共查询到20条相似文献,搜索用时 15 毫秒
1.
《Discrete Mathematics》2022,345(12):113077
In 2020, Bennett, Carrillo, Machacek and Sagan gave a polynomial generalization of the Narayana numbers and conjectured that these polynomials have positive integer coefficients for and for . In 2020, Sagan and Tirrell used a powerful algebraic method to prove this conjecture (in fact, they extend and prove the conjecture for more than just the type A case). In this paper we give a combinatorial proof of a formula satisfied by the Lucas-Narayana polynomials described by Bennett et al. This gives a combinatorial proof that these polynomials have positive integer coefficients. A corollary of our main result establishes a parallel theorem for the FiboNarayana numbers , providing a combinatorial proof of the conjecture that these are positive integers for . 相似文献
2.
Yasuhide Numata 《Journal of Algebraic Combinatorics》2007,26(1):27-45
Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between
certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators
to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations
of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized
Schur operators. We show that the commutation relation of generalized Schur operators implies Pieri's formula for generalized
Schur polynomials. 相似文献
3.
Sui Sun Cheng Shao Yuan Huang 《International Journal of Mathematical Education in Science & Technology》2013,44(7):950-964
Given a quadratic polynomial with complex coefficients, necessary and sufficient conditions are found in terms of the coefficients such that all its roots have absolute values <1. 相似文献
4.
Katsunori Iwasaki Hiroyuki Kawamuko 《Proceedings of the American Mathematical Society》1999,127(1):29-33
We establish a combinatorial formula of Leibniz type, which is an identity for a certain differential polynomial. The formula leads to new quadratic relations between Gegenbauer's orthogonal polynomials.
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We derive a new formula for the supersymmetric Schur polynomial s
(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for s
(x/y). This new expression gives rise to a determinantal formula for s
(x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz. 相似文献
7.
Yu-Feng YAO 《数学年刊B辑(英文版)》2020,41(1):49-60
Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra ■(m; 1) ■∧(n), where ■(m; 1) is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type... 相似文献
8.
In this paper we use the combinatorics of alcove walks to give uniform combinatorial formulas for Macdonald polynomials for all Lie types. These formulas resemble the formulas of Haglund, Haiman and Loehr for Macdonald polynomials of type GLn. At q=0 these formulas specialize to the formula of Schwer for the Macdonald spherical function in terms of positively folded alcove walks and at q=t=0 these formulas specialize to the formula for the Weyl character in terms of the Littelmann path model (in the positively folded gallery form of Gaussent and Littelmann). 相似文献
9.
Masao Ishikawa 《The Ramanujan Journal》2008,16(2):211-234
In the open problem session of the FPSAC’03, R.P. Stanley gave an open problem about a certain sum of the Schur functions.
The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we
express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear
Algebra 39:285–305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist’s
Pfaffian identities (J. Algebr. Comb. 5:135–148, 1996). Then we give a proof of Stanley’s open problem in Sect. 4. At the end of this paper we present certain corollaries obtained
from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize
Stanley’s open problem.
相似文献
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We give a combinatorial proof of the first Rogers-Ramanujan identity by using two symmetries of a new generalization of Dyson's rank. These symmetries are established by direct bijections. 相似文献
12.
Jeb F. Willenbring 《Transactions of the American Mathematical Society》2002,354(11):4393-4419
Consider a symmetric pair of linear algebraic groups with , where and are defined as the +1 and -1 eigenspaces of the involution defining . We view the ring of polynomial functions on as a representation of . Moreover, set , where is the space of homogeneous polynomial functions on of degree . This decomposition provides a graded -module structure on . A decomposition of is provided for some classical families when is within a certain stable range.
The stable range is defined so that the spaces are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of is interpreted as a -analog of the Kostant-Rallis theorem.
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YoungBin Choe 《Discrete Mathematics》2008,308(24):5944-5953
14.
We give a new proof of Fitzgerald's criterion for primitive polynomials over a finite field. Existing proofs essentially use the theory of linear recurrences over finite fields. Here, we give a much shorter and self-contained proof which does not use the theory of linear recurrences. 相似文献
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Soojin Cho 《Journal of Combinatorial Theory, Series A》2007,114(7):1199-1219
There are well-known reduction formulas for the universal Schubert coefficients defined on Grassmannians. These coefficients are also known as the Littlewood-Richardson coefficients in the theory of symmetric functions. We restate the reduction formulas combinatorially and provide a combinatorial proof for them. 相似文献
17.
Emilio Defez 《Applied Mathematics Letters》2013,26(8):899-903
This paper centers on the derivation of a Rodrigues-type formula for the Gegenbauer matrix polynomial. A connection between Gegenbauer and Jacobi matrix polynomials is given. 相似文献
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CHU YuMing XIA WeiFeng & ZHAO TieHong Department of Mathematics Huzhou Teachers College Huzhou China Institut de Mathmatiques Universit Pierre et Marie Curie Paris F- France 《中国科学 数学(英文版)》2010,(2)
The Schur convexity and concavity of a class of symmetric functions are discussed, and an open problem proposed by Guan in Some properties of a class of symmetric functions is answered. As consequences, some inequalities are established by use of the theory of majorization. 相似文献
20.
对x = (x1, x2,···, xn) ∈ (0,1)n 和 r ∈ {1, 2,···, n} 定义对称函数
Fn(x, r) = Fn(x1, x2,···, xn; r) =∏1≤i1∑j=1r(1+xi3/1- xi3)1/r,
其中i1, i2, ···, ir 是整数. 该文证明了Fn(x, r) 是(0,1)n 上的Schur凸、Schur乘性凸和Schur调和凸函数. 作为应用,利用控制理论建立了若干不等式. 相似文献