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《Discrete Mathematics》2020,343(8):111922
Tribonacci cubes are induced subgraphs of , obtained by removing all the vertices that contain more than two consecutive 1’s. In the present work, we give some enumerative properties related to . We show that the number of vertices of weight in is and express the number of edges of these graphs in terms of convolved Tribonacci numbers. We investigate the cube polynomials of Tribonacci cubes and determine the corresponding generating function. Finally, we give a formula for the number of induced -cubes in . 相似文献
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The paper deals with panchromatic 3-colorings of random hypergraphs. A vertex 3-coloring is said to be panchromatic for a hypergraph if every color can be found on every edge. Let denote the binomial model of a random -uniform hypergraph on vertices. For given fixed , and , we prove that if then admits a panchromatic 3-coloring with probability tending to 1 as , but if is large enough and then does not admit a panchromatic 3-coloring with probability tending to 1 as . 相似文献
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《Discrete Mathematics》2020,343(4):111696
For a set the -neighbourhood of is , where denotes the usual graph distance on . Harper’s vertex-isoperimetric theorem states that among the subsets of given size, the size of the -neighbourhood is minimised when is taken to be an initial segment of the simplicial order. Aubrun and Szarek asked the following question: if is a subset of given size for which the sizes of both and are minimal for all , does it follow that is isomorphic to an initial segment of the simplicial order?Our aim is to give a counterexample. Surprisingly it turns out that there is no counterexample that is a Hamming ball, meaning a set that lies between two consecutive exact Hamming balls, i.e. a set with for some . We go further to classify all the sets for which the sizes of both and are minimal for all among the subsets of of given size. We also prove that, perhaps surprisingly, if for which the sizes of and are minimal among the subsets of of given size, then the sizes of both and are also minimal for all among the subsets of of given size. Hence the same classification also holds when we only require and to have minimal size among the subsets of given size. 相似文献
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《Indagationes Mathematicae》2022,33(2):421-439
We prove the irreducibility of integer polynomials whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscissae and , with ratio of the distances to these points depending on the canonical decomposition of and . In particular, we obtain irreducibility criteria for the case where and have few prime factors, and is either an Eneström–Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting. 相似文献
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The Hankel determinants of the convolution powers of Catalan numbers were considered by Cigler and Krattenthaler. We evaluate these determinants for by finding shifted periodic continued fractions, which arose in application of Sulanke and Xin’s continued fraction method. These include some of the conjectures of Cigler as special cases. We also conjecture a polynomial characterization of these determinants. The same technique is used to evaluate the Hankel determinants . Similar results are obtained. 相似文献
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Le Chen Yaozhong Hu David Nualart 《Stochastic Processes and their Applications》2019,129(12):5073-5112
This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: where is the space–time white noise, , , and . Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: . In some cases, the initial data can be measures. When , we prove the sample path regularity of the solution. 相似文献
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In this paper, we completely determine all necessary and sufficient conditions such that the polynomial , where , is a permutation quadrinomial of over any finite field of odd characteristic. This quadrinomial has been studied first in [25] by Tu, Zeng and Helleseth, later in [24] Tu, Liu and Zeng revisited these quadrinomials and they proposed a more comprehensive characterization of the coefficients that results with new permutation quadrinomials, where and finally, in [16], Li, Qu, Li and Chen proved that the sufficient condition given in [24] is also necessary and thus completed the solution in even characteristic case. In [6] Gupta studied the permutation properties of the polynomial , where and and proposed some new classes of permutation quadrinomials of .In particular, in this paper we classify all permutation polynomials of of the form , where , over all finite fields of odd characteristic and obtain several new classes of such permutation quadrinomials. 相似文献
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We find a combinatorial interpretation of Shareshian and Wachs’ -binomial-Eulerian polynomials, which leads to an alternative proof of their --positivity using group actions. Motivated by the sign-balance identity of Désarménien–Foata–Loday for the -Eulerian polynomials, we further investigate the sign-balance of the -binomial-Eulerian polynomials. We show the unimodality of the resulting signed binomial-Eulerian polynomials by exploiting their continued fraction expansion and making use of a new quadratic recursion for the -binomial-Eulerian polynomials. We finally use the method of continued fractions to derive a new -extension of the -positivity of binomial-Eulerian polynomials which involves crossings and nestings of permutations. 相似文献
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We consider four classes of polynomials over the fields , , , , , , , where . We find sufficient conditions on the pairs for which these polynomials permute and we give lower bounds on the number of such pairs. 相似文献
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《Discrete Mathematics》2020,343(6):111842
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In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials and . Then, he defined and to be the polynomials satisfying and . In this paper, we give a combinatorial interpretation of the coefficients of and prove a symmetry of the coefficients, i.e., . We give a combinatorial interpretation of and prove that is a polynomial in with non-negative integer coefficients. We also prove that if then all coefficients of except the coefficient of are non-negative integers. For all , the coefficient of in is , and when some other coefficients of are also negative. 相似文献