping-induced self-recovery phenomenon refers to the fundamental property of underactuated mechanical systems: if an unactuated cyclic variable is under a viscous damping-like force and the system starts from rest, then the cyclic variable will always move back to its initial condition as the actuated variables come to a stop. The regular momentum conservation phenomenon can be viewed as the limit of the damping-induced self-recovery phenomenon in the sense that the self-recovery phenomenon disappears as the damping goes to zero. This paper generalizes the past result on damping-induced self-recovery for the case of a single unactuated cyclic variable to the case of multiple unactuated cyclic variables. We characterize a class of external forces that induce new conserved quantities, which we call the damping-induced momenta. The damping-induced momenta yield first-order asymptotically stable dynamics for the unactuated cyclic variables under some conditions, thereby inducing the self-recovery phenomenon. It is also shown that the viscous damping-like forces impose bounds on the range of trajectories of the unactuated cyclic variables. Two examples are presented to demonstrate the analytical discoveries: the planar pendulum with gimbal actuators and the three-link planar manipulator on a horizontal plane.     

Ternary Numbers, Algebras of Reflexive Numbers and Berger Graphs

The Calabi-Yau spaces with SU(n) holonomy can be studied by the algebraic way through the integer lattice where one can construct the Newton reflexive polyhedra or the Berger graphs. Our conjecture is that the Berger graphs can be directly related with the n-ary algebras. To find such algebras we study the n-ary generalization of the well-known binary norm division algebras, , which helped to discover the most important “minimal” binary simple Lie groups, U(1), SU(2) and G(2). As the most important example, we consider the case n = 3, which gives the ternary generalization of quaternions (octonions), 3 ⁿ , n = 2, 3, respectively. The ternary generalization of quaternions is directly related to the new ternary algebra (group) which are related to the natural extensions of the binary su(3) algebra (SU(3) group). Using this ternary algebra we found the solution for the Berger graph: a tetrahedron.

“Why geniosis live so short? They wanna stay kids.”

Alexey Dubrovski: On leave from JINR, Russia. Guennadi Volkov: On leave from PNPI, Russia.     

Cyclic Hypomonotonicity,Cyclic Submonotonicity,and Integration

Rockafellar has shown that the subdifferentials of convex functions are always cyclically monotone operators. Moreover, maximal cyclically monotone operators are necessarily operators of this type, since one can construct explicitly a convex function, which turns out to be unique up to a constant, whose subdifferential gives back the operator. This result is a cornerstone in convex analysis and relates tightly convexity and monotonicity. In this paper, we establish analogous robust results that relate weak convexity notions to corresponding notions of weak monotonicity, provided one deals with locally Lipschitz functions and locally bounded operators. In particular, the subdifferentials of locally Lipschitz functions that are directionally hypomonotone [respectively, directionally submonotone] enjoy also an additional cyclic strengthening of this notion and in fact are maximal under this new property. Moreover, every maximal cyclically hypomonotone [respectively, maximal cyclically submonotone] operator is always the Clarke subdifferential of some directionally weakly convex [respectively, directionally approximately convex] locally Lipschitz function, unique up to a constant, which in finite dimentions is a lower C² function [respectively, a lower C¹ function].     

素数阶循环图和经典Ramsey数R（4，n）的三个新下界

研究了素数阶循环圈的基本性质，提出了寻求有效参数构造正则循环圈的新方法，得到了3个经典Ramsey数的新下界：R（4，17）≥164，R（4，18）≥182，R（4，22）≥282．这前2个结果填补了关于Ramsey数综述[2]的上下界表中的2个空白，第3个结果超过了目前已知的最好下界R（4，22）≥258，     

U-Turn Alternating Sign Matrices,Symplectic Shifted Tableaux and their Weighted Enumeration

Alternating sign matrices with a U-turn boundary (UASMs) are a recent generalization of ordinary alternating sign matrices. Here we show that variations of these matrices are in bijective correspondence with certain symplectic shifted tableaux that were recently introduced in the context of a symplectic version of Tokuyama’s deformation of Weyl’s denominator formula. This bijection yields a formula for the weighted enumeration of UASMs. In this connection use is made of the link between UASMs and certain square ice configuration matrices.